On characterizations of a T_r− space: Continuity, Domination, and Alliance

ABSTRACT

This paper studies some properties of a topological space called ${\mathcal{T}}_r-$space. Specifically, we study the continuity of functions in this space, explore the most interesting sets, which are dominating sets, and alliances, and finally explore the existence of cut points in such a space. We study all these properties using the concept of $\mathit{\varphi }-$proximity.

KEYWORDS:

• Point-set topology
• nation
• decision spaces
• $\mathit{\varphi }-$proximity
• dominating sets
• alliance

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 54F05
• 54F65 secondary 54B99
• 91D30

1. Introduction

Recently, Mayila et al. (2023) constructed a topology called a ${\mathcal{T}}_{\mathit{r}}-$topology on a nation X using decision spaces as open sets and the concept of inclusiveness. The definition of a decision space is a modification of that suggested by Bossert (Bossert, 1998). According to Mayila et al. (2023), a decision space in a nation X is defined as a triple-ordered collection $(\mathit{U},S_{\mathit{U}},{\text{P}}_{\mathit{X}}(\mathit{U}))$ consisting of a set $\mathit{U}(\subset \mathit{X})$ of people, the set ${\text{P}}_{\mathit{X}}(\mathit{U})$ of decision authority of U in X, which is a set-valued function that gives the right, power, or obligation to make a decision and the duty to answer for its success or failure, and the set S_U of possible priority situations which may occur in the administrative section of U. The properties of decision spaces and topological operators were studied. Finally, the connectedness of ${\mathcal{T}}_{\mathit{r}}-$spaces was developed and studied. All of these were studied using the concept of $\mathit{\varphi }-$proximity.

In this paper, we use the same approach Mayila et al. (2023) used to define the continuity of functions on ${\mathcal{T}}_{\mathit{r}}-$spaces and study the one-to-one correspondence between these spaces. We will further study the idea of dominance and alliance in an international system setting, with the key goal being how one nation sits within the other. The paper will reach its conclusion by studying the existence of cut points in a $\mathit{\varphi }-$connected ${\mathcal{T}}_{\mathit{r}}-$space.

1.1. Preliminaries

This paper will use the following notations and terminologies. Unless stated otherwise, all preliminary results are taken from Mayila et al. (2023).

Definition 1.1.

Mayila et al. (2023, p. 2)

1. A decision space refers to a person or commission who has the authority by law or ordinance to make a final decision in approving or disapproving a particular matter within its administrative area.

2. A decision authority is the right, power, or obligation to make a decision and the duty to answer for its success or failure.

3. A decision index is the strength of the decision authority.

This definition was put into a mathematical language by Mayila et al. (2023). However, we will add a few lines to clarify the components of such a mathematical representation. Our presentation begins by formalizing the concept of a decision authority as defined in Definition 1.1 into a more rigorous mathematical language. A decision authority is a set of “IF [condition]-THEN [authority]” statements. The conditions are situations or facts, while the authority is a conclusion, a rule, or an action. Therefore, the value of an authority is a mapping of an observation: a situation. These mappings are words, symbols, and signs made by one or more people for the intention of being used by some set of people as a guide in decision-making (Baligh, 1990, p. 361). Mathematically, we define a decision authority mapping as follows:

Definition 1.2.

Let S be an $\mathit{n}-$dimensional set of conditions or situations, and $\mathit{s}=(s_1,s_2,\dots ,s_{\mathit{n}})\in \mathit{S}$ be a vector of facts, and $\mathit{a}=\{a_1,a_2,\dots \}\in \mathit{A}$ be a set of prescriptively allowed values of an authority variable, one of which is to be made into a fact, where A is a set of sets of decision values. Then

1. A decision authority mapping is a function ${\text{P}}_{\mathit{X}}:\mathit{S}\to \mathit{A}$ which maps an observation $\mathit{s}\in \mathit{S}$ into an appropriate authority ${\text{P}}_{\mathit{X}}(\mathit{a})\in \mathit{A}$.

2. A decision authority mapping ${\text{P}}_{\mathit{X}}:\mathit{S}\to \mathit{A}$ can be represented by

   (1.1) ${\text{P}}_{\mathit{X}}=\{(\mathit{s},\mathit{a}):\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{s}),\mathit{s}\in \mathit{S}\}.$(1.1)

Note 1.2.

In Equation (1.1)(1.1) ${\text{P}}_{\mathit{X}}=\{(\mathit{s},\mathit{a}):\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{s}),\mathit{s}\in \mathit{S}\}.$(1.1) , we observe the following;

1. The first component s of every pair (s, a) in the set ${\text{P}}_{\mathit{X}}$ is a vector $\mathit{s}=(s_1,\dots ,s_{\mathit{n}})$ of values of facts (existing situations). The component values s_i of s are the components of the situation.

2. The second component $\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{s})$ is a set $\mathit{a}=\{a_1,a_2,\dots \}$ of actions, one of which is to become a fact for the corresponding situation s. The values a_j in set a are the authorities, one of which is to be made a rule (action) for a particular situation.

Example 1.

Consider a statement “If a spy is arrested, take him to high court”. This statement is a single-element authority mapping. It has one element in the domain and one in its range. Let $\mathit{s}=$ a spy is arrested, and $\mathit{a}=$ take him to high court. Then, by definition 1.2, $\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{s})$, and ${\text{P}}_{\mathit{X}}=\{(\mathit{s},\mathit{a})\}=\{(\text{if }\mathit{s},\text{do }\mathit{a})\}$. Therefore, the entire statement is a single element of the mapping ${\text{P}}_{\mathit{X}}$.

Some authority mappings are complex as they may have a then segment made up of many elements connected by and or by or. These kinds of mappings will be represented using a collection of simple unit authority mappings, which we defined (see definition 1.2).

Example 2.

Consider a complex authority mapping which states, “If a public official commits a corruption offence, then either take him to jail for life or confiscate all his material properties, but not both.” In this statement, the element situation s is a public official commits a corruption offence, and the set, a, of prescriptively allowed values of an authority variable includes: take him to jail for life but do not confiscate all his material properties; do not take him to jail for life but confiscate all his material properties. One of these elements of a has to be made a fact. Let $\mathit{p},\mathit{q},\mathit{r}$ be the propositions that $\mathit{p}=$ a public official commits a corruption offence, $\mathit{q}=$ take him to jail for life, and $\mathit{r}=$ confiscate all his material properties. In mathematical logic, a given complex statement is represented as: $(\mathit{p}⟹(\mathit{q}\vee \mathit{r}))\wedge (\mathrm{\neg q}\wedge \mathrm{\neg r})$, and the simplification gives $((\mathit{p}⟹\mathit{r})\wedge \mathrm{\neg q})\wedge ((\mathit{p}⟹\mathit{q})\wedge \mathrm{\neg r})$. The predicates of the latter representation are $(\mathit{p}⟹\mathit{r},\mathrm{\neg q})$ and $(\mathit{p}⟹\mathit{q},\mathrm{\neg r})$. Therefore, the decision authority mapping for a situation p is given by

${\text{P}}_{\mathit{X}}(\mathit{p})=\{\begin{array}{ll}\text{do }\mathit{r}& \text{if not q,}\\ \text{do }\mathit{q}& \text{if not r}.\end{array}$

and the mapping is ${\text{P}}_{\mathit{X}}=\{((\mathit{p},\mathit{r}),(\mathrm{\neg q})),((\mathit{p},\mathit{q}),(\mathrm{\neg r}))\}$.

The definition of a decision authority mapping gives the value of an action in the form of words. Since the same decision authority in the same situation can be exercised by different people sitting in different authority positions in the chain, we need to quantify the decision authority by assigning a numerical value to each element of a set of prescriptively allowed values of a decision variable. These numbers will be used as measures of the strength of the decision authority someone has in his or her administrative position over a particular situation. Mayila (Mayila et al., 2023) called these numerical values decision indices.

To capture this setting, we redefine, without distorting the original meaning, the decision authority exercised by some set, U, of people in their administrative positions in various situations. With the exception of 2^U being a superset of a set U, the letters and symbols in the definition below carry the same meanings as in definition 1.2.

Definition 1.3.

A decision authority exercised by a set U of people in their administrative positions on a set S of situations is a mapping ${\text{P}}_{\mathit{X}}:2^{\mathit{U}}\setminus \mathrm{\varnothing }\times \mathit{S}\to \mathit{A}$ that assigns to every nonempty element $(\mathit{u},\mathit{s})\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }\times \mathit{S}$ a set a $(\in \mathit{A})$ of actions or rules, one of which must be made a fact.

With this new definition, Equation (1.1)(1.1) ${\text{P}}_{\mathit{X}}=\{(\mathit{s},\mathit{a}):\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{s}),\mathit{s}\in \mathit{S}\}.$(1.1) gives ${\text{P}}_{\mathit{X}}=\{((\mathit{u},\mathit{s}),\mathit{a}):\mathit{a}={\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s}),(\mathit{u},\mathit{s})\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }\times \mathit{S}\}$. Since a is a set of decision values, which are words; then, we assign a number to each element a_j of $\mathit{a}=\{a_1,a_2,\dots \}$ that shows the strength of someone to make a decision. These numbers, as noted earlier, are decision indices. Therefore, we define the decision index, denoted by ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})$, of an element $\mathit{u}\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }$ in a situation $\mathit{s}\in \mathit{S}$ as a strength or magnitude of ${\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s})$. The decision index ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})$ is non-negative, and for simplicity, we will restrict its range to a closed and bounded unit interval $[0,1]$ and define it by ${\rho }_{\mathit{X}}:2^{\mathit{U}}\setminus \mathrm{\varnothing }\times \mathit{S}\to [0,1]$. If ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})=0$, u has no authority over the situation s. If ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})=1$, then u has the authority ${\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s})$ which is final on a situation s, and if $0<{\rho }_{\mathit{X}}(\mathit{u},\mathit{s})<1$, u has authority ${\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s})$ which allows him or her to make decisions which are not final, implying that the decisions he or she makes have to be channelled to the higher-ups for approval.

We have laid a more than sufficient foundation for a decision authority. Since our purpose in this section is not to discuss decision-making processes and the execution of decision rules, we will not pursue this further. However, the discussion provides a setting towards defining a more rigorous mathematical representation of a decision space than it was given by Mayila (Mayila et al., 2023).

Definition 1.4.

Let U be a subset of X, ${\text{P}}_{\mathit{X}}(\mathit{U})$ be a set of decision authorities of U, and S_U be a set of possible situations in the administrative area of U. Mathematically, we define the following:

1. A decision taken by $\mathit{u}\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }$ over a situation $\mathit{s}\in S_{\mathit{U}}$ using a decision authority ${\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s})$ is a function

   (1.2) $\mathit{\varphi }:2^{\mathit{U}}\setminus \mathrm{\varnothing }\times S_{\mathit{U}}\times {\text{P}}_{\mathit{X}}\to \mathit{D}$(1.2)

   which maps a collection $(\mathit{u},\mathit{s},{\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s}))$ into a decision element $\mathit{d}=\mathit{\varphi }(\mathit{u},\mathit{s},{\text{P}}_{\mathit{X}}(\mathit{u},\mathit{s}))\in \mathit{D}$, where D is a set of decisions.

2. A decision space in a nation X is a triple ordered collection $(\mathit{U},S_{\mathit{U}},{\text{P}}_{\mathit{X}}(\mathit{U}))$ comprising the set U of people, the set S_U of possible situations that may occur in the administrative section of U, and the set of decision authorities ${\text{P}}_{\mathit{X}}(\mathit{U})$ of U in X on $S_{\mathit{U}}$.

   If there is no confusion that may arise, the decision space $(\mathit{U},S_{\mathit{U}},{\text{P}}_{\mathit{X}}(\mathit{U}))$ will be denoted by U, and from now on, we will call the collection $(\mathit{X},S_{\mathit{X}},{\text{P}}_{\mathit{X}}(\mathit{X}))$ a nation space. For the sake of brevity and simplicity, $(\mathit{X},S_{\mathit{X}},{\text{P}}_{\mathit{X}}(\mathit{X}))$ will be denoted simply by X.

It is important to note that the decision authority mapping ${\text{P}}_{\mathit{X}}:2^{\mathit{U}}\setminus \mathrm{\varnothing }\times S_{\mathit{U}}\to \mathit{A}$ is the heart of the decision space U.

Definition 1.5.

Mayila et al. (2023, p. 3)

Let x be a point and U be a nonempty decision space in the nation X. Then

1. $\mathit{x}\in \mathit{X}$ is a direct member of U if x is contained directly in U (x is a leader). This person attends the decision-making processes in U and has the right to vote for a decision.

2. $\mathit{x}\in \mathit{X}$ is an indirect member of U if there exists at least one direct member $\mathit{y}\in \mathit{U}$ who represents x in U. This person x has no right to vote for a decision even if s/he attends the meetings of U (${\rho }_{\mathit{U}}(\mathit{x},\mathit{s})=0$ for all $\mathit{s}\in S_{\mathit{U}}$). We will represent by $\mathit{y}{\sim }_{\mathit{U}}\mathit{x}$ the phrase “y represents x”.

