On a nation as a topological space

ABSTRACT

This paper introduces point-set topology into international interactions. Nations are sets of people who interact if there is a well-defined function between them. To do all these, we need to have the structure that describes how such nations interact. This calls for a topology. The kind of topology we construct in this perspective is made up by decision spaces. We first begin by developing a mathematical representation of a decision space, and use such spaces to develop a topology on a nation. Subsequently, we revisit some properties of the interior, closure, limit, and boundary points with respect to this topology and the new concept of $\varphi -$proximity. Finally, we define and develop $\varphi -$connectedness of subspaces of a nation.

KEYWORDS:

• point-set topology
• nation
• decision space
• $\varphi -$proximity

1. Intoduction

General topologists for a long time faced many questions about the importance of abstract topological spaces (Alharthi, 2016), (Kachapova, 2014), (Phillips, 2013). For example, what can a topologist do apart from teaching? Questions of this kind have been difficult for instructors to answer. The answers are almost always about the application of general topology in other branches of mathematics, mainly in analysis.

Our daily life in communities (nation) is influenced by the decisions made by the people whom we have given them the authority to decide various matters on our behalf in our own societies. Rare cases may happen when each member of a community may directly vote for a decision; probably when the size of the nation is small enough to be accommodated. If the size gets large, representation in decision making is necessary. The decisions made by these representatives directly affect the community being represented. Have we ever thought about mathematical representations of the interactions between decision makers? How does one by influence someone to make a decision? Can sociologists clearly explain the happening of complex espionage, frauds, or rebellion in the chain of command? Of course, at the eve of negotiation, mathematicians have devised some methods in game theory. However, such theories do not represent the consistency of decisions (see Brams, 2000) and (Syll, 2018), continuity, or one-to-one correspondence between interacting societies.

These concerns are best explained by topology. Our purpose in this paper is not about developing a topology concerning the decision making process but rather to investigate something else and, this is about turning a nation into a topological space; the point of description being a decision making body (or to what we will describe from now on as a decision space). A nation in this work will mean a set of people united by common descent, culture, or languages who inhabit a particular state or territory with a defined system of decision spaces. By definition, a decision space refers to a person, commission, or council that has the authority by law or ordinance to make a final decision in approving or disapproving the matter within its administrative area.

In this paper, we develop the mathematical representation of a decision space and its properties, develop a topology on a nation, explore some properties of topological operators (interior, closure, and boundary) and finally investigate the connectedness of subspaces in a nation with respect to this topology.

1.1. Preliminaries

We will not take long time in writing down and referencing basic definitive facts which are so well known in topology. However, we would like to point out that the axioms of topology, axioms for a basis for topology and the concept of subspaces will be used in developing the topology and defining the subspace topology, respectively. For those who are new to general topology and the aforementioned basic topological terms, we recommend to go after Munkress (Munkres, 2000 page 75-89), Croom (Croom, 2008 page 99-122), Adams and Franzosa (Adams & Franzosa, 2008), and Kelley (Kelley, 2017) for a better understanding of the same.

However, the following definition will be used in characterizing decision spaces.

Definition 1.1.

(i) 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.

(ii) Decision Authority is the right, power, or obligation to make a decision and the duty to answer for its success or failure.

(iii) Decision Index is the strength of the decision authority.

2. Mathematical representation of a decision space

A decision space was first introduced and used by Bossert (Bossert, 1998). Bossert described it as having three elements viz are; who decides?, over what? and, with how much choice? To make a decision, there must be a situation that arises from the environment that needs the attention of an actor (person) who acts according to the decision index one has. This implies that every authoritative decision depends on the state of the person(s), his decision authority, and the situation. The decision authority or index establishes demarcation between decision spaces and shows which space answers to who or what situation to approve or otherwise. We buy the components suggested by Bossert and make a little modification. The place of with how much choice? will be replaced by with what authority?. The reason behind this reformulation is due to the fact that the term with how much choice? refers to a set of decisions one needs to choose over others. We claim that this set is dependent. An authoritative decision depends on the state of a person, the matter to which a decision is to be made, and the decision authority that allows that person to act on the matter within the requirements of that decision authority. Therefore, the phrase with how much choice? is a set of possible decisions, where each of its elements is an image of the state of a person, decision authority, and the particular situation.

Kurt (Kurt, 1936) formulated a beautiful function in behavioral sciences that is used today in determining the behavior of an individual in various environments and situations. Let $P$ and $E$ represent the state of a person and that of environment, respectively, in which a person is. If $B$ stands for behavior or any kind of psychological mental event, then according to Kurt (Kurt, 1936), $B$ may be treated as a function of $P$ and $E$: $B=f(PE)$. We make a little change to this formula. A behavior will be replaced by an authoritative decision as a dependent co-domain. The domain will include the product of three sets: the state of a person, the decision authority, and the set of possible situations. These three elements constitute what we call a new version of a decision space, or simply a decision space. We are now in a position to present what we propose to be the mathematical representation of the decision space.

Let $X$ be a nation and

C_X = Set of decision powers or rights in $X$,

U = Set of decision makers in $X$ with their respective states,

P_X(U) = Set of decision authorities of $U$ in $X$,

S_U = Set of possible situations that can arise in the administrative area of $U$, and

D = The set of possible decisions $U$ can take.

We define the decision authority on $U$ in $X$ as the vector valued-function ${\mathrm{P}}_X:U\mapsto {\mathbb{R}}^n$ which gives the decision rights of every individual $u\in U$. The magnitude or strength of ${\mathrm{P}}_X(u)$ will be denoted by ${\rho }_X(u)$ and called decision index of $U$.

Then we represent the set $D$ of decisions as the function

(2.1) $\varphi :U\times {\mathrm{P}}_X(U)\times S_U\to D$(2.1)

and we define it by

(2.2) $\varphi \left(p,{\mathrm{P}}_X\{p\},s\right)=d,\mathrm{f}\mathrm{o}\mathrm{r}p\in U,{\mathrm{P}}_X\{p\}\subseteq {\mathrm{P}}_X(U),s\in S_U,d\in D.$(2.2)

Thus, mathematically, we define a decision space in a nation $X$ as a triple ordered collection $(U,{\mathrm{P}}_X(U),S_U)$ comprising the set $U$ of people, the decision authority ${\mathrm{P}}_X(U)$ of $U$ in $X$ and the set $S$ of possible situations that may occur in the administrative section of $U$. If there is no confusion that may arise, the decision space $(U,{\mathrm{P}}_X(U),S_U)$ will be denoted by $U$.

Since the decision space represents a particular social group in a nation, the quest on its membership comes into play. The definition in the following gives a clear distinction between those who influence decision making directly or indirectly.

Definition 2.1 (Neighborhood in decision space) Let $x$ be a point and $U$ be a nonempty decision space in the nation $X$. Then

(i) $x\in X$ 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 a right to vote for a decision.

(ii) $x\in X$ is an indirect member of $U$ if there exists at least one direct member $y\in U$ who represents $x$ in $U$.

This person $x$ has no right to vote for a decision even if s/he attends the meetings of $U$. We will represent by $y{\sim }_{U}x$ the phrase y represents x.

