# A New Closed Set in Topological Spaces

Authors: A. A. Azzam
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Our purpose of this work is to implement a class ofsg^-closed sets, which is property placed among the classes of semiclosed sets and gs-closed sets. The relations with other concepts directly or indirectly joined with generalized closed sets are inspected. In addition, as an application, using the notion of sg^-closed sets, we give a brief expansion of a new space named Tsg^-space.


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## Body

## 1. Introduction and Preliminaries

Topological space has become one of the most important spaces that help to solve many of the contemporary problems. It is the reference for measuring the descriptive rather than quantitative. Also, closed sets are essential matters in a space that carries topology. For instance, one can know the topology on a set by using either the axioms for the closed sets or the Kuratowski closure axioms. In 1970, Levine [1] initiated the study of so-called generalized closed sets. By definition, a subset B of the space Y which carries topology τ is named generalized closed set if the closure of any subset B of Y is included in each open superset of B. This notion has been studied extensively in recent years by many topologists because generalized closed sets are not only natural generalizations of closed sets. Furthermore, the study of generalized closed sets also extends new depictions of some known classes of spaces, for instance, the class of extremely disconnected spaces. In 1987, Bhattacharyya and Lahiri [2] introduced a new class of sets named semigeneralized closed sets using semiopen sets of Levine [3] which obtained various properties corresponding to [1]. In 1990, Arya and Nour [4] defined the generalized semiclosed sets. Dontchev [5] has introduced gsp-closed sets by generalizing semi-preopen sets. Generalized closed sets in bitopological spaces have been introduced by Fukutake [6]. More recently, Kumar [7] introduced g^-closed sets in topological spaces. John and Sundaram [8] introduced and studied the concept of g∗-closed sets and T1/2∗-space concerning bitopological space. Noiri and Rajesh [9] introduced generalized closed sets with respect to an ideal in bitopological spaces. Noiri and popa [10] studied the relation between ∗-closed sets and I-g-closed sets in ideal topological spaces. In 2018, Al-Saadi [11] discussed the concept of strongly g∗-closed sets and strongly T1/2∗-spaces in bitopological spaces. In the present work, we introduce the notion of sg^-closed sets and find some basic properties for it. We are also showing that this class lies among the class of semiclosed sets and the class of gs- closed sets. Applying the sets, we introduce a new space called Tsg^-space. Let us review some of the standard facts on near closed sets as the next.Definition 1.
A subsetB of a space Y which carries topology τ is named as follows:(i)
Semiclosed [12] if IntClB⊆B,(ii)
Semiopen [3] if Y\B is semiclosed or equivalently, if B⊆ClIntB,(iii)
Semi-preclosed [13] if IntClIntB⊆B,(iv)
Semi-preopen [13] if Y\B is semi-preclosed or equivalently, if B⊆ClIntClB.(v)
Generalized closed [1] if ClB⊆Y whenever B⊆U and U is open set in Y,τ. The semiclosure [12, 14] of a subset B of Y,τ, denoted by sClYB, briefly sClB, is defined to the intersection of all semiclosed sets containing B. The semi-interior of B [12], denoted by sIntB, is defined by the union of all semiopen sets contained in B. A number of definitions and depictions have been handled in [1, 2, 4, 5, 7] with respect to generalized closed sets or Tg^-space.