3. U is a direct neighborhood of x if x is a direct member of U.

4. U is an indirect neighborhood of x if x is an indirect member of U.

5. U is a neighborhood of x if x is either a direct or indirect member of U.

Definition 1.6.

Mayila et al. (2023, p. 3)

Let X be a nation and $\mathit{x}\in \mathit{X}$. Then

1. X is said to be inclusive at x if there exists a decision space G in X that represents $\mathit{x}.$

2. X is inclusive if it is inclusive at all points.

Lemma 1.7.

Mayila et al. (2023, p. 3)

Every nonempty decision space in an inclusive nation X is the union of its one-point subsets composed by its direct members.

The following result is a consequence of Lemma 1.7, and its proof is easy to sketch.

Corollary 1.8.

Every nonempty subset of a decision space is a decision space.

The following proposition is an immediate consequence of the definitions 1.3, 1.4, lemma 1.7, and corollary 1.8.

Corollary 1.9.

If U is a decision space in a nation space X, then for each $\mathit{u}\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }$, there exists at least one situation $\mathit{s}\in S_{\mathit{U}}$ such that ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})>0$.

Proof.

The proof here is very direct. Let $\mathit{u}\in 2^{\mathit{U}}\setminus \mathrm{\varnothing }$, and suppose in contrast that ${\rho }_{\mathit{X}}(\mathit{u},\mathit{s})=0$ for all $\mathit{s}\in S_U$. This makes u a non-decision maker in U, and hence, by Lemma 1.7 and Corollary 1.8, this is a contradiction.□

Definition 1.10.

Mayila et al. (2023, p. 4)

Let U and V be two distinct nonempty decision spaces in a ${\mathcal{T}}_{\mathit{r}}-$ space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$. Then

1. U is superior to V if V answers to U and we write $\mathit{V}\le \mathit{U}$.

2. U and V are equivalent if $\mathit{U}\le \mathit{V}$ and $\mathit{V}\le \mathit{U}$ and we denote this by $\mathit{U}\equiv \mathit{V}$.

Definition 1.11.

(Maximal and Minimal element) Let S be an arbitrary ordered set. Then

1. An element $\mathit{m}\in \mathit{S}$ is said to be a maximal element of S if $\mathit{m}\ge \mathit{s}$ for all $\mathit{s}\in \mathit{S}$.

2. Element $\mathit{l}\in \mathit{S}$ is a minimal element of S if $\mathit{l}\le \mathit{s}$ for all $\mathit{s}\in \mathit{S}$.

3. A set S is bounded if it has both maximal and minimal elements.

The following proposition gives the boundedness of a ${\mathcal{T}}_{\mathit{r}}-$ space:

Proposition 1.12.

Mayila et al. (2023, p. 4)

The following statements hold for every ${\mathcal{T}}_{\mathit{r}}-$ space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$;

1. Every ${\mathcal{T}}_{\mathit{r}}-$ space is linear.

2. Every ${\mathcal{T}}_{\mathit{r}}-$ space has a maximal and minimal decision space.

Corollary 1.13.

Mayila et al. (2023, p. 5)

Every nonempty decision space in the space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ has minimal and maximal elements.

Definition 1.14.

(ϕ-proximity)

Let $\mathit{U}=(\mathit{U},{\rho }_{\mathit{X}}(\mathit{U}),S_{{\mathit{E}}_{\mathit{U}}})$ and $\mathit{V}=(\mathit{V},{\rho }_{\mathit{X}}(\mathit{V}),S_{{\mathit{F}}_{\mathit{V}}})$ be two nonempty decision spaces in administrative sections E and F, respectively, in the space $(\mathit{X},{\mathcal{T}}_r)$. Then

1. The administrative intersection of U and V will be denoted by ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{V}}$ and defined by

   $\begin{array}{l}{\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{V}=\{\mathit{x}\in \mathit{U}\cup \mathit{V}:{\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}(\mathit{U})\text{and}}\\ {\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}(\mathit{V})\}.\end{array}$

2. If $\mathit{U}\le \mathit{V}$ (definition 1.10), we write

   1. ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{V}=\mathit{U}}$

   2. ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cup }\mathit{V}=\mathit{V}}$

3. If $\mathit{U}\equiv \mathit{V}$ (definition 1.10), we write ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{V}=\mathit{U}}$ (or = V).

4. U and V are said to be $\mathit{\varphi }-$near to each other, denoted as in Naimpally and Peters (2013: 67–68) by $\mathit{U}{\delta }_{\mathit{\varphi }}\mathit{V}$ if their administrative intersection is nonempty. Otherwise, U and V are $\mathit{\varphi }-$far, adapting denotion of $\mathit{U}{\underset{―}{\delta }}_{\mathit{\varphi }}\mathit{V}$.

5. E and F are said to be $\mathit{\varphi }-$near to each other if there exists a decision space in their union that answers to both.

Based on ϕ-proximity, Mayila (Mayila et al., 2023) defined the interior, closure, limit and boundary points of the subsets of the ${\mathcal{T}}_{\mathit{r}}-$space as follows:

Definition 1.15.

Mayila et al. (2023, p. 5–6)

Let A be a subset of a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$. Then

1. A point $\mathit{x}\in \mathit{A}$ is a ϕ-interior point of A if there exists a decision space U in X such that x is a direct member of U and $\mathit{U}\subset \mathit{A}$. The set of all $\mathit{\varphi }-$interior points of A will be called ϕ-interior of A in X, and we shall denote it by $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{A}$.

   It is worth noting that the properties of $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{A}$ mimic the well-known properties of interiors of sets under spatial or Cantorian set operations.

2. A person $\mathit{x}\in \mathit{X}$ is a $\mathit{\varphi }-$contact point of A if every decision space U_x that directly contains x is administratively near to A. $\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{A}$ will be used to denote the set of contact points of A in X under $\mathit{\varphi }-$proximity. Let $\{U_{\mathit{\alpha }\in \mathit{J}}:|\mathit{J}|<\mathrm{\infty }\}$ be an indexed family of all decision spaces in X, each of which contains x directly. Then, by definition, a person $\mathit{x}\in \mathit{X}$ is a $\mathit{\varphi }-$contact person of A only if

   ${\displaystyle U_{\mathit{\alpha }}\underset{\mathit{\varphi }}{\cap }\mathit{A}\ne \mathrm{\varnothing }\text{for all}\mathit{\alpha }\in \mathit{J}.}$

3. A person $\mathit{x}\in \mathit{X}$ is a $\mathit{\varphi }-$limit point of A in X if, for every direct neighborhood U_x of x, $U_{\mathit{x}}-\{\mathit{x}\}$ is administratively near A. Let $\{U_{\mathit{\alpha }\in \mathit{J}}:|\mathit{J}|<\mathrm{\infty }\}$ be an indexed family of all decision spaces in X where each contains x directly. Then, by definition, a person $\mathit{x}\in \mathit{X}$ is a $\mathit{\varphi }-$limit point of A if

   $\begin{array}{l}{\displaystyle (U_{\mathit{\alpha }}-\{\mathit{x}\})\underset{\mathit{\varphi }}{\cap }\mathit{A}\ne \mathrm{\varnothing }\text{for all}\mathit{\alpha }\in \mathit{J}.}\end{array}$

4. A person $\mathit{x}\in \mathit{X}$ is a $\mathit{\varphi }-$boundary point of A in X if every direct neighborhood of x is administratively near to both A and the complement of A in X.

   The set of all $\mathit{\varphi }-$boundary points of A in X is called the $\mathit{\varphi }-$boundary of A in X, and will be denoted by ${\mathit{Frt}}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{A}.$

Mayila (Mayila et al., 2023) then used this definition 1.15 and the concept of $\mathit{\varphi }-$proximity (see definition 1.14) and established various results regarding the properties of these new defined topological operators. The following are some of the results which were established, some of which will be used in our study. Their proofs are omitted here, unless it is a new preliminary result.

Theorem 1.16

Mayila et al. (2023, p. 6)

A subset E of a ${\mathcal{T}}_{\mathit{r}}-$space X is inclusive if and only if int${}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ is nonempty.

Theorem 1.17

Mayila et al. (2023, p. 7)

Let E be an administrative section of the space $(\mathit{X},{\mathcal{T}}_r)$ and x be any point in X. If ${\rho }_{\mathit{X}}(\mathit{G})\le {\rho }_{\mathit{X}}\{\mathit{x}\}$ for every non-empty decision space G contained in E, then x is a $\mathit{\varphi }-$contact point of E. The converse does not hold.

Corollary 1.18.

Mayila et al. (2023, p. 7)

If x is a maximal element of the subset E of the ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$, then x is a $\mathit{\varphi }-$frontier point of E.

Corollary 1.19. 

Mayila et al. (2023, p. 7)

If $E_1,E_2$ are two nonempty subsets of a subset E of the ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$, then the intersection of their $\mathit{\varphi }-$closures in E is nonempty.

The following theorem gives a topological meaning of a point being $\mathit{\varphi }-$near to the set in a ${\mathcal{T}}_{\mathit{r}}-$space:

Theorem 1.20

Let x be a point and E be the subset of a space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$, respectively. Then, $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$ if and only if $\mathit{x}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$.

Proof.

Suppose that $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$. Then, by definition 1.14, we have ${\displaystyle \{\mathit{x}\}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }.}$ Therefore, ${\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$ for each direct neighborhood U_x of $\mathit{x}\in \mathit{X}$. By definition of a $\mathit{\varphi }-$contact point, we have $\mathit{x}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. Conversely, suppose that x is a $\mathit{\varphi }-$contact point of E, then for each direct neighborhood U_x of x, we get ${\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }.}$ If $\mathit{x}\in {\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{E}}$ for each U_x, then ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}\mathit{E}$. This implies that $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$. If $\mathit{x}\notin {\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{E}}$, then there is an element $\mathit{y}\in {\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{E}}$ with the property that ${\rho }_{\mathit{X}}\{\mathit{y}\}\le {\rho }_{\mathit{X}}{U}_{\mathit{x}}$ and ${\rho }_{\mathit{X}}\{\mathit{y}\}\le {\rho }_{\mathit{X}}\mathit{E}$. This further implies that $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{y}$ and $\mathit{y}{\delta }_{\mathit{\varphi }}\mathit{E}$. By the property of transitivity, we have $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$.

The consideration of the $\mathit{\varphi }-$proximity of an element to a set in a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ leads to the following theorem:

Theorem 1.21

Let p be a point and $\mathit{E},\mathit{F}$ be subsets of a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}},{\delta }_{\mathit{\varphi }})$ endowed with $\mathit{\varphi }-$proximity ${\delta }_{\mathit{\varphi }}$. Then, the following statements hold:

1. $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ implies E is nonempty.

2. ${\displaystyle \{\mathit{p}\}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$ implies $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$.

3. $\mathit{p}{\delta }_{\mathit{\varphi }}(\mathit{E}\cup \mathit{F})$ if and only if $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ or $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{F}$.

4. If $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ and $\mathit{e}{\delta }_{\mathit{\varphi }}\mathit{F}$ for each $\mathit{e}\in \mathit{E}$, then $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{F}$.

Proof.

1. Given $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$, then ${\displaystyle \{\mathit{p}\}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$. Then, there is an element $\mathit{y}\in \{\mathit{p}\}\cup \mathit{E}$ that is $\mathit{\varphi }-$near to both $\{\mathit{p}\}$ and E. This logically implies that E is nonempty. Otherwise, y would be $\mathit{\varphi }-$far from E, contradicting the fact that y is $\mathit{\varphi }-$proximally (or administratively) contained in E.

2. This follows from the definition 1.14.

3. Suppose that $\mathit{p}{\delta }_{\mathit{\varphi }}(\mathit{E}\cup \mathit{F})$. Then, for each direct neighborhood U_p of $\mathit{p}\in \mathit{X}$, we have ${\displaystyle U_{\mathit{p}}\underset{\mathit{\varphi }}{\cap }(\mathit{E}\cup \mathit{F})\ne \mathrm{\varnothing }}$. By elementary set theory, we have ${\displaystyle U_{\mathit{p}}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$ or ${\displaystyle U_{\mathit{p}}\underset{\mathit{\varphi }}{\cap }\mathit{F}\ne \mathrm{\varnothing }}$ for each U_p. By definition of $\mathit{\varphi }-$contact point, we get $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ or $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{F}$. The latter statement further implies that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ or $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{F}$ (see Theorem 1.20). Conversely, if $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ or $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{F}$, then for each open neighborhood U of p, we have ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$ or ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{F}\ne \mathrm{\varnothing }}$. Using the distributive property of sets, we get ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }(\mathit{E}\cup \mathit{F})\ne \mathrm{\varnothing }}$ for each U. By theorem 1.20, $\mathit{x}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{E}\cup \mathit{F})$, and hence $\mathit{p}{\delta }_{\mathit{\varphi }}(\mathit{E}\cup \mathit{F})$.

4. Given that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ and $\mathit{e}{\delta }_{\mathit{\varphi }}\mathit{F}$ for each $\mathit{e}\in \mathit{E}$. Then, by theorem 1.20, we have $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, and $\mathit{e}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{F}$ for each $\mathit{e}\in \mathit{E}$. The latter implies that $\mathit{E}\subset \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{F}$ so that $\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\subset \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{F}$. Therefore, $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{F}$, and by theorem 1.20, it follows that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{F}$.

