Minimal spaces with a mathematical structure

Peer review under responsibility of University of Bahrain.

Abstract

This paper will discuss, grill topological space which is not only a space for obtaining a new topology but generalized grill space also gives a new topology. This has been discussed with the help of two operators in minimal spaces.

Keywords:

• Grill topological space
• Filter
• Grill minimal space
• Kuratowski closure axioms
• -local function

1 Introduction

The concept of grill is well known in topological spaces. Choquet (see Choquet, 1947) introduced this notion in 1947. After that mathematicians like Thrown (see Thrown, 1973) and Chattopadhyay and Thron (1977) and Chattopadhyay et al. (1983)" have developed this study at closure spaces, compact spaces, proximity spaces, uniform spaces and many other spaces. Grill topological space (see Choquet, 1947) is a much more new concept in the literature. It was introduced by Roy and Mukherjee (2007) in 2007. Authors like Al-Omari and Noiri (2011, 2012b, 2013)", Hatir and Jafari (2010) and Modak (2013a,b,c,d) have studied this field in detail. They have concentrated their study on two operators and generalized sets on this space and obtained different topologies. Modak has shown that new topology can be made from various types of generalized spaces in Modak (2013b,c)".

In this paper we shall define two operators on Alimohammady and Roohi’s minimal space (see Alimohammady and Roohi, 2005). We also divide the properties of these two operators into two parts. Again we shall try to obtain a new topology with the help of this minimal space with a grill on the same space. However Roy and Mukherjee (2007) and Al-Omari and Noiri (2012a,b)" have considered grill topological space for obtaining a new topology. Actually throughout this paper, we are trying to catch the essential space with a grill on the same space that gives a new topology.

2 Preliminaries

Following are the preliminaries for this paper:

Definition 2.1

Alimohammady and Roohi, 2005

A family is said to be minimal structure on X if .

In this case is called a minimal space. Throughout this paper means minimal space.

Example 2.2

Alimohammady and Roohi, 2005

Let be a topological space. Then (Levine, 1963), (Mashhour et al., 1982) and (Njastad, 1965) are the examples of minimal structures on X.

Recall the definition of grill:

Definition 2.3

Choquet, 1947

A subcollection (not containing the empty set) of is called a grill on X if satisfies the following conditions:

1.	and implies ;

2.	and implies that or .

Example 2.4

Let X be a nonempty set. Then the filter (Thron, 1966) and the grill (Choquet, 1947) do not form a minimal structure on X.

Definition 2.5

(Alimohammady and Roohi, 2005) A set is said to be an m – open set if . is an m – closed set if . We set

,

.

The Proposition 2.6 of Alimohammady and Roohi (2005) can be restated by the following:

Proposition 2.6

Let be a minimal space. Then for ,

1.	and if A is an m – open set.

2.	and if A is an m – closed set.

3.

and if .

4.

and .

5.

and .

6.

and .

7.	if and only if every m – open set containing .

8.

and .

Definition 2.7

Alimohammady and Roohi, 2005

A minimal space enjoys the property I if the finite intersection of m – open sets is an m – open set.

Example 2.8

Let X be a nonempty set. Let M be the m – structure (see Al-Omari and Noiri, 2012a) on X. Then the space is an example of a minimal space with the property I.

A minimal space with grill on X is called a grill minimal space and denoted as .

3 -operator

In this section we obtain a topology from the minimal structure and the grill.

Definition 3.1

Let be a grill minimal space. A mapping is defined as follows:

for all for each , where .

The mapping is called -local function.

3.1 Properties of -operator

Here we have divided the properties of into two parts. Some of the properties hold at grill minimal space and other properties hold at minimal space with property I.

Theorem 3.2

Let be a grill minimal space. Then

1.

.

2.

, if .

3.

for and .

4.

for .

5.

for .

6.	for is an m – closed set.

7.

for .

8.	, where is a grill on X with .

Proof

(1)	Obvious from definition of .

(2)	Obvious from definition of .

(3)	Let . Then for all . Again it is obvious that (from definition of grill). Hence .

(4)	Let , then from Proposition 2.6, there is an such that . Implies that . Hence .

(5)	Let and , then . Let Then and . Therefore , and hence . Thus .

(6)	Proof is obvious from Proposition 2.6 and above Property.

(7)	From Property 4, . Again from Property 5, . So, .

(8)	Obvious from definition grill.□

The Authors, Roy, Muhkerjee, Al-Omari, Noiri, Hatir and Jafiri have considered grill topological space for above theorem. But we have shown that the grill minimal space is the sufficient space for the same.