## 2. Basic Properties ofsg^-Closed Sets

In section2, we give a brief exposition of sg^-closed sets, some of their properties, and relations with other known classes of subsets.Definition 2.
LetY be a nonempty set that carries topology τ; a subset B of Y is called sg^-closed if sClB⊆G whenever B⊆G and G is a g^-open set in Y,τ.Theorem 1.
LetY be a nonempty set that carries topology τ. Then, the next declaration is verified.(i)
Each semiclosed set issg^-closed in Y,τ(ii)
Eachg-closed set is sg^-closed in Y,τ(iii)
Eachsg^-closed set is gs−closed and hence gsp−closed in Y,τ(iv)
Eachαg^-closed set is sg^-closed in Y,τProof.
(i)
LetB be a semiclosed set and G be any g^-open containing B. Since B is semiclosed, sClB=B for each subset B of Y. Therefore, sClB⊆G, and hence B is sg^-closed.(ii)
It is obvious from inclusionsClB⊂ClB for each subset B of Y,τ.(iii)
It is clear thatspClB⊆sClB for each subset B of Y,τ, and each open set is g^-open.(iv)
FromsClB⊆αClB for each subset B of Y,τ, the proving is clear.Remark 1.
Figure1 shows the connections results among sg^-closed and different types of other sets.Figure 1
Relationships between some classes of closed sets.In Remark1, the relationships cannot be reversible as the next instance shown.Example 1.
LetY=1,2,3 carry topology τ.(i)
Ifτ=ϕ,1,2,Y, then 1,3 is sg^-closed but neither semiclosed nor ∗g-closed.(ii)
Ifτ=ϕ,1,3,1,3,Y, then 1 is gs‐closed and gsp closed but not sg^-closed.(iii)
Ifτ=ϕ,1,3,1,Y, then 3 is sg^-closed but not ∗g-closed. Also, 1,2 is sg^-closed but not semiclosed.Remark 2.
The following examples from (i) to (iii) show that the concept ofsg^-closedness is independent from g^-closedness, g closedness, sg-closedness, and spg-closedness.(i)
LetY=1,2,3 and τ=ϕ,1,1,3,Y. In a space Y,τ, a subset 1,2 is sg^-closed but it is not spg-closed. Also, a subset 3 is sg^-closed but neither g‐closed nor g^-closed in Y,τ.(ii)
LetY,τ be the same space in Example 1(ii). A subset 2 is semiclosed and sg-closed but it is not sg^-closed.(iii)
Letτ be the usual topology on the real line ℝ. One can deduce that the open interval 1,2 is sg‐closed but not sg^-closed.Theorem 2.
LetB and C be two sg^-closed subsets of Y,τ. Then,(i)
The union of twosg^-closed sets is also sg^-closed in Y,τ(ii)
IfB is g^-open and sg^-closed, then B is semiclosed in Y,τ(iii)
IfB is sg^-closed in Y,τ, then sClB/B does not contain any nonempty g^-closed set(iv)
IfB is sg^-closed of Y,τ such that B⊆C⊆sClB, then C is also sg^-closed of Y,τ(v)
For eachx∈Y,x is g^-closed or X\x is sg^-closed of Y,τ(vi)
Every subset issg^-closed of Y,τ if and only if every g^-open set is semiclosedProof.
(i)
LetB and C be two sg^-closed sets and U be any g^-open set such that B∪C⊆U. Then, sClB∪C=sClB∪sClC⊆U.(ii)
SinceB is g^-open and sg^-closed, sClB⊆B. Hence, B is semiclosed.(iii)
Assume that there exists ag^-closed set F such that F⊆sClB/B. Then, B⊆X\sClB and so sClB⊆Y\F. Hence, we have that F⊆Y\sClB and F=ϕ because F⊆sClB.(iv)
LetU be a g^-open set such that C⊆U. Then, we have that sClB⊆U and ClC⊆sClB⊆U. Therefore, C is sg^-closed in Y,τ.(v)
Ifx is not g^-closed, then Y\x is not g^-open. Therefore, X\x is sg^-closed in Y,τ.(vi)
LetU be a g^-open set. Then, we have that sClU⊆U and hence U is semiclosed. Conversely, let B be a subset and U a g^-open set such that B⊆U. Then, sClB⊆sClU=U, and hence B is sg^-closed.Theorem 3.
A subsetB of Y,τ space is sg^-closed if and only if sClA/B contains no nonempty g^-closed set in Y,τ.Proof.
LetF be a nonempty g^-closed set of Y,τ such that F⊆sClB\B. Since Y\F is g^-open and B is sg^-closed, sClB⊆X\F, i.e., F⊆Y\sClB. Then, F⊆sClB∩Y\sClB=ϕ and hence F=ϕ, which is a contradiction.
Conversely, letG be a g^-open set such that A⊆G. Suppose that sClB is not contained in G. Then, by Remark 1, sClB∩X,G is nonempty g^-closed and sClB∩Y,G⊆sClB\B, which is a contradiction.Remark 3.
The intersection of twosg^-closed sets need not be sg^-closed as the following instance shows.Example 2.
LetY=1,2,3,4 with a topology τ=ϕ,1,2,1,2,1,2,3,Y. Put B=1,2,4 and C=1,3,4. Then, B and C are sg^-closed but B∩C=1,4 is not sg^-closed in Y,τ.


## 3.sg^-Open Sets

In the third section, we explain the definition ofsg^-open and related by sg^-closed with some examples.Definition 3.
A subsetB of a space Y which carries topology τ is named sg^-open if Y\B is sg^-closed.Remark 4.
For A subsetB of a space Y which carries topology τ, sClY\B=Y\sIntB.Theorem 4.
A subsetB is sg^-open in Y,τ if and only if F⊆sIntB, whenever F⊆B and F is g^-closed in Y¯,τ.Proof.
LetF be a g^-closed of Y,τ and F⊆B. Then, Y\F is g^-open and Y\B⊆Y\F. Since Y\B is sg^-closed, sClY\B⊆Y\F, that is, Y\sIntB⊆Y\F. Hence, F⊆sIntB.
Conversely, letG be a g^-open set of Y,τ and Y\B⊆G. Since Y\G is a g^-closed set contained in A, by hypothesis Y\G⊆sIntB, Y\sIntY⊆Y\G. By Remark 4, sClY\B⊆G. Hence, Y\A is sg^-closed, and so B is sg^-open.Theorem 5.
A subsetB is sg^-closed in Y,τ if and only if sClB\B is sg^-open.Proof.
LetF be a g^-closed set of Y,τ such that F⊆sClB\B. Since B is sg^-closed, by Theorem 4, F=ϕ that F⊂sIntsClB\B and sClB\B is sg^-open.
Reciprocally, letB⊆G, we have G as g^-open in Y,τ. Now, sClB∩Y,G⊂sClB∩Y,B=sClB\B and since sClB∩Y,G is g^-closed and sClBB is sg^-open, sClB∩Y,G⊂sIntsClB\B=ϕ. Then, sClB∩Y,G=ϕ or sClB⊆G, and hence B is sg^-closed.Example 3.
Let=1,2,3 with topology τ=ϕ,Y,1,1,2. In Y,τ, the set 2 is sg^-open.
As an application of the concept ofsg^-closed sets, we introduce the following space called Tsg^-space.Definition 4.
A spaceY which carries topology τ is called a Tsg^-space if each sg^-closed set is closed in Y.Remark 5.
The spacesTsg^ and Tg^ are independent as seen from the next instances.Example 4.
LetY=1,2,3 with topology τ=ϕ,Y,1. Then, the space Y,τ is Tg^-space but not a Tsg^-space.Example 5.
LetY=1,2,3 with τ=ϕ,Y,1,2,3. Then, Y,τ is Tsg^-space but not Tg^-space.


## 4. Conclusion

Popularizations of closed sets in point-set topology will give some new topological properties (for instance, separation axioms, compactness, connectedness, and continuity) and a brief expansion of a new space namedTsg^-space which have been found to be very beneficial in the study of certain objects of digital topology. Thus, we may stress once more the importance of sg^-closed sets as a branch of them and the possible application in computer [15–17] and quantum.



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