Definition 1.22.

Mayila et al. (2023, p. 8)

Let $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ be a ${\mathcal{T}}_{\mathit{r}}-$topological space. Then

1. A $\mathit{\varphi }-$separation of X is a pair $\mathit{U},\mathit{V}$ of spatially disjoint nonempty decision spaces in X such that $\mathit{U}\cup \mathit{V}{\sim }_{\mathit{X}}\mathit{X}$ and ${\displaystyle \mathit{U}\underset{\mathit{\varphi }}{\cap }\mathit{V}=\mathrm{\varnothing }}$.

2. A space with a $\mathit{\varphi }-$separation is said to be $\mathit{\varphi }-$separated.

3. A space without $\mathit{\varphi }-$separation is said to be $\mathit{\varphi }-$connected.

Since spatial nearness implies $\mathit{\varphi }-$nearness, then the following well-known result holds in $\mathit{\varphi }-$proximity.

Lemma 1.23.

Let Y be a subspace of $(\mathit{X},\mathcal{T})$. Then, Y is separated if and only if there exist two nonempty disjoint subsets A and B of Y whose union is Y, none of which contains the contact point of the other. The space Y is connected if such separation does not exist.

The statement and proof of Lemma 1.23 is found in Hocking and Young (1961, pp. 15-16) and Munkers (2000, pp. 148-149).

Theorem 1.24

Mayila et al. (2023, p. 8)

Every subspace of the ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ is $\mathit{\varphi }-$connected.

Corollary 1.25.

Mayila et al. (2023, p. 8)

A subset E of a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ is administratively connected if and only if the $\mathit{\varphi }-$interior of E in X is nonempty.

Theorem 1.26 

Mayila et al. (2023, p. 8)

Suppose that G is a nonempty decision space in $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ and that $\{E_{\mathit{\alpha }}\}$ is an indexed family of subsets of $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ such that ${\displaystyle E_{\mathit{\alpha }}\underset{\mathit{\varphi }}{\cap }\mathit{G}\ne \mathrm{\varnothing }}$ for each α. Then, the union $E^{\ast }=\mathit{G}\cup ({\cup }_{\mathit{\alpha }}{E}_{\mathit{\alpha }})$ is $\mathit{\varphi }-$connected.

Corollary 1.27.

Mayila et al. (2023, p. 9)

A ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ is $\mathit{\varphi }-$connected.

Definition 1.28.

Mayila et al. (2023, p. 10)

Let $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ be a ${\mathcal{T}}_{\mathit{r}}-$space. A chain of decision spaces, denoted by m-chain, is a finite sequence $\mathcal{M}=(G_1,G_2,G_3,\dots ,G_{\mathit{n}})\subset {\mathcal{T}}_{\mathit{r}}$ that satisfies the following axioms:

1. $G_{\mathit{k}}\ne \mathrm{\varnothing }$ for all $\mathit{k}\in \{1,2,3,\dots ,\mathit{n}\}$,

2. ${\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{G}_{\mathit{k}+1}\ne \mathrm{\varnothing }}$ for all $\mathit{k}=1,2,3,\dots ,\mathit{n}-1$,

3. Every $G_{\mathit{k}}=(G_{\mathit{k}},{\rho }_{\mathit{X}}(G_{\mathit{k}}),S_{{\mathit{G}}_{\mathit{k}}})\in \mathcal{M}$ has only one immediate successor $G_{\mathit{k}+1}\in \mathcal{M}$ for every $\mathit{s}\in S_{{\mathit{G}}_{\mathit{k}}}$.

Definition 1.29.

Mayila et al. (2023, p. 11)

Let ${\mathcal{M}}_1=\{G_{11},\dots ,\text{allowbreak}{G}_{1\mathit{m}}\}$ and ${\mathcal{M}}_2=\{G_{21},\dots ,G_{2\mathit{n}}\}$ be m-chains from a point x to a point y in a ${\mathcal{T}}_{\mathit{r}}-$space X. The m-chain ${\mathcal{M}}_1$ is said to go strictly straight through ${\mathcal{M}}_2$ provided that

1. each link $G_{1\mathit{j}}$ is strictly contained in some link $G_{2\mathit{k}}$ or lies within the two adjacent links $G_{2\mathit{k}}$ and $G_{2\mathit{k}+1}$

2. if $G_{1\mathit{j}}$ and $G_{1\mathit{l}}$ are both wholly contained in a set $G_{2\mathit{p}}$ for j < l, then for each integer $\mathit{k}\in (\mathit{j},\mathit{l})$, the link $G_{1\mathit{k}}$ lies within $G_{2\mathit{p}}$.

Theorem 1.30

Kinsey (1993, p. 30)

The topological equivalence is an equivalence relation.

Definition 1.31.

Adams and Franzosa (2008, p. 198)

Let X be a connected topological space. A cut set of X is a subset E of X such that X − E is disconnected. A cut point of a space X is a point $\mathit{p}\in \mathit{X}$ such that the set $\{\mathit{p}\}$ is a cut set of X. A cut set or cut point separates or disconnects X.

We have so far laid down a foundation for our study. We are now in a position to communicate the main findings of this paper.

2. Continuity and $\mathit{\varphi }-$continuity

If we are considering a function that is defined between two sets whose members are people, we need to be careful in defining the context of domain and co-domain. A person, as a point in a domain, is characterized by its attributes or by its state. The human being changes his mind, desires, needs, and any other states depending on the change in environments or situations. The function to a set of people may mean a change in the current state of a person into another state according to the requirements of that function. Since topology is a discipline that has been designed to formulate and treat continuity of functions (Singh, 2013, pp. 41–42; Latecki & Prokop, 1995; Kinsey, 1993, pp. 2–4; Alexandroff, 1961)), it is important to define the continuity of functions which show how points in two domains are interrelated. This will further enable us to study a one-to-one correspondence between the interacting nations and how one nation can sit within the other.

This section introduces the notion of continuous functions which preserve the $\mathit{\varphi }-$proximity of subsets of ${\mathcal{T}}_{\mathit{r}}-$spaces.

Definition 2.1.

Let $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$, $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$ be ${\mathcal{T}}_{\mathit{r}}-$spaces endowed with $\mathit{\varphi }-$proximity. A function $\mathit{f}:\mathit{X}\to \mathit{Y}$ is said to be $\mathit{\varphi }-$continuous at a point p of X if for each subset E of X, $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ implies $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$. The function f is $\mathit{\varphi }-$continuous if it is $\mathit{\varphi }-$continuous at every point of X.

Definition 2.1 can be extended further by replacing a single point p by another subset H of X. In this way, we say that the function $\mathit{f}:\mathit{X}\to \mathit{Y}$ is $\mathit{\varphi }-$continuous if for each subsets $\mathit{E},\mathit{H}$ of X, $\mathit{E}{\delta }_{\mathit{\varphi }}\mathit{F}$ implies $\mathit{f}(\mathit{E}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{F})$ .

If f is bijective and both f and f⁻¹ are $\mathit{\varphi }-$continuous, then f will be called a $\mathit{\varphi }-$homeomorphism and the spaces $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$ and $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$ are said to be proximally $\mathit{\varphi }-$homeomorphic with respect to f.

Example 3.

The identity function $\mathit{id}:(\mathit{X},{\mathcal{T}}_{\mathit{r}},{\delta }_{\mathit{\varphi }})\to (\mathit{X},{\mathcal{T}}_{\mathit{r}},\text{allowbreak}{\delta }_{\mathit{\varphi }})$, given by $\mathit{id}(\mathit{x})=\mathit{x}$ for each $\mathit{x}\in \mathit{X}$, is $\mathit{\varphi }-$continuous. This is easy to demonstrate. If x is a point in X and E is a subspace of X with $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$, then $\mathit{id}(\mathit{x}){\delta }_{\mathit{\varphi }}\mathit{id}(\mathit{E})$ since $\mathit{id}(\mathit{x})=\mathit{x}$ and $\mathit{id}(\mathit{E})=\mathit{E}$.

The following theorem gives some properties of $\mathit{\varphi }-$continuous functions between ${\mathcal{T}}_{\mathit{r}}-$spaces.

Theorem 2.2

Let $\mathit{f}:\mathit{X}\to \mathit{Y}$ be a function defined on spaces $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$, $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$. Then, the following statements are equivalent.

1. For every point $\mathit{p}\in \mathit{X}$ and subset E of X, $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ implies $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$.

2. For every subset E of X, $\mathit{f}(\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E})$ is contained in $\mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}(\mathit{f}(\mathit{E}))$.

3. For each $\mathit{\varphi }-$closed set C in Y, $f^{-1}(\mathit{C})$ is $\mathit{\varphi }-$closedin X.

4. For each decision space H in Y, $f^{-1}(\mathit{H})$ is a decision space in X.

Proof.

We show that $(\mathit{i})⟹(\mathit{ii})⟹(\mathit{iii})⟹(\mathit{iv})\text{allowbreak}⟹(\mathit{i}).$

$(\mathit{i})⟹(\mathit{ii})$: Assume that for every point p and every subset E of X, $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}⟹\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$. Then, for every decision space U covering p in X, we have $\mathit{U}{\delta }_{\mathit{\varphi }}\mathit{E}$. Thus $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ (see Theorem 1.20). A similar argument holds for f(p) being administratively near E. If V is a decision space in Y containing f(p), then $\mathit{V}{\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$, so that $\mathit{f}(\mathit{p})\in \mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}\mathit{f}(\mathit{E})$. Therefore, by hypothesis, if $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$ implies $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$, then $\mathit{p}\in \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ implies $\mathit{f}(\mathit{p})\in \mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}\mathit{f}(\mathit{E})$. Hence $\mathit{f}(\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E})\subset \mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}\mathit{f}(\mathit{E})$ for each $\mathit{E}\subset \mathit{X}$.

$(\mathit{ii})⟹(\mathit{iii})$: Let C be $\mathit{\varphi }-$closed in Y. To show that $f^{-1}(\mathit{C})$ is $\mathit{\varphi }-$closed in X, it is enough to show that $\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}{f}^{-1}(\mathit{C})$ is administratively contained in $f^{-1}(\mathit{C})$. By elementary set theory, $\mathit{f}(f^{-1}(\mathit{C}))\subset \mathit{C}.$ Since C is $\mathit{\varphi }-$closed in Y, $\mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}\mathit{f}(f^{-1}(\mathit{C}))\subset \mathit{C}$. By hypothesis (ii), $\mathit{c}{l}_{{\mathit{Y}}_{\mathit{\varphi }}}\mathit{f}(f^{-1}(\mathit{C})$ contains $\mathit{f}(\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}{f}^{-1}(\mathit{C}))$, so that C contains $\mathit{f}(\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}{f}^{-1}(\mathit{C}))$. By elementary set theory, we have that $f^{-1}(\mathit{C})$ contains $\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}{f}^{-1}(\mathit{C})$, as desired.

$(\mathit{iii})⟹(\mathit{iv})$: This is obvious.

$(\mathit{iv})⟹(\mathit{i})$: Suppose that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$, and G is a decision space that contains f(p) in Y. By hypothesis (iv), $f^{-1}(\mathit{G})$ is a decision space that contains p in X. Since $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$, then $f^{-1}(\mathit{G}){\delta }_{\mathit{\varphi }}\mathit{E}$. Therefore, ${\displaystyle f^{-1}(\mathit{G})\underset{\mathit{\varphi }}{\cap }\mathit{E}\ne \mathrm{\varnothing }}$. Let q be an element of ${\displaystyle f^{-1}(\mathit{G})\underset{\mathit{\varphi }}{\cap }\mathit{E}}$. Then, f(q) is an element of ${\displaystyle \mathit{G}\underset{\mathit{\varphi }}{\cap }\mathit{f}(\mathit{E})}$. Since G is an arbitrary decision space containing f(p) in Y, $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$.

The following theorem states the construction of various $\mathit{\varphi }-$continuous functions from one ${\mathcal{T}}_{\mathit{r}}-$space to another.

Theorem 2.3

Let $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$, $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$, and $(\mathit{Z},{\mathcal{T}}_{\mathit{r}2},\text{allowbreak}{\delta }_{\varphi }^{″})$ be ${\mathcal{T}}_{\mathit{r}-}$spaces.

1. If a function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ takes all points of X into a single point y₀ of Y, then f is $\mathit{\varphi }-$continuous.

2. If E is a subspace of X, the inclusion function $\mathit{i}:\mathit{E}\mapsto \mathit{X}$ is $\mathit{\varphi }-$continuous.

3. If the functions $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ and $\mathit{g}:\mathit{Y}\mapsto \mathit{Z}$ are $\mathit{\varphi }-$continuous, then the composition function $\mathit{g}\circ \mathit{f}:\mathit{X}\mapsto \mathit{Z}$ is $\mathit{\varphi }-$continuous.

4. If the function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ is $\mathit{\varphi }-$continuous, and if E is a subspace of X, then $\mathit{f}{|}_{\mathit{E}}:\mathit{E}\mapsto \mathit{Y}$ is $\mathit{\varphi }-$continuous.