(iii) $U$ is a direct neighborhood of $x$ if $x$ is a direct member of $U$.

(iv) $U$ is an indirect neighborhood of $x$ if $x$ is an indirect member of $U$.

(v) $U$ is a neighborhood of $x$ if $x$ is either a direct or indirect member of $U$.

It is now obvious from the definition 2.1 $(i)$ that if $U$ is a decision space in $X$, then we have the following properties.

(i) Each direct member $x$ of $U$ is a decision maker who has some powers ${\rho }_X\{x\}$ to approve or disprove some issues in $U$.

(ii) There is a point $y$ in $U$ with the property that ${\rho }_X\{y\}\ge {\rho }_X\{u\}$ for each $u\in U$. This point is called the maximal element of $U$.

A good decision space is the one which represents everyone in its administrative section. In this work, we consider an inclusive nation. This guarantees the representation of every member of a community to decision spaces and leaves no one unrepresented. Throughout this paper, whenever a nation is mentioned, it will represent an inclusive nation. The following definition gives the meaning of this line.

Definition 2.2. Let $X$ be a nation and $x\in X$. Then

(i) $X$ is said to be inclusive at $x$ if there exists a decision space $G$ in $X$ that represents $x.$

(ii) $X$ is inclusive if it is inclusive at all points.

2.1. A nation as a topological space

We define the topology in the nation $X$, which with it we can study the connectivity, separability, compactness, and continuity of functions between nations. The topology that we construct comprises of decision spaces in $X$. We will call this topology a representative topology. Before we present this, we will need the following lemma;

Lemma 2.3. Every nonempty decision space in an inclusive nation $X$ is the union of its one-point subsets composed by its direct members.

Proof. Let $U$ be an arbitrary nonempty decision space in $X$. If $U$ contains only one direct member, the result follows immediately. Suppose $U$ contains more than one direct member, and $x$ is any of its such members. By definition 2.1 $(i)$, $x$ has the right to vote for a decision in $U$. This implies that $\{x\}$ is a decision space. If $x$ is allowed to vary over $U$, then we have $U={\bigcup }_{x\in U}\{x\}$.

The following theorem gives the basis for the representative topology.

Theorem 2.4.

Let $X$ be an inclusive space, and $\mathcal{G}$ be the collection of all decision spaces in $X$. Then $\mathcal{G}$ is a basis for the representative topology $\mathcal{T}$ on $X$.

Theorem 2.4 can be easily proven: if $x$ is any point in $X$, then either $x$ is a decision maker or a commoner. If $x$ is a decision maker in $X$, then $\{x\}$ is a decision space that contains $x$. If $x$ is a commoner in $X$ and since $X$ is inclusive (definition 2.2), then there exists at least one representative, say $y$, who represents $x$ in the decision spaces in $X$. This implies that there exists a decision space $G_y^x\in \mathcal{G}$ that directly contains $y$, a representative of $x$. Thus, $x$ is an indirect member of $G_y^x$ in the sense that $y$ makes his decision in $G_y^x$ based on the views and opinions of $x$. The second axiom holds trivially because if $G_1$ and $G_2$ are intersecting decision spaces, then every point in common constitutes a decision space (Lemma 2.3).

Now for any inclusive nation $X$, we have a representative topology $\mathcal{T}$ that is generated by the collection of all decision spaces in $X$. With this topology, the nation $X$ is a topological space. We will call this a ${\mathcal{T}}_r-$topological space, or simply ${\mathcal{T}}_r-$space and denote it by $(X,{\mathcal{T}}_r)$, where ${\mathcal{T}}_r$ is a representative topology on the inclusive space $X$.

Example 1.

Let $V$ be an inclusive set of people who forms a village at a certain area and let $S$ be a set of possible situations such as defence and security, finance, planning, catastrophes, and so on in $V$. Suppose that $V$ has been partitioned into smaller $n$ administrative sections called suburbs $E_i$, $i=1,2,\dots ,n$, and that there are decision spaces which have been given authority by members of the village to decide on situations on behalf of the villagers. For each $E_i$, let $G_i$ denote the general assembly in $E_i$ that includes all adult residents (as may be defined). Assume that $G_i$ elects members of $E_i$ to form the suburb council, the main executive system of decision spaces in $E_i$. Let the members of this council include the Chairperson (president) $h_i$ of the suburb, the Chie Executive Officer $e_i$ of the suburb, and other members of the suburb elected to form various committees $K_{E{i}_j}$ depending on the size $m$ of the set $S$ of priority situations. All these decision spaces in a suburb council answer to the village council which includes the Village Chairperson $H$ (head of the village) who is elected by the village general assembly $G$, Suburb Chairpersons $h_i$, other members of the village elected by $G$ who form various committees $K_{V_j},j=1,\dots ,m$ where $m=|S|$, and the Village Chief Executive Officer (also a village council secretary) $e$ who is an appointee of $H$. At either level, the general assembly is a parliament where residents are given the opportunity to be heard, while the council is a government that is the main executive decision space in the village administration. The village chairperson presides the village general assembly while the suburb chairperson chairs the suburb general assembly. The set $C=\{\{H\},\{h_1\},\{h_2\},\dots ,\{h_n\},\{e\},K_{V_1},\dots ,K_{V_m}\}$ forms the village council. Similarly, the set $C_{E_i}=\left\{\{h_i\},\{e_i\},\underset{j=1}{\overset{m}{\cup }}{K}_{E_{ij}}\right\}$ forms a council in the suburb $E_i$. Then the collection

$\mathcal{B}=C\cup \bigcup _{i=1}^n{C}_{E_i}$

$=\left\{\{H\},\{e\},\{h_1\},\{h_2\},\dots ,\{h_n\},K_{V_1},\dots ,K_{V_m},\bigcup _{i=1}^n\{e_i\},\bigcup _{i=1}^n\left(\bigcup _{j=1}^m{K}_{E_{ij}}\right)\right\}$

of decision spaces in $V$ satisfies both conditions for a basis. Because $V$ is inclusive and $V={\cup }_{i=1}^n{E}_i$, then for each $i$, $\{h_i\}{\sim }_V{E}_i$ so that ${\cup }_{i=1}^n\{h_i\}$${\sim }_{V}V$. Therefore, each element $v$ of $V$ is contained in some element $\{h_i\}$ of $\mathcal{B}$. The second axiom holds trivially because the intersection of decision spaces gives another decision space. This is because every one-point subset of a decision space is itself a decision space (Lemma 2.3). Thus, the pair $(V,{\mathcal{T}}_r)$ is a ${\mathcal{T}}_r-$space whose representative topology ${\mathcal{T}}_r$ is generated by $\mathcal{B}$.

2.2. Linearity of the system of decision spaces

It is important to note that in the system of decision making process, there is a level for each decision space to make a final decision on some matters. This is only possible if the matter in question can be handled in that administrative area. Otherwise, the next decision space higher in rank will take over, and the process will continue this way until the matter is resolved. When a decision is made, it will be made and passed down to the level at which the matter began. This implies that the decision-making process is linear. To understand this, we shall first prepare a setting that will be used in characterizing such process.

The following definition states the inclusion between decision spaces in terms of superiority and subordinates.

Definition 2.5.