Second type properties of -operator are:

Theorem 3.3

Let be a grill minimal space and enjoys the property I. Then

1.

for .

2.

for and .

3.

for .

4.	for with .

Proof

(1)	From Theorem 3.2(3), . For reverse inclusion, suppose that . Then there are such that and hence . Now and , so, . Therefore . Hence the result.

(2)	From Theorem 3.2(3), . For reverse inclusion, suppose and . Then and , implies . So . This implies that . Thus .

(3)	Here, (from Theorem 3.3 (1)) (from Theorem 3.2(3)). Thus . Again, (from Theorem 3.2(3)). This implies that . Hence .

(4)	From Theorem 3.3(1), (from Theorem 3.2(2)). Again from Theorem 3.2(3), . Also from Theorem 3.3(3), . This implies that , since . Thus .□

Now we shall give an example, which shows that the condition I on grill minimal space is an essential condition for the above theorem.

Example 3.4

Let and . Then the space does not enjoy the property I. Now, open sets containing a are: open sets containing b are: open sets containing c are: open sets containing d are: . Consider , then . Now . Hence .

Let be a grill minimal space. We define a map by , for all . Then we have:

Theorem 3.5

The above map satisfies Kuratowski Closure axioms.

Proof

From Theorem 3.2, , and obviously . Now (from Theorem 3.3(1)) = . Again for any (from Theorem 3.3(1)) = (from Theorem 3.2(7) = . □

If is a grill on X and is a minimal space enjoys the property I, then from Kuratowski Closure operator , we get an unique topology on X which is given by following:

Theorem 3.6

Let be a grill minimal space and enjoys the property I. Then is a topology on X, where .

3.2 Properties of the topology

Theorem 3.7

(a)	If and are two grills on X with , then .

(b)	If is a grill on a set X and , then B is closed in .

(c)	For any is -closed.

Proof

(a)	Let . Then . Thus , since is closed set. Implies that (from Theorem 3.2(8)). So , and hence .

(b)	It is obvious that, for . Then = B. Hence B is -closed.

(c)	We have, . Thus is -closed.□

Here we find a simple open base for the topology on X.

Theorem 3.8

Let be a grill minimal space. Then and is an open base for .

Proof

Let and . Then is - closed so that , and hence . Then and so there exists such that . Let , then and . Thus . It now suffices to observe that is closed under finite intersections. Let , that is and . Then and . Now, , proving ultimate that is an open base for . □

Corollary 3.9

For any grill and any minimal structure on .

4 -operator

In this section we shall introduce another operator on grill minimal space. We shall also discuss the properties of the same operator in the front of the topology which has been obtained in the previous section. At first we shall give the definition:

Definition 4.1

Let be a grill minimal space. An operator is defined as follows for every : there exists a such that and observe that .

The properties of -Operator has two types. One type of property holds in grill minimal space. Another type of property holds in restricted minimal space.

Theorem 4.2

Let be a grill minimal space. Then following properties hold:

1.	If , then is -open.

2.	If , then .

3.	If , then .

4.	If , then .

5.	Let , then if and only if .

Proof

1.	Obvious from definition.

2.	Obvious from Theorem 3.2(7).

3.	If , then is -closed which implies and hence .

4.	This follows from (1) and (3).

5.	This follows from the facts: (i) . (ii) .□

Corollary 4.3

Let be a grill minimal space. Then for every .

Theorem 4.4

Let be a grill minimal space and enjoys the property I. Then

1.	If , then .

2.	If , then .

3.	If , then (where denote the interior operator of ).

4.	If and , then .

5.	If and , then .

6.	If and , then .

Proof

1.	It is obvious from Theorem 4.2(2), and . Hence . Now, let . Then there exists such that and . Let and we have and (from definition of -operator). Thus (from definition of grill), and hence . We have shown that . Hence the prove is completed.

2.	We know from Theorem 3.3(1), if . Then .

3.	If , then and there exists a such that . Then by Theorem 3.8, is a -open neighborhood of x and . Conversely suppose that , there exists a basic -open neighborhood of x where and , such that which implies that and hence . Hence .

4.	(since ) = .

5.	(from 4)= .

6.	Assume . Let and . Observe that (from definition of grill). Also observe that . Thus (from (4) and (5)).□

Now we shall show that the property I is an essential condition for the previous theorem.

Example 4.5

Here we consider the Example 3.4. Let and . Then and . Hence we have:

.

Therefore we conclude that the topological space is not the only space for discussing the properties of -Operator. Al-Omari and Noiri (2012a) is also a suitable space for the same. Moreover grill minimal space is also a suitable space.

Notes

Peer review under responsibility of University of Bahrain.