5. Let the function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ be $\mathit{\varphi }-$continuous. If Z is a ${\mathcal{T}}_{\mathit{r}}-$space that contains Y as a subspace, then the function $\mathit{h}:\mathit{X}\mapsto \mathit{Z}$ is $\mathit{\varphi }-$continuous.

Proof.

1. Let $\mathit{p}\in \mathit{X}$ and E be any subset of X with the property that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$. We show that $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$. Since $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$, there is a point y of $\{\mathit{p}\}\cup \mathit{E}$ such that $\mathit{y}\in {\displaystyle \{\mathit{p}\}\underset{\mathit{\varphi }}{\cap }\mathit{E}}$ and $\mathit{f}(\mathit{y})\in {\displaystyle \mathit{f}(\{\mathit{p}\})\underset{\mathit{\varphi }}{\cap }\mathit{f}(\mathit{E})}$. Since f is constant, there is y₀ in Y such that $\mathit{f}(\mathit{x})=y_0$ for every $\mathit{x}\in \mathit{X}$. This implies that $y_0\in {\displaystyle \{y_0\}\underset{\mathit{\varphi }}{\cap }\{y_0\}}$. Therefore, $\mathit{f}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$.

2. Suppose that $\mathit{p}\in \mathit{E}$ and $\mathit{A}\subset \mathit{E}$ such that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{A}$. Since $\mathit{i}(\mathit{a})=\mathit{a}$ for each $\mathit{a}\in \mathit{E}$, then $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{A}$ implies $\mathit{i}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{i}(\mathit{A})$.

3. Let x be any point in X and E be a subset of X such that $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$. We show that $\mathit{g}\circ \mathit{f}(\mathit{x}){\delta }_{\varphi }^{″}\mathit{g}\circ \mathit{f}(\mathit{E})$. Since $\mathit{x}{\delta }_{\mathit{\varphi }}\mathit{E}$ and f is $\mathit{\varphi }-$continuous, then $\mathit{f}(\mathit{x}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{E})$. Similarly, the $\mathit{\varphi }-$continuity of g implies that $\mathit{g}(\mathit{f}(\mathit{x})){\delta }_{\varphi }^{″}\mathit{g}(\mathit{f}(\mathit{E}))$, as desired.

4. Note that the restricted function $\mathit{f}{|}_{\mathit{E}}:\mathit{E}\mapsto \mathit{Y}$ is a composition of the function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ and the inclusion function $\mathit{i}:\mathit{E}\mapsto \mathit{Y}$, both of which are $\mathit{\varphi }-$continuous. By (iii), $\mathit{f}{|}_{\mathit{E}}$ is $\mathit{\varphi }-$continuous

5. Since Y is a subspace of Z, the inclusion function $\mathit{i}:\mathit{Y}\mapsto \mathit{Z}$ is $\mathit{\varphi }-$continuous (refer to Theorem 2.3(ii)). But the function h is the composition of the two $\mathit{\varphi }-$continuous functions, i and f. That is, $\mathit{h}=\mathit{i}\circ \mathit{f}:\mathit{X}\mapsto \mathit{Z}$. By Theorem 2.3(iii), h is $\mathit{\varphi }-$continuous.

Since a representation in decision-making is necessary, it is important to consider the continuity of a function over only a small set, such that the continuity of a function on such a small set will mean the continuity on the whole space. The following theorem establishes the continuity of a function on a small set, which implies continuity on the whole set.

Theorem 2.4

Let $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$ and $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$ be ${\mathcal{T}}_{\mathit{r}}-$spaces and $\mathit{h}:\mathit{X}\mapsto \mathit{Y}$ be a function. Suppose that $\{G_{\mathit{\alpha }}:\mathit{\alpha }\in \{1,\dots ,\mathit{n}\}\}$ is a collection of decision spaces in X such that ${\displaystyle \underset{\mathit{\alpha }=1}{\overset{\mathit{n}}{\cup }}{G}_{\mathit{\alpha }}{\sim }_{\mathit{X}}\mathit{X}}$ and that the restriction function $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}:G_{\mathit{\alpha }}\mapsto \mathit{Y}$ is $\mathit{\varphi }-$continuous for each index α. Then, the function $\mathit{h}:\mathit{X}\mapsto \mathit{Y}$ is $\mathit{\varphi }-$continuous.

Proof.

Let p be any point and E be any subset of X with the property that $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$. We need to show that $\mathit{h}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{h}(\mathit{E})$. Let V be a decision space in Y that contains h(p). Since $h^{-1}(\mathit{V})$ is a subset of X, then by elementary set theory, we have

(2.1) $h^{-1}(\mathit{V})=h^{-1}(\mathit{V})\cap \mathit{X}.$(2.1)

Since ${\displaystyle \underset{\mathit{\alpha }=1}{\overset{\mathit{n}}{\cup }}{G}_{\mathit{\alpha }}{\sim }_{\mathit{X}}\mathit{X}}$, Equation (2.1)(2.1) $h^{-1}(\mathit{V})=h^{-1}(\mathit{V})\cap \mathit{X}.$(2.1) gives

(2.2) $h^{-1}(\mathit{V})\cap {\displaystyle \underset{\mathit{\alpha }=1}{\overset{\mathit{n}}{\cup }}{G}_{\mathit{\alpha }}{\sim }_{\mathit{X}}{h}^{-1}(\mathit{V})\cap \mathit{X}.}$(2.2)

The simplification of Equation (2.2)(2.2) $h^{-1}(\mathit{V})\cap {\displaystyle \underset{\mathit{\alpha }=1}{\overset{\mathit{n}}{\cup }}{G}_{\mathit{\alpha }}{\sim }_{\mathit{X}}{h}^{-1}(\mathit{V})\cap \mathit{X}.}$(2.2) gives

(2.3) ${\displaystyle \bigcup _{\mathit{\alpha }=1}^{\mathit{n}}\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}^{-1}(\mathit{V}){\sim }_{\mathit{X}}{h}^{-1}(\mathit{V}).}$(2.3)

Therefore, for $\mathit{h}(\mathit{p})\in \mathit{V}$, we have ${\displaystyle \bigcup _{\mathit{\alpha }=1}^{\mathit{n}}\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}^{-1}(\{\mathit{h}(\mathit{p})\}){\sim }_{\mathit{X}}\mathit{p}}$. This implies that $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}^{-1}(\{\mathit{h}(\mathit{p})\}){\sim }_{\mathit{X}}\mathit{p}$ for some fixed index $\mathit{\alpha }\in \{1,\dots ,\mathit{n}\}$. Since $\mathit{p}{\delta }_{\mathit{\varphi }}\mathit{E}$, then $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}^{-1}(\{\mathit{h}(\mathit{p})\}){\delta }_{\mathit{\varphi }}\mathit{E}$. By the $\mathit{\varphi }-$continuity of $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}:G_{\mathit{\alpha }}\mapsto \mathit{Y}$, we have $\{\mathit{h}(\mathit{p})\}\text{allowbreak}{\delta }_{\varphi }^{\prime }\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}(\mathit{E})$ or just

(2.4) $\mathit{h}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}(\mathit{E}).$(2.4)

Since $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}$ is well defined on G_α, the fact that Equation (2.4)(2.4) $\mathit{h}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}(\mathit{E}).$(2.4) holds implies that either E is properly contained in G_α or E is equal to G_α. In either case, $\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}(\mathit{E})=(\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}){|}_{\mathit{E}}(\mathit{E})=\mathit{h}{|}_{\mathit{E}}(\mathit{E})=\mathit{h}(\mathit{E})$ (refer to the definition of restriction of a function), and thus Equation (2.4)(2.4) $\mathit{h}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{h}{|}_{{\mathit{G}}_{\mathit{\alpha }}}(\mathit{E}).$(2.4) gives $\mathit{h}(\mathit{p}){\delta }_{\varphi }^{\prime }\mathit{h}(\mathit{E})$.□

Theorem 2.5

(Continuous Invariants)

Let $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ be a $\mathit{\varphi }-$continuous function between ${\mathcal{T}}_{\mathit{r}}-$spaces $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$ and $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$ endowed with $\mathit{\varphi }-$proximities.

1. If X is inclusive, then f(X) is inclusive in Y.

2. If X is $\mathit{\varphi }-$connected, then the image f(X) is $\mathit{\varphi }-$connected in Y.

Proof.

1. Suppose that f(X) is not inclusive. Then, there exists at least one element of f(X) that is unrepresented in Y. Let y₀ be such a point of f(X). Then, $y_0{\underset{―}{\delta }}_{\mathit{\varphi }}^{\prime }\mathit{G}$ for each decision space G in Y. Since $y_0\in \mathit{f}(\mathit{X})$, there is an element x₀ of X such that $y_0=\mathit{f}(x_0)$. Thus, $y_0{\underset{―}{\delta }}_{\mathit{\varphi }}^{\prime }\mathit{G}$ implies $\mathit{f}(x_0){\underset{―}{\delta }}_{\mathit{\varphi }}^{\prime }\mathit{G}$. Since f is $\mathit{\varphi }-$continuous, $f^{-1}(\mathit{G})$ is a decision space in X that is administratively far from x₀ for each decision space $\mathit{G}\subset \mathit{Y}$. That is, $x_0{\underset{―}{\delta }}_{\mathit{\varphi }}{f}^{-1}(\mathit{G})$. This is a contradiction because, by hypothesis, X is inclusive, and therefore x₀ cannot be unrepresented. So, f(X) is inclusive.

2. Suppose, on the contrary, that f(X) is $\mathit{\varphi }-$separated. Let $G_1,G_2$ be a $\mathit{\varphi }-$separation of f(X) such that $G_1\cup G_2{\sim }_{\mathit{Y}}\mathit{f}(\mathit{X})$ and $G_1{\underset{―}{\delta }}_{\mathit{\varphi }}{G}_2$. By $\mathit{\varphi }-$continuity of f, we have that $f^{-1}(G_1)$ and $f^{-1}(G_2)$ are decision spaces which form a $\mathit{\varphi }-$separation of X. This is a contradiction because it is given that X is $\mathit{\varphi }-$connected.

Theorem 2.6

The ${\mathcal{T}}_{\mathit{r}}-$spaces $(\mathit{X},{\mathcal{T}}_{\mathit{r}1},{\delta }_{\mathit{\varphi }})$ and $(\mathit{Y},{\mathcal{T}}_{\mathit{r}2},{\delta }_{\varphi }^{\prime })$ are $\mathit{\varphi }-$homeomorphic if there exists a bijective function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ from X onto Y with the property that $\mathit{E}{\delta }_{\mathit{\varphi }}\mathit{F}$ in X if and only if $\mathit{f}(\mathit{E}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{F})$ in Y, for all subsets E and F of X.

Proof.

To prove this, it is only enough to show that the function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ and its inverse $f^{-1}:\mathit{Y}\mapsto \mathit{X}$ are both $\mathit{\varphi }-$continuous. Since it is given that $\mathit{E}{\delta }_{\mathit{\varphi }}\mathit{F}⟺\mathit{f}(\mathit{E}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{F})$ for all subsets $\mathit{E},\mathit{F}$ of X, then the forward implication $\mathit{E}{\delta }_{\mathit{\varphi }}\mathit{F}⟹\mathit{f}(\mathit{E}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{F})$ implies that f is $\mathit{\varphi }-$continuous (definition 2.1). Similarly, the backward implication $\mathit{f}(\mathit{E}){\delta }_{\varphi }^{\prime }\mathit{f}(\mathit{F})⟹\mathit{E}{\delta }_{\mathit{\varphi }}\mathit{F}$ implies that f⁻¹ is, by the same definition 2.1, $\mathit{\varphi }-$continuous too. Hence, the conclusion.

In general topology, we have two kinds of properties: hereditary and topological properties. A topological property or topological invariant is a property of a space such that any space homeomorphic to this space has that property (see Adams and Franzosa (2008, pp. 140–151), Croom (2008, pp. 118–119)). Hereditary property, as the name suggests, is the property of a space that is inherited by all of its subspaces. It is too obvious that the $\mathit{\varphi }-$connectedness is a topological property. We will show in the next theorem that the property of inclusiveness is also a topological property.

Theorem 2.7

The inclusive property of ${\mathcal{T}}_{\mathit{r}}-$spaces is a topological property.

Proof.

Let $(\mathit{X},{\mathcal{T}}_{r_1},{\delta }_{\mathit{\varphi }})$ be an inclusive space and $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_2},{\delta }_{\varphi }^{\prime })$ be a ${\mathcal{T}}_r-$space that is $\mathit{\varphi }-$homeomorphic to X. Let $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ be a $\mathit{\varphi }-$homeomorphism and y be a point in Y. Since f is bijective, then $f^{-1}(\mathit{y})$ is a point in X. Since X is inclusive (see definition 1.6), there is a decision space G in X such that $\mathit{G}{\sim }_{\mathit{X}}{f}^{-1}(\mathit{y})$. By $\mathit{\varphi }-$homeomorphism of f, f(G) is a decision space in Y and $\mathit{f}(\mathit{G}){\sim }_{\mathit{Y}}\mathit{y}$. Therefore, Y is inclusive, and hence inclusiveness is a topological property.