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

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

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

Following the superiority system of the decision making process, it is obvious that the space $(X,{\mathcal{T}}_r)$ is an ordered pair with respect to a relation $\le$.

Remark 2.6. The following statements hold in an ordered ${\mathcal{T}}_r-$ space $(X,{\mathcal{T}}_r)$ with the order relation $\le$ as defined in definition 2.5;

1. $U\le V$ if and only if ${\rho }_X(U)\le {\rho }_X(V)$.

2. A space $(X,{\mathcal{T}}_r)$ is a poset with respect to the relation $\le$.

Definition 2.7. (Maximal and Minimal element) Let $S$ be an arbitrary ordered set. Then

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

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

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

The following proposition gives the boundedness of a ${\mathcal{T}}_r-$ space.

Proposition 2.8. The following statements hold for every ${\mathcal{T}}_r-$ space $(X,{\mathcal{T}}_r)$;

1. Every ${\mathcal{T}}_r-$ space is linear.

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

Proof.

(i) It is enough to show that every element in ${\mathcal{T}}_r$ has a predecessor (or successor). Let $U$ be a nonempty element of ${\mathcal{T}}_r$. If ${\rho }_X(U)$ allows $U$ to make final decisions for whatever matter that arises in its administrative area, then $U$ is maximal (Definition 2.5). If not, then there is another decision space $V\in {\mathcal{T}}_r$ such that $U\le V$ and $U$ answers to $V$. If the decision authority ${\mathrm{P}}_X(V)$ does not allow $V$ to handle the matter, the next higher decision space in the sequence of decision making must take over and so on. Since the ${\mathcal{T}}_r-$ space is an ordered finite set, there exists a decision space $M\in {\mathcal{T}}_r$ such that $M\ge U$ for all $U\in {\mathcal{T}}_r$. Then $M$ makes the final decision.

If $U=\mathrm{\varnothing }$, then the result follows trivially because $\mathrm{\varnothing }\le V$ for all $V\in {\mathcal{T}}_r$.

(ii) We know that ${\mathcal{T}}_r$ is a linearly ordered set and since ${\mathcal{T}}_r$ is finite ($X$ is finite), then it is bounded. This implies that it has a maximal and minimal element (Definition 2.7).

The following corollary, which can be easily shown, is an immediate result from Proposition 2.8.

Corollary 2.9.

Every nonempty decision space in the space $(X,{\mathcal{T}}_r)$ has minimal and maximal element.

3. Interior, Closure, Limit and Boundary Points in a ${\mathcal{T}}_r$-Space

Since the membership of the decision spaces depends on whether the member takes vote or does not in the decision-making process, it is necessary to revisit and redefine the interior, closure and boundary of subsets in a ${\mathcal{T}}_r-$ space. Before we present the main theme of this part, it is important to note that although decision spaces may spatially not intersect, they can still intersect administratively. The administrative intersection of two decision spaces is that decision space that answers to both decision spaces. The answering decision space does not necessarily need to be spatially included in both spaces to which it answers to. This brings us to another set theory of proximity: nearness of sets. This has been intensively studied and developed by Naimpally and Peters (Naimpally & Peters, 2013), Naimpally and Warrack (Naimpally & Warrack, 1970), Naimpally (Naimpally, 2009a)- (Naimpally, 2009b), Peters and Naimpally (Peters & Naimpally, 2012), Peters (Peters, 2007a)- (Peters, 2007b), and Smirnov (Smirnov, 1952). They discuss two types of proximity: spatial and descriptive proximity. These work best when dealing with sets under usual Cantorian set operations. As noted earlier, our case is somewhat peculiar as the communicating decision spaces necessarily need not have a point in common. Therefore, spatial and descriptive proximity do not suit to explain the authoritative interactions of decision spaces. To serve the purpose of this study, we develop a new type of proximity called administrative proximity. This will be denoted by $\varphi$-proximity, and defined as follows:

Definition 3.1.

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

(a) The administrative intersection of $U$ and $V$ will be denoted by $U\underset{\varphi }{\cap }V$ and defined by

$U\underset{\varphi }{\cap }V=\{x\in U\cup V:{\rho }_X\{x\}\le {\rho }_X(U)\mathrm{a}\mathrm{n}\mathrm{d}{\rho }_X\{x\}\le {\rho }_X(V)\}.$

(b) If $U\le V$ (definition 2.5), we write

• (i) $U\underset{\varphi }{\cap }V=U.$

• (ii) $U\underset{\varphi }{\cup }V=V.$

(c) If $U\equiv V$ (definition 2.5), we write $U\underset{\varphi }{\cap }V=U$ (or $=V$).

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

(e) $E$ and $F$ are said to be administratively near to each other if there exists a decision space in their union that answers to both.

Note 3.1. The decision spaces $U$ and $V$ may be spatially disjoint ($U\cap V$=$\mathrm{\varnothing }$) and still $\underset{\varphi }{\cap }V$ can be nonempty. However, $U\cap V\ne \mathrm{\varnothing }$ implies $\underset{\varphi }{\cap }V\ne \varnothing$.

At this point, we would like to point out that it is not difficult to show that $\varphi -$proximity satisfies Efremovič proximity (see Naimpally (2013, page 67-68) and Peters (2012, page 538-539) for axioms of Efremovič proximity).

We are now set to define, based on $\varphi$-proximity, the interior, closure, limits and boundary points of the subsets of the ${\mathcal{T}}_r-$space.

Definition 3.2. (Interior, Closure, Boundary). Let $A$ be a subset of a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$. Then

(i) A point $x\in A$ is a $\varphi$-interior point of $A$ if there exists a decision space $U$ in $X$ such that $x$ is a direct member of $U$ and $U\subset A$. The set of all $\varphi -$interior points of $A$ will be called $\varphi$-interior of $A$ in $X$ and we shall denote it by $in{t}_{X_{\varphi }}A$. The properties of $in{t}_{X_{\varphi }}A$ mimic the well-known properties of interiors of sets under spatial or Cantorian set operations.

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

$U_{\alpha }\underset{\varphi }{\cap }A\ne \mathrm{\varnothing }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in J.$

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

$(U_{\alpha }-\{x\})\underset{\varphi }{\cap }A\ne \mathrm{\varnothing }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in J.$

(iv) A person $x\in X$ is a $\varphi -$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 boundary points of A in X is called the $\varphi -$boundary of $A$ in $X$, and will be denoted by $Fr{t}_{X_{\varphi }}A.$

Remark 3.3. (i) If $x\in c{l}_{X_{\varphi }}A$, then every element that $x$ represents stands as an indirect $\varphi -$contact point to $A$. Any point will touch $A$ if such point communicates $x$ through the points it represents.

(ii) For any administrative section $E$ of a space $(X,{\mathcal{T}}_r)$, $\varphi -$interior of $E$ in $X$ (covers) represents the whole $E$; that is, int${ }_{X_{\varphi }}E{\sim }_{X}E$.

Theorem 3.4.

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

Proof.

Suppose that $E$ is inclusive. Then, for each point $e$ in $E$, there is a nonempty decision space $G\subset E$ such that $G\sim e$. This implies that $G\subset \mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E$ and the result follows. The converse follows from Remark 3.3

3.2. Properties of $\varphi -$closure, $\varphi -$boundary and $\varphi -$limits of Administrative Sections of $(\mathit{X},{\mathcal{T}}_{\mathit{r}})$. In this section we develop some properties of the contact and boundary points of an administrative section (or any subset) of the ${\mathcal{T}}_r$-space. The interior (decision spaces) and the contact persons of any social subset in a nation play a great role in enabling the interactions within and outside territorial boundaries of a nation. The $\varphi -$contact and $\varphi -$limit points in a nation are related. The following proposition states this relationship.

Proposition 3.5.

Let $A$ be a subset of the ${\mathcal{T}}_r-$ space $(X,T_r)$. Then

1. Every $\varphi -$limit point to $A$ is a $\varphi -$contact point to $A$.

2. Every $\varphi -$boundary point to $A$ is a $\varphi -$contact point to both $A$ and $A^c=X-A.$

Proof. (a) Let $x\in X$ be a $\varphi -$limit point to $A$ and $\{U_{\alpha \in J}:|J|<\mathrm{\infty }\}$ be the collection of all decision spaces which directly contain $x$. Then $(U_{\alpha }-\{x\}){\cap }_{\varphi }A\ne \mathrm{\varnothing }$ for all $\alpha \in J.$ Since $U_{\alpha }-\{x\}\subset U_{\alpha }$ for all $\alpha$, then we have $U_{\alpha }{\cap }_{\varphi }A\ne \mathrm{\varnothing }$ for all $\alpha \in J.$ Thus, $x\in c{l}_{X_{\varphi }}A$.

(b) This obviously follows from the definition of a $\varphi -$boundary.

There is an undeniable relationship between the limit points and the closure of a set. It is an important relationship that is so well known, and the concept of $\varphi -$proximity upholds this relationship. The proposition below states the named relation between the two sets in question.

Proposition 3.6.

Let $A$ be a subset of a ${\mathcal{T}}_r-$ space $(X,{\mathcal{T}}_r)$ and $A{ }^{\mathrm{\prime }}$ be the set of all $\varphi -$limit persons of $A$, then $c{l}_{X_{\varphi }}A=A\cup A{ }^{\mathrm{\prime }}$.

Proof.

To prove this, we need to show that $c{l}_{X_{\varphi }}A\subset A\cup A{ }^{\mathrm{\prime }}$ and $A\cup A{ }^{\mathrm{\prime }}\subset c{l}_{X_{\varphi }}A$.

Now, let $x\in c{l}_{X_{\varphi }}A$. Then, for every direct neighborhood $U_{\alpha }$ of $x$ in $X$, $U_{\alpha }$ is administratively near to $A$ (definition 3.2). If $x$ is an element of $A$, then we have $x\in A\cup A{ }^{\mathrm{\prime }}$. If $x$ is neither an element of $A$ nor of $U_{\alpha }{\cap }_{\varphi }A$, then $(U_{\alpha }{\cap }_{\varphi }A)-\{x\}\ne \mathrm{\varnothing }$ for all $\alpha$. By the definition of $\varphi -$limits, we have $x\in A{ }^{\mathrm{\prime }}$ and therefore $x\in A\cup A{ }^{\mathrm{\prime }}$.

On the contrary, we show that $A\cup A\hspace{0.17em}\subset c{l}_{X_{\varphi }}A$. Let $x$ be any direct member of $A\cup A{ }^{\mathrm{\prime }}$. If $x\in A$, then $x\in c{l}_{X_{\varphi }}A$ $(A\subset c{l}_{X_{\varphi }}A)$. If $x\in A{ }^{\mathrm{\prime }}$, then $x\in c{l}_{X_{\varphi }}A$ (proposition 3.5).

The theorem below gives the relationship between the contact and interior points of any subset of the space $(X,{\mathcal{T}}_r)$.

Theorem 3.7.

Let $E$ be an administrative section of the space $(X,{\mathcal{T}}_r)$. Then the $\varphi -$closure of the $\varphi -$interior of $E$ in $X$ is administratively equal to the $\varphi -$closure of the entire section $E$ in $X$.

Proof.

In terms of elements, it is clear that $in{t}_{X_{\varphi }}E\subseteq E$ and hence $\mathrm{c}{\mathrm{l}}_{X_{\varphi }}(\mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E)\subseteq c{l}_{X_{\varphi }}E$. Conversely, we show that $E\subseteq \mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E$. Let $x\in E$. Then $x$ is either a decision maker or a commoner in $E$. If $x$ is a decision maker in $E$, then $x\in \mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E$. If $x$ is a commoner in $E$, there is an indirect neighborhood $G_x$ of $x$ in $E$ that represents $x$ ($X$ is inclusive). This implies that $x$ is an indirect member to $\mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E$. This further implies that int${ }_{X_{\varphi }}E$ covers $E$. Thus, $c{l}_{X_{\varphi }}E\subseteq \mathrm{c}{\mathrm{l}}_{X_{\varphi }}(\mathrm{i}\mathrm{n}{\mathrm{t}}_{X_{\varphi }}E)$.

The following theorem establishes the criteria for the decision space that does or does not belong to the particular administrative section to be the contact point of such section.

Theorem 3.8.

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

Proof.

Let $(G_x)$ be the family of direct neighborhoods of $x$ in $X$. Then $\{x\}\subset G_x$ for all $G_x$. Since ${\rho }_X(G)\le {\rho }_X\{x\}$ for every decision space $G\subset E$, by the transitivity property we have that for every $G_x$, ${\rho }_X(G)\le {\rho }_X\{G_x\}$ for all $G\subset E$. By definition 3.1, we have that for every $G_x$, $G{\cap }_{\varphi }{G}_x=G$ for all $G\subset E$. This further implies that $G_x{\cap }_{\varphi }E\ne \mathrm{\varnothing }$ for all $G_x$. Therefore, $x\in c{l}_{X_{\varphi }}E$.

It is important to determine the existence of $\varphi -$frontier points for any subset $E$ of a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$. The most important of these points is that which lies in the $\varphi -$interior of $E$. The corollary below, which is partly an immediate result of Theorem 3.8, gives such point(s).

Corollary 3.9.

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

Proof.

From Theorem 3.8, it can be easily shown that $x$ is a $\varphi -$contact point of $E$.

Our task will be to show that $x$ is $\varphi -$contact point of $X-E$. Let $s$ be a situation that arises in $E$ and suppose that ${\rho }_X\{x\}$ does not allow $x$ to solve the matter, then its immediate successor $y\in E^{\star }\subseteq X-E$ will step in. This implies that ${\rho }_X\{x\}\le {\rho }_X\{y\}$. Then, every direct neighborhood $G_{\alpha }$ of $x$ administratively intersects all direct neighborhoods $G_y$ of $y$ in $X-E$ through $x$. Thus, $G_{\alpha }$ administratively intersects $X-E$ for all $\alpha$ and, therefore, $x$ is a $\varphi -$contact point of $X-E$.

Corollary 3.10.

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

Proof.

This is easy to prove. From Corollary 3.9 and Theorem 3.8, the maximal element of int${ }_{X_{\varphi }}E$ is a $\varphi -$ contact point for all subsets of $E$.