By Theorem 1.16, it can be easily shown that the inclusive property is a hereditary property. Because if X is a ${\mathcal{T}}_{\mathit{r}}-$space and E is its nonempty subspace, then for any point $\mathit{x}\in \mathit{E}$, there is a decision space $\mathit{G}\subset \mathit{X}$ such that $\mathit{G}{\sim }_{\mathit{X}}\mathit{x}$ (X is inclusive). Under the subspace topology, we have ${\displaystyle \mathit{G}\underset{\mathit{\varphi }}{\cap }\mathit{E}{\sim }_{\mathit{E}}\mathit{x}}$.

3. $\mathit{\varphi }-$dominating sets and $\mathit{\varphi }-$alliance

One of the properties of nonempty open sets in ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ is the existence of a point in each such set whose decision index is greater than all in that set. We call such a point a maximal element (see Corollary 1.13). In characterizing the ${\mathcal{T}}_{\mathit{r}}-$space in the international systems, it is important to consider some special kinds of sets of the same stance as maximal elements. These sets are dominating sets and alliances. Their importance is essential in formulating the connection between one nation and another. In this section, we will explore these kinds of sets.

Definition 3.1.

Let $\mathit{X}=(\mathit{X},{\mathcal{T}}_{\mathit{r}},{\delta }_{\mathit{\varphi }})$ be a ${\mathcal{T}}_{\mathit{r}}-$space and E a subset of X. A set E is called a $\mathit{\varphi }-$dominating set of X if for every point p outside E and every situation $\mathit{s}\in S_{\mathit{X}}$, there is a direct point q in E such that ${\rho }_{\mathit{X}}(\mathit{p},\mathit{s})\le {\rho }_{\mathit{X}}(\mathit{q},\mathit{s})$.

Definition 3.1 clearly asserts that if E is a $\mathit{\varphi }-$dominating set, then every point in X − E is $\mathit{\varphi }-$near to some points in E. This definition considers the nearness of points outside E to E but does not talk about the nearness of points within E itself. In order to eliminate the necessity of points within E being $\mathit{\varphi }-$far from each other, we need to extend our definition of $\mathit{\varphi }-$dominance so that the $\mathit{\varphi }-$nearness to E must consider all points in the entire space X.

Definition 3.2.

A subset E of a space $(\mathit{X},{\mathcal{T}}_{\mathit{r}},{\delta }_{\mathit{\varphi }})$ is called a total $\mathit{\varphi }-$dominating set of X if for every point $(\mathit{p},\mathit{s})\in \mathit{X}\times S_{\mathit{X}}$, there is a direct point q in E such that ${\rho }_{\mathit{X}}(\mathit{p},\mathit{s})\le {\rho }_{\mathit{X}}(\mathit{q},\mathit{s})$.

From these two definitions, it follows that the total $\mathit{\varphi }-$dominating set is the $\mathit{\varphi }-$dominating set. However, the converse does not necessarily hold. The two definitions lead to the following result:

Theorem 3.3

Every ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ has a total $\mathit{\varphi }-$dominating set.

Proof.

Since every ${\mathcal{T}}_{\mathit{r}}-$space has a maximal decision space (Proposition 1.12), let G be such a space. We claim that G is a total $\mathit{\varphi }-$dominating set of X. By definition of G, we have ${\rho }_{\mathit{X}}(\mathit{H})\le {\rho }_{\mathit{X}}(\mathit{G})$ for all $\mathit{H}\in {\mathcal{T}}_{\mathit{r}}$. Since every point in X is represented by a decision space, it follows that for each point $\mathit{x}\in \mathit{X}$, there is an element $\mathit{g}\in \mathit{G}$ such that ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}\{\mathit{g}\}$.

Apart from the maximal decision space being the total $\mathit{\varphi }-$dominating set of X, the entire space X itself is a total $\mathit{\varphi }-$dominating set. This can be easily verified. In space X, there can be many sets with the property of $\mathit{\varphi }-$dominance. It is always more difficult to work on larger sets than smaller ones. We seek to identify the minimal one, for which no other subset has this property. The following theorem establishes this result:

Theorem 3.4

A maximal decision space in a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$ is a minimal total $\mathit{\varphi }-$dominating set in X.

Proof.

Let G be a maximal decision space in a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$. By theorem 3.3, G is a total $\mathit{\varphi }-$dominating set. Our task that remains is to show that G is a minimal total $\mathit{\varphi }-$dominating set. If G contains only one direct element, then G is minimal because a single-point set contains no nonempty proper subsets. If G has more than one direct element, we show that it is still a minimal total $\mathit{\varphi }-$dominating set. Suppose, on the contrary, that G is not a minimal total $\mathit{\varphi }-$dominating set of X and let $G^{\ast }$ be a total $\mathit{\varphi }-$dominating set of X that is properly contained in G. Suppose further that there is a situation s in the administrative section of X that needs to be resolved. Since G is a maximal decision space in X, there must be an element x directly contained in G that resolves the situation s. If x is directly contained in G but outside $G^{\ast }$ and since x is maximal in the m-chain formed due to situation s, it follows that there is no element y in $G^{\ast }$ such that ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}\{\mathit{y}\}$. This contradicts the assertion that $G^{\ast }$ is a total $\mathit{\varphi }-$dominating set of X. Similarly, if x was a direct member of $G^{\ast }$, there would be an element p directly contained in $\mathit{G}-G^{\ast }$ such that ${\rho }_{\mathit{X}}\{\mathit{x}\}>{\rho }_{\mathit{X}}\{\mathit{p}\}$. This would violate the maximality of G (see definition 1.11) and its property of total $\mathit{\varphi }-$dominance. Therefore, it follows that in either case, $G^{\ast }$ does not exist, and the result follows immediately.□

In the international system, no nation exists in isolation. Chances are that each nation is necessarily involved in interactions with others. Interactions may come with $\mathit{\varphi }-$domination. A $\mathit{\varphi }-$domination of one nation by another may mean a $\mathit{\varphi }-$dominating nation is sitting within a $\mathit{\varphi }-$dominated nation by having too much influence in the decision-making processes. This kind of relationship brings us to the topological concept of embedding one space into another.

Definition 3.5.

An injective function $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ between ${\mathcal{T}}_{\mathit{r}}-$spaces $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_1},{\delta }_{\mathit{\varphi }})$ and $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_2},{\delta }_{\varphi }^{\prime })$ is called a $\mathit{\varphi }-$embedding of the space X into the space Y if X is $\mathit{\varphi }-$homeomorphic to its image f(X) as the subspace of Y.

Before we state the necessary and sufficient conditions for one nation to sit within and dominate another, it is important to note that the decision index of the dominating nation is always greater than or equal to that of the nation under dominance. The following theorem establishes the criteria for this dominance.

Theorem 3.6

Let $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ be a $\mathit{\varphi }-$embedding of the space $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_1},{\delta }_{\mathit{\varphi }})$ into the space $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_2},{\delta }_{\varphi }^{\prime })$. The space X totally $\mathit{\varphi }-$dominates Y if and only if f(X) totally $\mathit{\varphi }-$dominates Y.

Proof.

Suppose that X totally $\mathit{\varphi }-$dominates Y. We show that f(X) totally $\mathit{\varphi }-$dominates Y. This is equivalent to showing that for each point $\mathit{y}\in \mathit{Y}$, there is a point $\mathit{f}(\mathit{x})\in \mathit{f}(\mathit{X})$ for some unique point $\mathit{x}\in \mathit{X}$ such that ${\rho }_{\mathit{Y}}\{\mathit{f}(\mathit{x})\}\ge {\rho }_{\mathit{Y}}\{\mathit{y}\}$. Let y be an arbitrary point of Y. Since X totally $\mathit{\varphi }-$dominates Y, there is a point x directly contained in some decision space G in X such that ${\rho }_{\mathit{X}}\{\mathit{x}\}\ge {\rho }_{\mathit{Y}}\{\mathit{y}\}$. Since X is $\mathit{\varphi }-$embedded in Y, then X is topologically identical to its image f(X) in Y under $\mathit{\varphi }-$proximity (Definition 3.5). Therefore, x is $\mathit{\varphi }-$proximally equivalent to $\mathit{f}(\mathit{x})\in \mathit{G}\subset \mathit{f}(\mathit{X})$ (see definition 1.10), and so ${\rho }_{\mathit{X}}\{\mathit{x}\}\equiv {\rho }_{\mathit{Y}}\{\mathit{f}(\mathit{x})\}$. This implies that ${\rho }_{\mathit{Y}}\{\mathit{f}(\mathit{x})\}\ge {\rho }_{\mathit{Y}}\{\mathit{y}\}$. Conversely, suppose that f(X) totally $\mathit{\varphi }-$dominates Y and that X does not totally $\mathit{\varphi }-$dominate Y. Then, there is at least one point $\mathit{y}\in \mathit{Y}$ such that $\mathit{y}{\underset{―}{\delta }}_{\mathit{\varphi }}\mathit{G}$ for all $\mathit{G}\in {\mathcal{T}}_{{\mathit{r}}_1}$. Equivalently, $\mathit{y}{\underset{―}{\delta }}_{\mathit{\varphi }}\mathit{x}$ for all $\mathit{x}\in \mathit{X}$. Since X is $\mathit{\varphi }-$embedded in Y, then $\mathit{x}\in \mathit{X}$ is uniquely mapped to $\mathit{f}(\mathit{x})\in \mathit{f}(\mathit{X})$ with the property that $\{\mathit{x}\}\equiv \{\mathit{f}(\mathit{x})\}$. Because f(X) totally $\mathit{\varphi }-$dominates Y, then for all $\mathit{y}\in \mathit{Y}$, ${\rho }_{\mathit{Y}}\mathit{f}(\mathit{X})\ge {\rho }_{\mathit{Y}}\{\mathit{y}\}$. The $\mathit{\varphi }-$homeomorphism and the bijective property between X and f(X) give ${\rho }_{\mathit{X}}\mathit{X}\ge {\rho }_{\mathit{Y}}\{\mathit{y}\}$. This implies that $\mathit{y}{\delta }_{\mathit{\varphi }}\mathit{X}$, or $\mathit{y}{\delta }_{\mathit{\varphi }}\bigcup _{\mathit{x}\in \mathit{X}}\{\mathit{x}\}$. Therefore, $\mathit{y}{\delta }_{\mathit{\varphi }}\mathit{x}$ for some $\mathit{x}\in \mathit{X}$. This contradicts the assumption that $\mathit{y}{\underset{―}{\delta }}_{\mathit{\varphi }}\mathit{x}$ for each $\mathit{x}\in \mathit{X}$.□

In Cantor’s set theory setting, it is known that if $\mathit{X},\mathit{Y},\mathit{Z}$ are topological spaces, and if X is embeddable in Y, and Y is embeddable in Z, then X is embeddable in Z (Croom 2008, p. 127). We will show that this result holds for $\mathit{\varphi }-$proximal ${\mathcal{T}}_r-$spaces. This result will be needed in extending the $\mathit{\varphi }-$dominance of one space into the other.

Theorem 3.7

Let $(\mathit{X},{\mathcal{T}}_{{\mathit{r}}_1},{\delta }_{\mathit{\varphi }}),(\mathit{Y},{\mathcal{T}}_{{\mathit{r}}_2},{\delta }_{\varphi }^{\prime })$, and $(\mathit{Z},{\mathcal{T}}_{{\mathit{r}}_3},\text{allowbreak}{\delta }_{\varphi }^{″})$ be ${\mathcal{T}}_{\mathit{r}}-$spaces equipped with $\mathit{\varphi }-$proximities ${\delta }_{\mathit{\varphi }}$, ${\delta }_{\varphi }^{\prime }$, and ${\delta }_{\varphi }^{″}$, respectively. If X is $\mathit{\varphi }-$embeddable in Y, and Y is $\mathit{\varphi }-$embeddable in Z, then X is $\mathit{\varphi }-$embeddable in Z.

Proof.

Let $\mathit{f}:\mathit{X}\mapsto \mathit{Y}$ be a $\mathit{\varphi }-$embedding of X into Y and $\mathit{g}:\mathit{Y}\mapsto \mathit{Z}$ be a $\mathit{\varphi }-$embedding of Y into Z. By definition 3.5, the restrictions $f^{\prime }:\mathit{X}\mapsto \mathit{f}(\mathit{X})$ and $g^{\prime }:\mathit{Y}\mapsto \mathit{g}(\mathit{Y})$ on ranges of f and g are both $\mathit{\varphi }-$homeomorphisms. We claim that a composition function $\mathit{g}\circ \mathit{f}:\mathit{X}\mapsto \mathit{Z}$ is a $\mathit{\varphi }-$embedding of X into Z, and the restriction $\mathit{g}\circ \mathit{f}{|}_{g^{\prime }(\mathit{f}(\mathit{X}))}:\mathit{X}\mapsto g^{\prime }(\mathit{f}(\mathit{X}))$ defined by

(3.1) $\begin{array}{l}\mathit{h}=\mathit{g}\circ \mathit{f}{|}_{g^{\prime }(\mathit{f}(\mathit{X}))}=i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\end{array}$(3.1)

is a $\mathit{\varphi }-$homeomorphism. In this Equation (3.1)(3.1) $\begin{array}{l}\mathit{h}=\mathit{g}\circ \mathit{f}{|}_{g^{\prime }(\mathit{f}(\mathit{X}))}=i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\end{array}$(3.1) , the functions $i_{\mathit{g}(\mathit{f}(\mathit{X}))}$ and $i_{\mathit{f}(\mathit{X})}$ are inclusions. We show that h and h⁻¹ are both $\mathit{\varphi }-$continuous. Since $\mathit{h}=i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }$ is a composition of $\mathit{\varphi }-$continuous functions, it is $\mathit{\varphi }-$continuous. Similarly, $h^{-1}=f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}$ is also $\mathit{\varphi }-$continuous (see Theorem 2.3 and the fact that f^ʹ and g^ʹ are $\mathit{\varphi }-$homeomorphisms). We next show that h and h⁻¹ are inverses of each other.