The following properties of $\varphi -$closure and $\varphi -$frontier of the administrative section in a ${\mathcal{T}}_r-$space are easy to prove. We shall skip their proofs.

Proposition 3.11.

Let $E_1$ and $E_2$ be two administrative sections in a space $(X,{\mathcal{T}}_r)$. Then

1. If $E_1$ and $E_2$ administratively intersect, then $c{l}_{X_{\varphi }}{E}_1\cap c{l}_{X_{\varphi }}{E}_2\ne \mathrm{\varnothing }$.

2. If $E_1\subset E_2$, then $c{l}_{X_{\varphi }}{E}_1\subset c{l}_{X_{\varphi }}{E}_2$.

3. If $E_1$ and $E_2$ are spatially disjoint and ${\rho }_X(in{t}_{X_{\varphi }}{E}_1)\equiv {\rho }_X(in{t}_{X_{\varphi }}{E}_2)$, then $c{l}_{X_{\varphi }}{E}_1\cap c{l}_{X_{\varphi }}{E}_2\ne \mathrm{\varnothing }$.

4. If $A$ is a subset of the subspace $E$ of $(X,{\mathcal{T}}_r)$, then $c{l}_{E_{\varphi }}A=c{l}_{X_{\varphi }}A\cap E$.

3.3. Subspace of a ${\mathcal{T}}_r$-space. Let $A$ be a nonempty subset of a space $(X,{\mathcal{T}}_r)$. Here, we need to establish a topology on $A$. Since the representative topology ${\mathcal{T}}_r$ is induced by the relationship of the decision spaces in $X$, there is a naturally induced relationship of the decision spaces in $A$. That is, a person $a\in A$ is a direct or indirect member to the decision space $G\subset A$ if and only if $a$ is a direct or indirect member to $G$ in $X$. Equivalently, a decision space $G\subset A$ is a neighborhood of a person $a\in A$ if and only if it ($G$) is a neighborhood of $a$ in $X$. In other words, the topology ${\mathcal{T}}_{rA}$ on $A$ induced by ${\mathcal{T}}_r$ is the collection $\{G\cap A:G\in {\mathcal{T}}_r\}$. The topology ${\mathcal{T}}_{rA}$ is a subspace topology on $A$ and the pair $(A,{\mathcal{T}}_{rA})$ is called a subspace of the ${\mathcal{T}}_r-$ space $(X,{\mathcal{T}}_r)$.

Remark 3.12. In the subspace $(A,{\mathcal{T}}_{rA})$ of a space $(X,{\mathcal{T}}_r)$, the following statements hold;

(a) A subset $B$ of $A$ is a decision space (open) in $A$ if and only if $B=G\cap A$ for some decision space (open set) $G$ in $X$.

(b) A subset $C$ of $A$ is $\varphi -$closed in $A$ if and only if $C=A\cap F$ for some $\varphi -$closed set $F$ in $X$.

4. $\varphi -$connectedness of Administrative Sections in $(X,{\mathcal{T}}_r)$

In order to make a decision, each decision space in a chain of decision making process has to play its part as the decision authority of that particular nation instructs. There must always be a way to communicate within and outside of the particular administrative section of the nation. In this part, it is our aim to explore whether such sections can be broken into non-empty disjoint subsections which administratively do or do not communicate with each other. This brings us to the vast, rich and most developed concept of connectedness and separatedness (disconnectedness) of subsets in the topological spaces.

The following definition and lemma will be used as tools to determine the $\varphi -$ connectedness and $\varphi -$ separatedness of subsets of the ${\mathcal{T}}_r-$space.

Definition 4.1.

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

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

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

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

Since spatial nearness implies $\varphi -$nearness (note 3.1), then the following well-known result holds in $\varphi -$proximity.

Lemma 4.2.

Let $Y$ be a subspace of $(X,\mathcal{T})$. 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 proof of Lemma 4.2 is found in Hocking (Hocking, 1961 page 15-16) and Munkres (Munkres, 2000 page 148–149).

These settings are enough to get to the bottom of this section. The following theorem states the connectedness, called $\varphi -$connectedness, of administrative subspaces of the ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$.

Theorem 4.3.

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

Proof.

Let $E$ be a subspace of a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$. Suppose, on the contrary, that $E$ is not $\varphi -$connected. Then by Lemma 4.2, there are two nonempty disjoint subsets $E_1$ and $E_2$ of $E$ whose union covers $E$ and $(c{l}_{X_{\varphi }}{E}_1\cap E_2)\cup (E_1\cap c{l}_{X_{\varphi }}{E}_2)$ is empty. Since $E$ is a subspace, int${ }_{X_{\varphi }}E\ne \mathrm{\varnothing }$. The $\varphi -$interior of $E$ in $X$ has a maximal element (Corollary 2.9). Let $x$ be a maximal element of int${ }_{X_{\varphi }}E$. Then $x$ is a $\varphi -$contact point of both subsets $E_1$ and $E_2$ (Theorem 3.8, Corollary 3.9 and 3.10). If $x$ is in $E_1$, then $E_1\cap c{l}_{X_{\varphi }}{E}_2\ne \mathrm{\varnothing }$ (the same applies if $x$ is in $E_2$). This is a contradiction. Therefore, such two subsets $E_1$ and $E_2$ cannot exist, and if they do not exist, then $E$ has no $\varphi -$separation.

Theorem 4.3 establishes the $\varphi -$connectedness of an administrative section of the space $(X,{\mathcal{T}}_r)$. It does not generalize the criteria for $\varphi -$connectedness of an arbitrary subset of such a space. The corollary below states the necessary and sufficient condition for $\varphi -$connectedness of any subset of $(X,{\mathcal{T}}_r)$.

Corollary 4.4.

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

Proof.

Suppose that $in{t}_{X_{\varphi }}E=\mathrm{\varnothing }$, then there is no maximal element of $E$ as $c{l}_{X_{\varphi }}(in{t}_{X_{\varphi }}E)=\mathrm{\varnothing }$. Thus, for any spatially disjoint subsets $E_1$ and $E_2$ of $E$, we find that both $E_1$ and $E_2$ are administratively far from each other, and thus $c{l}_{X_{\varphi }}{E}_1\cap E_2=\mathrm{\varnothing }=E_1\cap c{l}_{X_{\varphi }}{E}_2$. Thus $E$ has a $\varphi -$separation. We have, by means of contrapositive, that $E$ is $\varphi -$connected provided that $in{t}_{X_{\varphi }}E\ne \mathrm{\varnothing }$. The converse follows from Theorem 4.3.

Theorem 4.5.

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

From this theorem, it is worthy to note that the subsets $G$ and $E_{\alpha }$ of $(X,{\mathcal{T}}_r)$ are each $\varphi -$connected (Theorem 4.3).

Proof of Theorem 4.5.

Since $E^{\ast }$ may be taken as the subspace of $X$, there is no harm in supposing that $E^{\ast }=X$. Suppose that $E^{\ast }$ is $\varphi -$separated. Let $G_1,G_2$ be a $\varphi -$separation of $E^{\ast }$. Since every $E_{\alpha }$ is $\varphi -$connected, for each $\alpha$, $E_{\alpha }$ is administratively contained in $G_1$ or in $G_2$. Likewise, $G$ (which is $\varphi -$ connected) is contained entirely in $G_1$ or in $G_2$. Suppose that $G$ is contained entirely in $G_1$. Since $E_{\alpha }{\cap }_{\varphi }G\ne \mathrm{\varnothing }$, then for each $\alpha$, $E_{\alpha }$ administratively meets $G_1$, and hence $E_{\alpha }$ is fully contained in $G_1$. This further implies that $E^{\ast }=G\cup ({\cup }_{\alpha }{E}_{\alpha })$ is entirely contained in $G_1$, leaving $G_2$ empty. This contradicts the fact that $G_2$ is non-empty.

Corollary 4.6.

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

Example 2.

Consider the space $V=\underset{i=1}{\overset{n}{\cup }}{E}_i$ as in Example 1 together with a basis

$\mathcal{B}=\left\{\{H\},\{e\},\{h_1\},\{h_2\},\dots ,\{h_n\},K_{V_1},\dots ,K_{V_m},\bigcup _{i=1}^n\{e_i\},\bigcup _{i=1}^n\left(\bigcup _{j=1}^m{K}_{E_{ij}}\right)\right\}$

for a representative topology ${\mathcal{T}}_r$ on $V$. Then, the space $(V,{\mathcal{T}}_r)$ is $\varphi -$connected. It is easy to show this. Suppose $A,B$ are any disjoint nonempty decision spaces in $V$ such that $A\cup B{\sim }_{V}V$ and $A{\underset{\_}{\delta }}_{\varphi }B$. Since $H$ is a maximal element in $V$, then $H\in c{l}_{V_{\varphi }}A\cap c{l}_{V_{\varphi }}B$ (see Theorem 3.8). If $H$ is covered in $A$, then $A$ would contain the $\varphi -$contact point $H$ of $B$. Same argument holds if $H$ was administratively contained in $B$. By Lemma 4.2, $V$ is not $\varphi -$separated. Therefore, there cannot exist such decision spaces in $V$ whose union represents $V$ and are themselves administratively far.

If $G$ is a one-point set in Theorem 4.5, then $G$ would be covered in each $E_{\alpha }$. Thus the union $E^{\ast }=G\cup ({\cup }_{\alpha }{E}_{\alpha })$ would reduce to $E^{\ast }={\cup }_{\alpha }{E}_{\alpha }$. Theorem 4.5 would reduce to:

Theorem 4.7.

The union $E^{\ast }={\cup }_{\alpha }{E}_{\alpha }$ of subsets $E_{\alpha }$ of $(X,{\mathcal{T}}_r)$ is $\varphi -$connected if ${\cap }_{\varphi }{E}_{\alpha }\ne \mathrm{\varnothing }$.

The theorem below states a necessary condition for the $\varphi -$connectedness of the union of a sequence of administrative sections of the space $(X,{\mathcal{T}}_r)$ that intersect pairwise.

Theorem 4.8.

Let $I$ be a finite set of indices and suppose that $(E_{\alpha }{)}_{\alpha \in I}$ is a sequence of administrative sections (subspaces) of $(X,{\mathcal{T}}_r)$ such that

$E_{\alpha }\underset{\varphi }{\cap }{E}_{\alpha +1}\ne \mathrm{\varnothing }.$

Then ${\displaystyle \underset{\alpha \in I}{\cup }{E}_{\alpha }}$ is $\varphi -$connected.

Proof.

Let $Y=\bigcup _{\alpha \in I}{E}_{\alpha }$. Suppose, in contrast, that $Y$ is $\varphi -$disconnected. Then $Y$ has a $\varphi -$separation. There is a pair $G_1,G_2$ of nonempty disjoint decision spaces in $X$ such that $G_1\cup G_2\sim Y$ and $G_1{\cap }_{\varphi }{G}_2=\mathrm{\varnothing }$. Let $p\in E_{\alpha }{\cap }_{\varphi }{E}_{\alpha +1}$. Then $p\in in{t}_{X_{\varphi }}{E}_{\alpha }\cup in{t}_{X_{\varphi }}{E}_{\alpha +1}$ with ${\rho }_X\{p\}\le {\rho }_X(in{t}_{X_{\varphi }}{E}_{\alpha })$ and ${\rho }_X\{p\}\le {\rho }_X(in{t}_{X_{\varphi }}{E}_{\alpha +1})$.

Suppose that $p\in in{t}_{X_{\varphi }}{E}_{\alpha }$. Then $p$ is covered in $G_1$ or in $G_2$. Similarly, since $in{t}_{X_{\varphi }}{E}_{\alpha }$ is $\varphi -$connected, then it either lies wholly in $G_1$ or in $G_2$. Suppose that $p$ is contained in $G_1$, then $in{t}_{X_{\varphi }}{E}_{\alpha }$ can never be contained in $G_2$, otherwise $G_2$ would contain a point $p$ of $G_1$, which is impossible because $G_1\cap G_2=\mathrm{\varnothing }$ and $G_1{\cap }_{\varphi }{G}_2=\mathrm{\varnothing }.$ Therefore, $in{t}_{X_{\varphi }}{E}_{\alpha }$ is entirely covered by $G_1$, implying that $E_{\alpha }$ is wholly covered by $G_1$. Since $\{p\}$ answers to both $in{t}_{X_{\varphi }}{E}_{\alpha }$ and $in{t}_{X_{\varphi }}{E}_{\alpha +1}$, then there exists $G_{\alpha +1}\subset in{t}_{X_{\varphi }}{E}_{\alpha +1}$ for which ${\rho }_X\{p\}\le {\rho }_X(G_{\alpha +1})$. $G_{\alpha +1}$ cannot be contained in $G_2$ because $G_2$ would administratively contain point $p$ that is already covered in $G_1$, violating the fact that $G_1{\cap }_{\varphi }{G}_2=\mathrm{\varnothing }.$ Therefore, $G_1$ contains $G_{\alpha +1}$. Since $in{t}_{X_{\varphi }}{E}_{\alpha +1}$ is connected, it cannot be covered by $G_2$, otherwise $G_2$ would cover $G_{\alpha +1}$ that is already covered in $G_1$, which is impossible. Therefore $in{t}_{X_{\varphi }}{E}_{\alpha +1}$ is wholly contained in $G_1$. Since $in{t}_{X_{\varphi }}{E}_{\alpha +1}$ administratively covers $E_{\alpha +1}$, then $G_1\sim E_{\alpha +1}$ and thus $G_1\sim E_{\alpha }\cup E_{\alpha +1}$. If $\alpha$ is allowed to vary over $I$, we have $G_1\sim \bigcup _{\alpha \in I}{E}_{\alpha }$. This leaves $G_2$ empty, contradicting the fact that $G_2$ must not be empty. Therefore, $Y=\bigcup _{\alpha \in I}{E}_{\alpha }$ has no $\varphi -$separation.

4.1. Chain of Decision Spaces. From the linearity property of a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$ (Proposition 2.8), it is evident that the decision spaces work in chain for every situation that may arise in their administrative sections. It is the particular situation that determines such a chain. This kind of chain serves as a means of communication that links citizens to decision spaces and vice versa.