$\begin{array}{ll}\mathit{h}\circ h^{-1}& =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\circ f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\\ & \circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ (f^{\prime }\circ f^{\prime -1})\circ i_{f(X)}^{-1}\circ g^{\prime -1}\\ & \circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ i_{\mathit{f}(\mathit{X})}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ (i_{\mathit{f}(\mathit{X})}\circ i_{f(X)}^{-1})\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ (g^{\prime }\circ g^{\prime -1})\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ i_{\mathit{g}(\mathit{Y})}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\\ & =i_{g^{\prime }(\mathit{f}(\mathit{X}))},\end{array}$

and

$\begin{array}{ll}{h}^{-1}\circ \mathit{h}& =f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\\ & \circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ (i_{g^{\prime }(\mathit{f}(\mathit{X}))}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1})\circ g^{\prime }\\ & \circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ i_{g^{\prime }(\mathit{f}(\mathit{X}))}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ g^{\prime -1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ (g^{\prime -1}\circ g^{\prime })\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ i_{\mathit{Y}}\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{f(X)}^{-1}\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\\ & =f^{\prime -1}\circ f^{\prime }\\ & =i_{\mathit{X}}.\end{array}$

Therefore, h and h⁻¹ are inverses of each other, and therefore the function $\mathit{h}=i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }:\mathit{X}\mapsto g^{\prime }\circ \mathit{f}(\mathit{X})$ is a $\mathit{\varphi }-$homeomorphism. This implies that the function $\mathit{g}\circ \mathit{f}:\mathit{X}\mapsto \mathit{Z}$ is a $\mathit{\varphi }-$embedding of X into Z.□

The result of the theorem 3.7 can be generalized to any finite number of topological spaces. Technically speaking, the homeomorphisms established between the embedded space and the subspace of the embedding space are constructed using the properties of composite functions and their bijective behaviour. Suppose that $(X_{\mathit{i}}{)}_{1\le \mathit{i}\le \mathit{n}}$ is a finite sequence of topological spaces with the property that there is an embedding $f_{\mathit{i}}:X_{\mathit{i}}\mapsto X_{\mathit{i}+1}$ of X_i into $X_{\mathit{i}+1}$ for each $\mathit{i}=1,2,\dots ,\mathit{n}-1$. Then, X₁ is embeddable in X_n. The homeomorphism between X₁ and the subspace of X_n is a restriction of a function $f_{\mathit{n}-1}\circ f_{\mathit{n}-2}\circ \cdots \circ f_2\circ f_1:X_1\mapsto X_{\mathit{n}}$ on its range and it is given by $F_{\mathit{n}-2}:X_1\mapsto f_{n-1}^{\prime }(f_{n-2}^{\prime }(f_{n-3}^{\prime }(\dots (f_3^{\prime }(f_2^{\prime }(f_1(X_1))))\dots )))$ where

(3.2) $\begin{array}{l}\begin{array}{l}{F}_{\mathit{n}-2}=i_{f_{n-1}^{\prime }{f}_{n-2}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-1}^{\prime }\circ i_{f_{n-2}^{\prime }{f}_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-2}^{\prime }\\ & \circ i_{f_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-3}^{\prime }\circ \\ i_{f_{n-4}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ \cdots \circ i_{f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_2^{\prime }\circ i_{{\mathit{f}}_1(X_1)}\circ f_1^{\prime },(\mathit{n}\ge 3),\end{array}\end{array}$(3.2)

and the function $f_i^{\prime }:X_{\mathit{i}}\mapsto f_{\mathit{i}}(X_{\mathit{i}})$ is a homeomorphism between embedded space X_i and the subspace $f_{\mathit{i}}(X_{\mathit{i}})$ of the embedding space $X_{\mathit{i}+1}$. Equation (3.2)(3.2) $\begin{array}{l}\begin{array}{l}{F}_{\mathit{n}-2}=i_{f_{n-1}^{\prime }{f}_{n-2}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-1}^{\prime }\circ i_{f_{n-2}^{\prime }{f}_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-2}^{\prime }\\ & \circ i_{f_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-3}^{\prime }\circ \\ i_{f_{n-4}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ \cdots \circ i_{f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_2^{\prime }\circ i_{{\mathit{f}}_1(X_1)}\circ f_1^{\prime },(\mathit{n}\ge 3),\end{array}\end{array}$(3.2) , for instance, yields Equation (3.1)(3.1) $\begin{array}{l}\mathit{h}=\mathit{g}\circ \mathit{f}{|}_{g^{\prime }(\mathit{f}(\mathit{X}))}=i_{g^{\prime }(\mathit{f}(\mathit{X}))}^{-1}\circ g^{\prime }\circ i_{\mathit{f}(\mathit{X})}\circ f^{\prime }\end{array}$(3.1) when n = 3, with $f_1=\mathit{f}$ and $f_2=\mathit{g}$. This result is very powerful, and with it at hand, we can picture the dominating m-chain that goes through m-chains of decision spaces in nations. Suppose that X_i totally $\mathit{\varphi }-$dominates $X_{\mathit{i}+1}$ for $\mathit{i}=1,2,\dots ,\mathit{n}-1$ and $f_{\mathit{i}}:X_{\mathit{i}}\to X_{\mathit{i}+1}$ is a $\mathit{\varphi }-$embedding of X_i into $X_{\mathit{i}+1}$, then by Theorem 3.6, $f_{\mathit{i}}(X_{\mathit{i}})$ totally $\mathit{\varphi }-$dominates $X_{\mathit{i}+1}$. This further implies that $f_{\mathit{i}}(X_{\mathit{i}})$ totally $\mathit{\varphi }-$dominates $f_{\mathit{i}+1}(X_{\mathit{i}+1})$. Since X₁ is $\mathit{\varphi }-$homeomorphic to $f_1(X_1)$, then the sequence ${\mathcal{M}}_{{\mathit{f}}_{\mathit{i}}}=(f_{\mathit{i}}(X_{\mathit{i}}):\mathit{i}=1,\dots ,\mathit{n}-1)$ forms a total $\mathit{\varphi }-$dominating m-chain that goes straight (see definition 1.29) through the sequence $(X_{\mathit{i}}:\mathit{i}=2,\dots ,\mathit{n})$ and that is induced by $\mathit{\varphi }-$embeddings f_i . There is another m-chain that goes strictly straight through $M_{{\mathit{f}}_{\mathit{i}}}$ and that is given by subspaces of $f_{\mathit{i}}(X_{\mathit{i}})$ for each X_i which are in one-to-one correspondence to the space X₁. Since $X_1\equiv f_1(X_1)\equiv f_2^{{}^{\prime }}\circ f_1(X_1)\equiv f_3^{{}^{\prime }}\circ f_2^{{}^{\prime }}\circ f_1(X_1)\equiv \cdots \equiv f_{\mathit{n}-2}^{{}^{\prime }}\circ f_{\mathit{n}-3}^{{}^{\prime }}\circ \cdots \circ f_2^{{}^{\prime }}\circ f_1(X_1)\equiv f_{\mathit{n}-1}^{{}^{\prime }}\circ f_{\mathit{n}-2}^{{}^{\prime }}\circ \cdots \circ f_2^{{}^{\prime }}\circ f_1(X_1)$ and for each $\mathit{k}\ge 1$, $f_{\mathit{k}}^{{}^{\prime }}\circ f_{\mathit{k}-1}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1)\subset f_{\mathit{k}}(X_{\mathit{k}})\subset X_{\mathit{k}+1}$, then the chain ${\mathcal{M}}^{\ast }=(f_1(X_1),f_2^{{}^{\prime }}\circ f_1(X_1),f_3^{{}^{\prime }}\circ f_2^{{}^{\prime }}\circ f_1(X_1),\dots ,f_{\mathit{n}-2}^{{}^{\prime }}\circ f_{\mathit{n}-3}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1),f_{\mathit{n}-1}^{{}^{\prime }}\circ f_{\mathit{n}-2}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1))$ is an m-chain that goes strictly straight through ${\mathcal{M}}_{{\mathit{f}}_{\mathit{i}}}$ if and only if ${\displaystyle f_{\mathit{k}}^{{}^{\prime }}\circ f_{\mathit{k}-1}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1)\bigcap _{\mathit{\varphi }}{f}_{\mathit{k}+1}^{{}^{\prime }}\circ f_{\mathit{k}+2}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1)\ne \mathrm{\varnothing }}$ for each k. Because X₁ is $\mathit{\varphi }-$homeomorphic to $f_1(X_1)$, and $f_1(X_1)$ is $\mathit{\varphi }-$homeomorphic to $f_2^{{}^{\prime }}\circ f_1(X_1)$ and so on, then by Theorem 1.30, all elements in ${\mathcal{M}}^{\ast }$ are pairwise $\mathit{\varphi }-$homeomorphic, and, therefore, at any finite step $\mathit{k}\ge 3$, there is always a $\mathit{\varphi }-$homeomorphism $F_{\mathit{k}-2}:X_1\to f_{\mathit{k}-1}^{{}^{\prime }}\circ f_{\mathit{k}-2}^{{}^{\prime }}\circ \dots f_2^{{}^{\prime }}\circ f_1(X_1)$ (see Equation (3.2)(3.2) $\begin{array}{l}\begin{array}{l}{F}_{\mathit{n}-2}=i_{f_{n-1}^{\prime }{f}_{n-2}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-1}^{\prime }\circ i_{f_{n-2}^{\prime }{f}_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-2}^{\prime }\\ & \circ i_{f_{n-3}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_{n-3}^{\prime }\circ \\ i_{f_{n-4}^{\prime }\dots f_2^{\prime }{f}_1(X_1)}^{-1}\circ \cdots \circ i_{f_2^{\prime }{f}_1(X_1)}^{-1}\circ f_2^{\prime }\circ i_{{\mathit{f}}_1(X_1)}\circ f_1^{\prime },(\mathit{n}\ge 3),\end{array}\end{array}$(3.2) ) that completes the diagram $f_1^{{}^{\prime }}:X_1\to f_1(X_1),i_{{\mathit{f}}_2^{{}^{\prime }}{f}_1(X_1)}^{-1}\circ f_2^{{}^{\prime }}\circ i_{{\mathit{f}}_1(X_1)}:f_1(X_1)\to f_2^{{}^{\prime }}\circ f_1(X_1),i_{{\mathit{f}}_3^{{}^{\prime }}{f}_2^{{}^{\prime }}{f}_1(X_1)}^{-1}\circ f_3^{{}^{\prime }}\circ i_{{\mathit{f}}_2^{{}^{\prime }}{f}_1(X_1)}:f_2^{{}^{\prime }}\circ f_1(X_1)\to f_3^{{}^{\prime }}\circ f_2^{{}^{\prime }}\circ f_1(X_1),\dots ,i_{{\mathit{f}}_{\mathit{k}-1}^{{}^{\prime }}{f}_{\mathit{k}-2}^{{}^{\prime }}\dots f_2^{{}^{\prime }}{f}_1(X_1)}^{-1}\circ f_{\mathit{k}-1}^{{}^{\prime }}\circ i_{{\mathit{f}}_{\mathit{k}-2}^{{}^{\prime }}{f}_{\mathit{k}-3}^{{}^{\prime }}\dots f_2^{{}^{\prime }}{f}_1(X_1)}:f_{\mathit{k}-2}^{{}^{\prime }}\circ f_{\mathit{k}-3}^{{}^{\prime }}\circ \cdots \circ f_2^{{}^{\prime }}\circ f_1(X_1)\to f_{\mathit{k}-1}^{{}^{\prime }}\circ f_{\mathit{k}-2}^{{}^{\prime }}\circ \cdots \circ f_2^{{}^{\prime }}\circ f_1(X_1)$.

It may happen that nations join forces for a common cause or situation. This may lead to the formation of an alliance. An alliance is generally thought of as a treaty or formal agreement between two or more parties made in order to unite for a common situation (see Haynes and Hedetniemi (2001, p. 52), Snyder (1990, p. 104)). This can be simply put as the union of allying spaces. This union will be assumed to be disjoint: a condition that eliminates the necessity of one point being directly contained in more than one ${\mathcal{T}}_{\mathit{r}}-$spaces. To study an alliance topologically on the basis of $\mathit{\varphi }-$proximity, we need to develop the topology of such a disjoint union. Now, let $\mathit{X}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ be the union of an indexed family of the set $\{(X_{\mathit{i}},{\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}})\}$ of ${\mathcal{T}}_{\mathit{r}}-$spaces. Since each X_i is a subset of X, there is an inclusion function

$\begin{array}{l}{f}_{\mathit{i}}:X_{\mathit{i}}\mapsto \bigcup _{\mathit{j}=1}^{\mathit{n}}{X}_{\mathit{j}}\end{array}$

defined by $f_{\mathit{i}}(\mathit{x})=(\mathit{x},\mathit{i})$. The coordinate i in (x, i) identifies with the function f_i. Our task here is to build an open set (or a decision space) in X from X_i. By the $\mathit{\varphi }-$continuity of f_i (Theorem 2.3), a subset G of X is open in X if and only if $f_i^{-1}(\mathit{G})$ is open in X_i for each index i. Since $f_i^{-1}(\mathit{G})\subset \mathit{X}$, then $f_i^{-1}(\mathit{G})={\displaystyle f_i^{-1}(\mathit{G})\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}={\displaystyle \mathit{G}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}}}$ for each i. This implies that a subset G of X is open in X if and only if the administrative intersection of G with each X_i is an element of ${\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}$. We will now show, in the proposition 3.8, that the collection of sets of this kind in X constitutes a representative topology on X.

Proposition 3.8.

Let $\{(X_{\mathit{i}},{\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}})|\mathit{i}=1,2,\dots ,\mathit{n}\}$ be an indexed family of distinct ${\mathcal{T}}_{\mathit{r}}-$spaces and $\mathit{X}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$. Define the set $\mathcal{T}$ by $\mathcal{T}=\{\mathit{G}\subseteq \mathit{X}|{\displaystyle \mathit{G}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}\text{for all }\mathit{i}\}}$. Then, the collection $\mathcal{T}$ is a ${\mathcal{T}}_{\mathit{r}}-$topology on X.

Proof.

It is obvious that $\mathrm{\varnothing }$ and the whole set X are in $\mathcal{T}$, because if $\mathit{G}=\mathrm{\varnothing }$ or G = X, then for each i, ${\displaystyle \mathrm{\varnothing }\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}=\mathrm{\varnothing }\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}$ or ${\displaystyle \mathit{X}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}=X_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}$, respectively (${\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}$ is a ${\mathcal{T}}_{\mathit{r}}-$topology on X_i). Likewise, if $G_1,G_2,G_3,\dots$ are elements of $\mathcal{T}$, then ${\displaystyle G_{\mathit{j}}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}$ for each i. Since ${\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}$ is a topology, then $\bigcup _{\mathit{j}\in \mathit{J}}{\displaystyle G_{\mathit{j}}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}={\displaystyle \left(\bigcup _{\mathit{j}\in \mathit{J}}{G}_{\mathit{j}}\right)\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}}$. By definition, $\bigcup _{\mathit{j}\in \mathit{J}}{G}_{\mathit{j}}\in \mathcal{T}$. To show that $\mathcal{T}$ is closed under finite intersection, we suppose that $G_1,\dots ,G_{\mathit{m}}\in \mathcal{T}$, then by definition, ${\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}$ for every index i, $1\le \mathit{k}\le \mathit{m}$. This implies that ${\displaystyle \bigcap _{\mathit{\varphi },\mathit{k}=1}^{\mathit{m}}(G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}})={\displaystyle \left(\bigcap _{\mathit{\varphi },\mathit{k}=1}^{\mathit{m}}{G}_{\mathit{k}}\right)\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\in {\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}}}$. It follows by definition that ${\displaystyle \bigcap _{\mathit{\varphi },\mathit{k}=1}^{\mathit{m}}{G}_{\mathit{k}}\in \mathcal{T}}$.□

We are now in a position to define what we have been postponing: the $\mathit{\varphi }-$alliance.

Definition 3.9.

Let $\mathit{Y}=\{(X_{\mathit{i}},{\mathcal{T}}_{{\mathit{r}}_{\mathit{i}}}):1\le \mathit{i}\le \mathit{n}\}$ be a collection of ${\mathcal{T}}_{\mathit{r}}-$spaces and $X^{\ast }=\{(X_{\mathit{j}},{\mathcal{T}}_{{\mathit{r}}_{\mathit{j}}}):1\le \mathit{j}\le \mathit{m}\}$ be a subcollection of Y, where $2\le \mathit{m}\le \mathit{n}$. Suppose that S_i is a set of priority situations for each space X_i. Then, a $\mathit{\varphi }-$alliance of spaces $(X_{\mathit{j}},{\mathcal{T}}_{{\mathit{r}}_{\mathit{j}}})$ is a union $\mathit{X}=\bigcup _{\mathit{j}=1}^{\mathit{m}}{X}_{\mathit{j}}$ of ${\mathcal{T}}_{\mathit{r}}-$spaces such that there exists an element $\mathit{s}\in \bigcap _{\mathit{j}=1}^{\mathit{m}}{S}_{\mathit{j}}$ and a decision space $(\mathit{G},{\rho }_{\mathit{X}}\mathit{G},\{\mathit{s}\})\subset \mathit{X}$ with the property that $\mathit{G}{\delta }_{\mathit{\varphi }}{X}_{\mathit{j}}$ for each index j.

Definition 3.9 deduces that every space in a $\mathit{\varphi }-$alliance is $\mathit{\varphi }-$near to other members of the $\mathit{\varphi }-$alliance. This $\mathit{\varphi }-$proximity between spaces forming an alliance suggests the following result: the result which states the $\mathit{\varphi }-$connectedness of the $\mathit{\varphi }-$alliance.

Theorem 3.10

A $\mathit{\varphi }-$alliance is $\mathit{\varphi }-$connected with respect to a common priority situation.

Before we provide the proof of Theorem 3.10, it is important to note that the $\mathit{\varphi }-$connectedness of a $\mathit{\varphi }-$alliance is built on the essence of a set of common priority situation(s). These situations may be many in number; however, only one or a few situations among these many can cause the formation of a $\mathit{\varphi }-$alliance.

Proof.

Proof of Theorem 3.10

Let $\mathit{X}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ be a $\mathit{\varphi }-$alliance of ${\mathcal{T}}_{\mathit{r}}-$spaces $X_{\mathit{i}},\mathit{i}=1,\dots ,\mathit{n}$, and let $\mathit{s}\in \bigcap _{\mathit{i}=1}^{\mathit{n}}{S}_{\mathit{i}}$ be a common situation upon which the $\mathit{\varphi }-$alliance is made. Suppose that X is $\mathit{\varphi }-$separated with respect to s. Then, there exist two disjoint nonempty decision spaces $G_1=(G_1,{\rho }_{\mathit{X}}{G}_1,\{\mathit{s}\})$ and $G_2=(G_2,{\rho }_{\mathit{X}}{G}_2,\{\mathit{s}\})$ such that $G_1\cup G_2{\sim }_{\mathit{X}}\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ and $G_1{\underset{―}{\delta }}_{\mathit{\varphi }}{G}_2$ (refer Definition 1.22). Since X is a $\mathit{\varphi }-$alliance, there is a decision space $\mathit{G}\subseteq \mathit{X}$ for s such that $\mathit{G}{\delta }_{\mathit{\varphi }}{X}_{\mathit{i}}$ for each index i. The $\mathit{\varphi }-$connectedness of $\mathit{G},G_1$, and G₂ (see Theorem 1.24 and Corollary 1.25) implies that G either lies entirely in G₁ or in G₂ but not in both. If G entirely lies in G₁, then $\mathit{G}{\underset{―}{\delta }}_{\mathit{\varphi }}{G}_2$. Since G₂ is nonempty, there are some spaces $X_{\mathit{j}}\subset \bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ for some index $\mathit{j}\in \{1,2,\dots ,\mathit{n}\}$ such that $G_2{\sim }_{\mathit{X}}{X}_{\mathit{j}}$. Because G₂ is $\mathit{\varphi }-$far from G, then all X_j’s represented by G₂ will be $\mathit{\varphi }-$far from G. This contradicts the fact that G is $\mathit{\varphi }-$near to all X_i’s.

The definition of $\mathit{\varphi }-$alliance requires a common priority situation that brings together distinct ${\mathcal{T}}_{\mathit{r}}-$spaces. It is obvious that there might be some other common situations, but not all can cause nations to ally. The $\mathit{\varphi }-$alliance formed in a single common situation is weakly $\mathit{\varphi }-$connected. To have strong or complete $\mathit{\varphi }-$connectedness, the $\mathit{\varphi }-$alliance must be $\mathit{\varphi }-$connected for each priority situation common to all such spaces. Hence, the following theorem states the condition for an alliance to be strongly $\mathit{\varphi }-$connected.

Theorem 3.11

The $\mathit{\varphi }-$alliance is $\mathit{\varphi }-$connected if and only if it is $\mathit{\varphi }-$connected with respect to each common priority situation.

Proof.

We know by Theorem 3.10 that any $\mathit{\varphi }-$alliance is $\mathit{\varphi }-$connected with respect to a common priority situation. If the $\mathit{\varphi }-$alliance $\mathit{X}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ is $\mathit{\varphi }-$connected, then X has no $\mathit{\varphi }-$separation, and so for any common situation $\mathit{s}\in \bigcap _{\mathit{i}=1}^{\mathit{n}}{S}_{\mathit{i}}$, X is $\mathit{\varphi }-$connected. Conversely, suppose that X is $\mathit{\varphi }-$connected with respect to every element $s_{\mathit{m}}\in \bigcap _{\mathit{i}=1}^{\mathit{n}}{S}_{\mathit{i}}$. Then, for each s_m, there is a decision space $G_{\mathit{m}}\subset \mathit{X}$ such that ${\displaystyle G_{\mathit{m}}\underset{\mathit{\varphi }}{\cap }{X}_{\mathit{i}}\ne \mathrm{\varnothing }}$ for each i. This further implies that ${\displaystyle G_{\mathit{m}}\underset{\mathit{\varphi }}{\cap }({\cup }_{i=1}^n{X}_{\mathit{i}})\ne \mathrm{\varnothing }}$. Since each X_i is $\mathit{\varphi }-$connected (Corollary 1.27), it follows by Theorem 1.26 that $\mathit{X}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$ is $\mathit{\varphi }-$connected.

With these results, we give one living example of a $\mathit{\varphi }-$alliance and show that such an alliance is $\mathit{\varphi }-$connected.

Example 4.

Let us consider the political and military alliance, the North Atlantic Treaty Organization (NATO), the alliance formed primarily to keep and maintain peace and security in the North Atlantic area. The members of the alliance are resolved to unite their efforts for collective defence and the preservation of peace and security. Therefore, the members have constructed a number of principles to run the alliance. Among those principles, we only consider two principles, Articles 5 and 9 of the North Atlantic Treaty (see (NATO, 1949)). For the purpose of self-containment in this work, we briefly state them below:

Article 5

The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all and consequently they agree that, if such an armed attack occurs, each of them, in exercise of the right of individual or collective self-defence recognised by Article 51 of the Charter of the United Nations, will assist the Party or Parties so attacked by taking forthwith, individually and in concert with the other Parties, such action as it deems necessary, including the use of armed force, to restore and maintain the security of the North Atlantic area.

Article 9

The Parties hereby establish a Council, on which each of them shall be represented, to consider matters concerning the implementation of this Treaty. The Council shall be so organised as to be able to meet promptly at any time. The Council shall set up such subsidiary bodies as may be necessary; in particular it shall establish immediately a defence committee which shall recommend measures for the implementation of Articles 3 and 5.

For Article 3 and other articles not stated here, refer to (NATO, 1949). We claim that this alliance is $\mathit{\varphi }-$alliance and we show that it is $\mathit{\varphi }-$connected with respect to a priority situation (collective defence) upon which the alliance is formed. Before this, we first need to understand the embodiment of Articles 5 and 9. According to definition 1.2, Article 5 has two decision authority mappings:

1. If one or more parties of the alliance are armed attacked, then this attack is considered an armed attack against all.