Definition 4.9.

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

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

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

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

Axiom (iii) in definition 4.9 ensures the consistency in decision making and eliminates the contradicting decisions that may be decided by distinct decision spaces should multiple situational immediate successors be allowed.

The concept of an m-chain will be used to determine the connectedness of subsets of the space $(X,{\mathcal{T}}_r)$. Intuitively speaking, we say that a certain property defined for topological space is the property of a subset if such subset has that property as the subspace. Before we pursue this objective any further, we would like to stop a bit at the connectedness of the m-chain itself.

Theorem 4.10.

An m-chain is $\varphi -$connected.

Proof.

It is not difficult to show this result, as it is an immediate consequence of Theorem 4.8.

By definition 4.9, every element of the chain has only one immediate successor. The number of predecessors is not fixed. This implies that a particular situation can occur in different administrative sections and all corresponding decision spaces in such sections answer to one successor. This makes these different subsets communicate each other through this successor. It is only natural to ask whether the union of m-chains is $\varphi -$connected or otherwise. The following theorem establishes the $\varphi -$connectedness of union of such chains.

Theorem 4.11.

Let $I$ be a finite set of indices, $N$ be a finite subset of $\mathbb{N}$ and ${\left({\mathcal{M}}_{\alpha }\right)}_{\alpha \in I}=((G_n^{\alpha }{)}_{n\in N}{)}_{\alpha \in I}$ be the family of m-chains by the open sets $G_n^{\alpha }$ in the space $(X,{\mathcal{T}}_r)$. If $G$ is any nonempty decision space in $X$ with $G\underset{\varphi }{\cap }{\mathcal{M}}_{\alpha }\ne \mathrm{\varnothing }$ for all $\alpha \in I$, then $M=G\cup ({\cup }_{\alpha }{\mathcal{M}}_{\alpha })$ is $\varphi -$connected.

Proof.

We will prove this by contradiction. Suppose that $M=G\cup ({\cup }_{\alpha }{\mathcal{M}}_{\alpha })$ is $\varphi -$separated; then there is a pair $G_1,G_2$ of non-empty disjoint decision spaces in $X$ whose union covers the whole $M$ and $G_1{\cap }_{\varphi }{G}_2\ne \mathrm{\varnothing }$. Since $G\subset M$, then either $G\subset G_1$ or $G\subset G_2$. Suppose that $G\subset G_1$: then each ${\mathcal{M}}_{\alpha }$ is administratively near $G_1$. Since $G_1$ and $G_2$ are spatially disjoint and administratively far, then the $\varphi -$connected ${\mathcal{M}}_{\alpha }$ (theorem 4.10) does not lie in $G_2$. This implies that each ${\mathcal{M}}_{\alpha }$ is covered by $G_1$. Therefore, $M=G\cup ({\cup }_{\alpha }{\mathcal{M}}_{\alpha })\subset G_1$. This leaves $G_2$ empty, a contradiction.

Since m-chains are made up of decision spaces, their intersection comprises such spaces in common. Therefore, Theorem 4.11 can be rephrased as follows.

Theorem 4.12.

The union of collection of m-chains in the ${\mathcal{T}}_r-$space is $\varphi -$connected if and only if their administrative intersection is nonempty.

The following definition, analogous to the definition of path-connectedness, will be used in proving connectedness of subsets of the ${\mathcal{T}}_r-$space.

Definition 4.13.

Let $\mathcal{G}$ be the collection of decision spaces in a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$ which administratively cover $X$, $E$ the non-empty subset of $X$, and $x,y$ be any two points in $X$. Then

(i) An m-chain from point $x$ to $y$ in $X$ is a finite subcollection $\mathcal{M}=\{G_k:1\le k\le n,n\in \mathbb{N}\}$ of $\mathcal{G}$ such that $G_k\underset{\varphi }{\cap }{G}_{k+1}\ne \mathrm{\varnothing }$ for all $k=1,2,3,\dots ,n-1$, and $G_1$ and $G_n$ are neighborhoods of $x$ and $y$, respectively.

(ii) $E$ is an m-chain connected subset of $X$ if for every administrative cover $\mathcal{G}$ of $X$ and for every two points $x,y\in E$, there is an m-chain $\mathcal{M}\subseteq \mathcal{G}$ that connects $x$ and $y$.

(iii) The nonempty subsets $E_1$ and $E_2$ of $(X,{\mathcal{T}}_r)$ are m-chain separated in $X$ if there exists an administrative covering ${\mathcal{G}}^{\star }$ of $X$ by open sets in $X$ such that for every point $x\in E_1$ and every point $y\in E_2$, there does not exist an m-chain in ${\mathcal{G}}^{\ast }$ that connects $x$ and $y$.

Theorem 4.14.

Let $p$ be a point in a space $(X,{\mathcal{T}}_r)$, $\mathcal{H}$ be the administrative covering of $X$ and suppose that $M_p$ is the set containing all points $q$ of $X$ such that there is an m-chain of elements of $\mathcal{H}$ from $p$ to $q$. Then the set $M_p$ is nonempty, open, and closed in $X$.

Proof.

Since every point in $X$ belongs to a decision space and each of this decision space belongs to an m-chain, then $M_p$ contains $q$.

Openness of $M_p$: Let $q$ be an element of $M_p$ and $G_1,\dots ,G_n$ be an m-chain of elements of $\mathcal{H}$ from $p$ to $q$, then for each point $r$ covered by $G_n$ we have $G_1,\dots ,G_{n-1}$ or $G_1,\dots ,G_n$ is an m-chain from $p$ to $r$, since it may happen that $r$ lies in $G_{n-1}\cap G_n$ or in $G_{n-1}{\cap }_{\varphi }{G}_n$. By definition, it follows that $r$ is administratively contained in $M_p$. Thus, the whole $G_n$ is included in $M_p$, making $M_p$ the union of decision spaces in $X$.

Closedness of $M_p$: we need to show that the set $M_p$ contains all its limit points in $X$. Now let $y$ be a limit point of $M_p$. Since $\mathcal{H}$ administratively covers $X$, there exists a decision space $G_{\alpha }$ in $\mathcal{H}$ that contains $y$. Since $y$ is a limit point of $M_p$, it follows that $G_{\alpha }-\{y\}$ is administratively near $M_p$. There is an element $q$ in $G_{\alpha }-\{y\}$ or in $M_p$ such that ${\rho }_X\{q\}\le {\rho }_X(G_{\alpha }-\{y\})$ and ${\rho }_X\{q\}\le {\rho }_X(M_p)$. In both cases,$(\{q\},{\rho }_X\{x\},S_{\{q\}})$ is administratively contained in $G_{\alpha }$ and in $M_q$. By definition, there is an m-chain $G_1,\dots ,G_n$ of elements of $\mathcal{H}$ from the point $p$ to $q$. Since $q$ is also administratively contained in $G_n$, $G_n$ is administratively close to $G_{\alpha }$: the direct neighborhood of $y$. Thus, the sequence $G_1,\dots ,G_n,G_{\alpha }$ is an m-chain from point $p$ to $y$. Thus, $y$ is contained in $M_p$.