   Let $\mathit{s}=$ one or more parties are armed attacked, and $\mathit{a}=$ it is an attack against all. By definition 1.2,

   (3.3) $\begin{array}{l}\mathit{a}={\text{P}}_{\mathit{C}}(\mathit{s}).\end{array}$(3.3)

2. If an armed attack occurs, then each member of the alliance will assist the party or parties attacked to restore and maintain the security of the North Atlantic area. Let $\mathit{b}=$ each member of the alliance will assist the party or parties attacked to restore and maintain the security of the North Atlantic area, then the decision authority is

   (3.4) $\begin{array}{l}\mathit{b}={\text{P}}_C^{\prime }(\mathit{a}).\end{array}$(3.4)

   Combining Equations (3.3)(3.3) $\begin{array}{l}\mathit{a}={\text{P}}_{\mathit{C}}(\mathit{s}).\end{array}$(3.3) and (3.4(3.4) $\begin{array}{l}\mathit{b}={\text{P}}_C^{\prime }(\mathit{a}).\end{array}$(3.4) ) gives

   (3.5) $\begin{array}{l}\mathit{b}={\text{P}}_C^{\prime }({\text{P}}_{\mathit{C}}(\mathit{s})).\end{array}$(3.5)

   Therefore, by definition 1.2, Equation (3.5)(3.5) $\begin{array}{l}\mathit{b}={\text{P}}_C^{\prime }({\text{P}}_{\mathit{C}}(\mathit{s})).\end{array}$(3.5) mathematically summarizes Article 5 as follows:

   (3.6) $\begin{array}{l}{\text{P}}_{\mathit{C}}=\{(\mathit{s},\mathit{a}),(\mathit{a},\mathit{b})\}.\end{array}$(3.6)

Following the decision authority stipulated in Equation (3.6)(3.6) $\begin{array}{l}{\text{P}}_{\mathit{C}}=\{(\mathit{s},\mathit{a}),(\mathit{a},\mathit{b})\}.\end{array}$(3.6) , Article 9 establishes a decision space called a Council (also known as the North Atlantic Council), on which every member of the alliance is represented, and that which oversees the political and military process relating to security issues affecting the whole alliance by exercising the decision authority identified in Article 5. Let this council be denoted by $\mathit{C}=(\mathit{C},{\text{P}}_{\mathit{C}}(\mathit{C}),\{\mathit{s}\})$, and the alliance be denoted by N and defined as $\mathit{N}=\bigcup _{\mathit{i}=1}^{\mathit{n}}{X}_{\mathit{i}}$, where X_i is a member country (in our case, a ${\mathcal{T}}_{\mathit{r}}-$space) of N. Then, we can now show that this alliance is a $\mathit{\varphi }-$alliance and is $\mathit{\varphi }-$connected with respect to s.

• On N is a $\mathit{\varphi }-$alliance: This clearly follows consequently from Articles 5 and 6, and the definition 3.9. The $\mathit{\varphi }-$alliance forming situation is s, and the council C is a decision space that is proximally $\mathit{\varphi }-$near to every member X_i of N.

• On $\mathit{\varphi }-$connectedness of N: This result follows from Theorems 1.26 and 3.10. Since each X_i is a ${\mathcal{T}}_{\mathit{r}}-$space, it is $\mathit{\varphi }-$connected (Corollary 1.27). Therefore, $\mathit{N}={\cup }_{i=1}^n{X}_{\mathit{i}}$ is $\mathit{\varphi }-$connected with respect to s because there is a nonempty decision space $\mathit{C}\subset \mathit{N}$ such that $X_{\mathit{i}}{\delta }_{\mathit{\varphi }}\mathit{C}$ for each i.

Mayila et al. (2023, p. 9) established the $\mathit{\varphi }-$connectedness of a ${\mathcal{T}}_{\mathit{r}}-$space $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$, and we have just shown that the $\mathit{\varphi }-$alliance is also $\mathit{\varphi }-$connected. Now, let us recall the concept of a cut point or a cut set in definition 1.31. A connected space can be made disconnected if some points are removed from it. Since a ${\mathcal{T}}_r-$space is $\mathit{\varphi }-$connected, we need to identify the points or sets whose removal or deletion from X produces a $\mathit{\varphi }-$separated subspace.

Theorem 3.12

Let $(\mathit{X},{\mathcal{T}}_r,{\delta }_{\mathit{\varphi }})$ be a ${\mathcal{T}}_{\mathit{r}}-$space endowed with $\mathit{\varphi }-$proximity ${\delta }_{\mathit{\varphi }}$ and let E be a nonempty subset of X. Moreover, suppose that $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}{\sim }_{\mathit{X}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}{\sim }_{\mathit{X}}\mathit{X}-\mathit{E}-\mathit{V}$, where $\mathit{U}=\mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ and $\mathit{V}=(\mathit{X}-\mathit{E})\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. Then, a $\mathit{\varphi }-$boundary $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ is a cut set of X.

Proof.

We need to show that $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ $\mathit{\varphi }-$separates X. It suffices by definition 1.22 to show that there exist two disjoint nonempty decision spaces G₁ and G₂ in X such that $G_1\cup G_2{\sim }_{\mathit{X}}\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ and $G_1{\underset{―}{\delta }}_{\mathit{\varphi }}{G}_2$. We claim that such decision spaces are $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}$. Indeed, these sets are decision spaces in X. This follows from Lemma 1.7, and they are all nonempty because $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}{\sim }_{\mathit{X}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}{\sim }_{\mathit{X}}\mathit{X}-\mathit{E}-\mathit{V}$. To show that the sets $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}$ form a $\mathit{\varphi }-$separation of $\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, it is a basic condition, as Lemma 1.23 postulates, to show that neither of the two contains a $\mathit{\varphi }-$contact point (see definition 1.15) of the other. Specifically, we will show that

1. ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathrm{\varnothing }}$

2. ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathrm{\varnothing }}$

3. ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\underset{\mathit{\varphi }}{\cap }(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})=\mathrm{\varnothing }}$ and

4. $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\cup (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}){\sim }_{\mathit{X}}\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$

Now

1. Suppose, in contrast, that ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\ne \mathrm{\varnothing }}$, and let $\mathit{x}\in {\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}}$. Then, by definition 1.14, $\mathit{x}\in (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\cup \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ with a property that ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})$ and ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}(\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E})$. If $\mathit{x}\in \mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$ and U_x is any direct neighborhood of x, then ${\displaystyle U_{\mathit{x}}\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\ne \mathrm{\varnothing }}$. This means that $\mathit{x}\in \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. This is impossible because $\mathit{x}\in \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ would mean $\mathit{x}\in \mathit{E}\cap \mathit{Fr}{t}_{X_{\varphi }}\mathit{E}=\mathit{U}$: the elements which are removed from $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. So, $\mathit{x}\notin \mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$. If $\mathit{x}\in \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, we have that either $\mathit{x}\in \mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ or $\mathit{x}\in (\mathit{X}-\mathit{E})\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. Suppose $\mathit{x}\in \mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, then $\mathit{x}{\underset{―}{\delta }}_{\mathit{\varphi }}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E})$, contradicting the fact that ${\rho }_{\mathit{X}}\{\mathit{x}\}\le {\rho }_{\mathit{X}}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})$. Therefore, x cannot be in $\mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. Similarly, if $\mathit{x}\in (\mathit{X}-\mathit{E})\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathit{V}$, x would be $\mathit{\varphi }-$far from $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}$. This also contradicts the fact that $\mathit{x}{\delta }_{\mathit{\varphi }}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})$. Therefore, $\mathit{x}\notin (\mathit{X}-\mathit{E})\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$. Since in either case there is no point in X that is $\mathit{\varphi }-$near to both $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$ and $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, it follows that $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}){\underset{―}{\delta }}_{\mathit{\varphi }}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})$.

2. On ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathrm{\varnothing }}$: This result easily follows from (i) above because $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})$. By definition,

   (3.7) $\begin{array}{l}\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\cap \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E}).\end{array}$(3.7)

   Replacing E by X − E, Equation (3.7)(3.7) $\begin{array}{l}\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}=\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\cap \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E}).\end{array}$(3.7) gives $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})=\mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})\cap \mathit{c}{l}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{E})$, and the result follows immediately. Therefore,

   (3.8) $\begin{array}{ll}{\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}}& (\mathit{X}-\mathit{E})-\mathit{V})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}\\ & ={\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})\underset{\mathit{\varphi }}{\cap }\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})=\mathrm{\varnothing }.}\end{array}$(3.8)

3. On ${\displaystyle (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\underset{\mathit{\varphi }}{\cap }(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})=\mathrm{\varnothing }}$: This is very obvious. There is no point in X that is $\mathit{\varphi }-$near to both sets $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}$, because if they were, such points would be contained either in U or V: the set of deleted points.

4. On $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\cup (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}){\sim }_{\mathit{X}}\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$: Since it is given that $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}){\sim }_{\mathit{X}}\mathit{E}-\mathit{U}$ and $\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}){\sim }_{\mathit{X}}\mathit{X}-\mathit{E}-\mathit{V}$, then

   $\begin{array}{ll}& (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\cup (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}){\sim }_{\mathit{X}}(\mathit{E}-\mathit{U})\\ & \cup (\mathit{X}-\mathit{E}-\mathit{V})\\ & =(\mathit{E}-\mathit{E}\cap \mathit{Fr}{t}_{{\mathit{X}}_{{}_{\mathit{\varphi }}}}\mathit{E})\cup (\mathit{X}-\mathit{E}-(\mathit{X}-\mathit{E})\cap \mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E})\\ & =[\mathit{E}\cap (E^{\mathit{c}}\cup (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}})]\cup [(\mathit{X}-\mathit{E})\cap ((\mathit{X}-\mathit{E}{)}^{\mathit{c}}\\ & \cup (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}})]\\ & =[\mathit{E}\cap (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}})]\cup [(\mathit{X}-\mathit{E})\cap (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}})]\\ & =[\mathit{E}\cup (\mathit{X}-\mathit{E})]\cap (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}}\\ & =\mathit{X}\cap (\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}{)}^{\mathit{c}}\\ & =\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}.\end{array}$

Since $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})\cup (\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V}){\sim }_{\mathit{X}}\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ and $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U}){\underset{―}{\delta }}_{\mathit{\varphi }}(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})$, then the spaces $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}-\mathit{U})$ and $(\mathit{in}{t}_{{\mathit{X}}_{\mathit{\varphi }}}(\mathit{X}-\mathit{E})-\mathit{V})$ form a $\mathit{\varphi }-$separation of $\mathit{X}-\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$, and thus $\mathit{Fr}{t}_{{\mathit{X}}_{\mathit{\varphi }}}\mathit{E}$ is a cut set of X.

The result of Theorem 3.12 applies to a number of cases. In an m-chain $\mathcal{M}=\{G_1\dots ,G_{\mathit{n}}\}$ (see definition 1.28), it can be easily shown that a set ${\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{G}_{\mathit{k}+1}}$ is a cut set of $\mathcal{M}=\{G_1,G_2,\dots ,G_{\mathit{k}},G_{\mathit{k}+1},\dots ,G_{\mathit{n}-1},G_{\mathit{n}}\}$ for all k. This is because every point of ${\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{G}_{\mathit{k}+1}}$ is proximally $\mathit{\varphi }-$near to both G_k and $G_{\mathit{k}+1}$. The omission of such points results in holes in the links G_k and $G_{\mathit{k}+1}$. This $\mathit{\varphi }-$separates the chain and produces two $\mathit{\varphi }-$separated components: $C_1=\{G_1,G_2,\dots ,G_{\mathit{k}-1},G_{\mathit{k}}-{\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{G}_{\mathit{k}+1}\}}$ and $C_2=\{G_{\mathit{k}+1}-{\displaystyle G_{\mathit{k}}\underset{\mathit{\varphi }}{\cap }{G}_{\mathit{k}+1},G_{\mathit{k}+2},\dots ,G_{\mathit{n}-1},G_{\mathit{n}}\}}$. In the case of $\mathit{\varphi }-$alliance, the omission of a decision space in the alliance that is $\mathit{\varphi }-$near to all member spaces results in the $\mathit{\varphi }-$disconnectedness of such a $\mathit{\varphi }-$alliance.

4. Conclusion

In this paper, we have used $\mathit{\varphi }-$proximity to define continuous functions on ${\mathcal{T}}_{\mathit{r}}-$spaces. The $\mathit{\varphi }-$continuity so defined preserves the $\mathit{\varphi }-$nearness between individual points or between a point and a set. We then presented various results concerning $\mathit{\varphi }-$continuous functions on ${\mathcal{T}}_{\mathit{r}}-$spaces. This includes the fact that $\mathit{\varphi }-$continuous functions preserve $\mathit{\varphi }-$connectedness and the inclusive property of ${\mathcal{T}}_{\mathit{r}}-$spaces. We then studied the special sets in an international system: $\mathit{\varphi }-$dominating sets and $\mathit{\varphi }-$alliance. Using a concept of $\mathit{\varphi }-$proximity and $\mathit{\varphi }-$continuity on ${\mathcal{T}}_{\mathit{r}}-$spaces, we showed that a space X totally $\mathit{\varphi }-$dominates space Y if and only if there exists a $\mathit{\varphi }-$embedding $\mathit{f}:\mathit{X}\to \mathit{Y}$ such that the image f(X) of X under f totally $\mathit{\varphi }-$dominates Y. This result is very important as it states when and how one nation sits within the other and exercises its influence, decision authority, and dominance. We concluded our study by defining what we called a $\mathit{\varphi }-$alliance on ${\mathcal{T}}_{\mathit{r}}-$spaces, developing a ${\mathcal{T}}_{\mathit{r}}-$topology on it, establishing its $\mathit{\varphi }-$connectedness, and finally identifying the cut sets (or cut points) in a $\mathit{\varphi }-$connected space X, whose removal $\mathit{\varphi }-$separates the remaining subspaces.

Acknowledgements

We acknowledge the contribution from every single individual, including our fellow colleagues from our respective departments, who in one way or another contributed to the development and improvement of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research did not receive a grant from any funding agency.