Since the ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$ is administratively connected, then the only nonempty subset of $X$ that is both open and $\varphi -$closed in $X$ is $X$ itself. Therefore, the set $M_p$ is equal to the whole set $X$. It is now obvious that $(X,{\mathcal{T}}_r)$ is m-chain connected.

Theorem 4.15.

For every two points in a ${\mathcal{T}}_r-$space $(X,{\mathcal{T}}_r)$ and for each covering $\mathcal{H}$ of $X$, there is an m-chain of elements of $\mathcal{H}$ that connects such points.

Corollary 4.16.

Every subspace of $(X,{\mathcal{T}}_r)$ is m-chain connected.

Since every administrative section of a space $(X,{\mathcal{T}}_r)$ is $\varphi -$connected and m-chain connected as well, the following assertions hold, and their proofs can be easily shown as they follow the same algorithms as those of $\varphi -$connectedness.

(i) The union of m-chain connected subspaces of a space $(X,{\mathcal{T}}_r)$ which have a common point is m-chain connected.

(ii) The union of sequence of m-chain connected subspaces of $(X,{\mathcal{T}}_r)$ which adjacently administratively intersect pairwise is m-chain connected.

The $\varphi -$connectedness of $(X,{\mathcal{T}}_r)$ implies that the only $\varphi -$component in $X$ that contains any point of $X$ is $X$ itself. This further implies that $X$ is locally $\varphi -$connected in the sense that for every point $x$ in $X$ and for each decision space $G$ containing $x$, there is $\varphi -$connected decision space $G^{\ast }$ that administratively contains $x$ and is contained in $G$. We have already shown in Theorem 4.3 that the administrative sections, and hence all subsets of $X$ are $\varphi -$connected in $X$ and under their relative topologies derived from the representative topology of $X$. The fact that $(X,{\mathcal{T}}_r)$ is locally $\varphi -$connected also follows from the fact that the basis that forms ${\mathcal{T}}_r$ comprises $\varphi -$connected decision spaces. The subspaces of $X$ are also both locally $\varphi -$connected and locally m-chain connected. This can be proved in the same way the space $X$ is proved to have such properties.

The space $(X,{\mathcal{T}}_r)$ and its subspaces are also locally m-chain connected, for if $x$ is any point in $X$ and $G$ is any decision space administratively containing $x$ and, since $G$ is the union of one-point sets, there is a decision space $H$ made up of such one-point sets that administratively neighbors $x$ and is contained in $G$. Since all subsets of $X$ are $\varphi -$connected, then by Theorem 4.15 and Corollary 4.3, we have that $G$ and its nonempty subsets are m-chain connected.

Definition 4.17. Let ${\mathcal{M}}_1=\{G_{11},\dots ,G_{1m}\}$ and ${\mathcal{M}}_2=\{G_{21},\dots ,G_{2n}\}$ be m-chains from a point $x$ to a point $y$ in a ${\mathcal{T}}_r-$space $X$. The m-chain ${\mathcal{M}}_1$ is said to go strictly straight through ${\mathcal{M}}_2$ provided that

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

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

The $\varphi -$connectedness and locally $\varphi -$connectedness of a space $(X,{\mathcal{T}}_r)$ make it easier to establish a chain that strictly goes straight through the other.

Theorem 4.18.

Let $(X,{\mathcal{T}}_r)$ be a ${\mathcal{T}}_r-$space and $\mathcal{M}=\{G_1\dots ,G_n\}$ be an m-chain from a point $x$ to a point $y$ in $X$. Suppose that $\mathcal{U}$ is a family of nonempty decision spaces such that each link $G_k$ is a union of elements of $\mathcal{U}$ and directly contains more than one element. Then there is an m-chain that goes strictly straight through $\mathcal{M}$.

Proof.

Given that $\mathcal{M}$ is an m-chain, then $G_k{\cap }_{\varphi }{G}_{k+1}\ne \mathrm{\varnothing }$ for each $k$. Let $g_0=x,g_n=y$ and $g_k$ be an element of $G_k{\cap }_{\varphi }{G}_{k+1}$ for $k=1,2,\dots ,n-1.$ We know that each $G_k$ is $\varphi -$connected (Theorem 4.3 and Corollary 4.4), then by Theorem 4.15 and the fact that each $G_k$ contains more than one direct member, there is an m-chain ${\mathcal{M}}_k=\{V_{k1},\dots ,V_{k{m}_1}\}$ of elements of $\mathcal{U}$ from $g_{k-1}$ to $g_k$ with all links $V_{kj}$ of ${\mathcal{M}}_k$ administratively lying strictly within $G_k$ for $j=1,2,\dots ,m_1$. Since $g_k$ is administratively contained in both $G_k$ and $G_{k+1}$, there is an element $V_{(k+1)1}$ of $\mathcal{U}$ in $G_{k+1}$ such that ${\rho }_X\{g_k\}\le {\rho }_X(V_{(k+1)1})$. In ${\mathcal{M}}_k$, there is a link $V_{k{m}_1}$ that is administratively close to $V_{(k+1)1}$: the first link of ${\mathcal{M}}_{k+1}$. If this process is repeated for each $k$, the collection of all m-chains ${\mathcal{M}}_k$ forms an m-chain from $g_0=x$ to point $g_n=y$ that goes strictly straight through $\mathcal{M}$.

s we close up this part, it is important to note that the inclusive property of a space $X$, the property that is inherited by its subspaces, is necessary in maintaining the unity of $X$ and making the space to be administratively one.

5. Conclusion

In this paper, we have developed a mathematical representation of decision spaces and explored their properties, constructed in an inclusive nation the topology that is made up of such decision spaces. We then introduced a $\varphi -$proximity that describes the interactions between decision spaces, and we this proximity, we defined and proved various properties of topological operators (closure, boundary, limit and interior points). We finally investigated the connectedness of subsets of a nation with respect to $\varphi -$proximity. The reason behind the choice of decision spaces over other spaces to form a topology in a nation is that the earlier best describes the nation. The interaction between nations is best described by decision spaces. If we can toss our card correctly, the question of continuity of functions and homeomorphism between the interacting nations can be clearly explained, as the concept in topology is built from the bottom up. This can further serve as a mean of introducing general topology into the mathematical understanding of international relations.

Acknowledgements

The authors acknowledge the criticism, discussion and every kind of unconditional cooperation from the two departments to which authors are affiliated.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research did not receive grant from any funding agency.

Notes on contributors

Shega Mayila

Shega Mayila is an assistant lecturer of Mathematics at the University of Dodoma. He is currently pursuing his Ph.D. studies in Mathematics at the University of Dar-es-Salaam. His research interests are in the area of topology, analysis, and philosophy of mathematics.

Marco Mpimbo

Marco Mpimbo received his Ph.D. in Mathematics from Kent State University, USA. He is currently working at the University of Dar-es-Salaam in the Department of Mathematics as a senior lecturer. His research interests include topology, analysis, and operator theory.

Sylvester Rugeihyamu

Sylvester Rugeihyamu received his Ph.D. in Mathematics from National University of Ireland, Cork, Ireland. He is currently working at the University of Dar-es-Salaam as the senior lecturer and head of the Department of Mathematics. His research interests are complex analysis, mathematical education and STEM education.