A New Closed Set in Topological Spaces

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. %is 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 semipreopen 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, AlSaadi [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 s􏽢 g-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 gsclosed sets. Applying the sets, we introduce a new space called T s􏽢g-space. Let us review some of the standard facts on near closed sets as the next.


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. is 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 semipreopen 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 T * 1/2 -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 T * 1/2 -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 gsclosed sets. Applying the sets, we introduce a new space called T sg -space. Let us review some of the standard facts on near closed sets as the next.
and U is open set in (Y, τ). e semiclosure [12,14] of a subset B of (Y, τ), denoted by sCl Y (B), briefly sCl(B), is defined to the intersection of all semiclosed sets containing B. e semi-interior of B [12], denoted by sInt(B), 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 T g -space.

Basic Properties of sg-Closed Sets
In section 2, we give a brief exposition of sg-closed sets, some of their properties, and relations with other known classes of subsets.
Theorem 1. Let Y be a nonempty set that carries topology τ. en, the next declaration is verified.
the proving is clear.

□
Remark 1. Figure 1 shows the connections results among sg-closed and different types of other sets.
In Remark 1, the relationships cannot be reversible as the next instance shown.

Remark 2.
e following examples from (i) to (iii) show that the concept of sg-closedness is independent from g-closedness, g closedness, sg-closedness, and spg-closedness.
{ } is sg-closed but neither g-closed nor g-closed in (Y, τ). (ii) Let (Y, τ) be the same space in Example 1(ii). A subset 2 { } is semiclosed and sg-closed but it is not sg-closed. (iii) Let τ 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.
Let B and C be two sg-closed subsets of (Y, τ). en,    e intersection of two sg-closed sets need not be sg-closed as the following instance shows.    Proof. Let F be a g-closed of (Y, τ) and F⊆B. en, Y\F is g-open and Y\B ⊆ Y\F. Since Y\B is sg-closed, sCl(Y\B)⊆Y\F, that is, Y\sInt(B)⊆Y\F. Hence, F⊆sInt(B).

sg-Open Sets
Conversely, let G 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  Proof. Let F be a g-closed set of (Y, τ) such that F⊆sCl(B)\B. Since B is sg-closed, by eorem 4, F � ϕ that F ⊂ sInt(sCl(B))\B and sCl(B)\B is sg-open.
As an application of the concept of sg-closed sets, we introduce the following space called T sg -space.

Remark 5.
e spaces T sg and T g are independent as seen from the next instances.

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 named T sg -space which have been found to be very beneficial in the study of certain objects of digital topology. us, we may stress once more the importance of sg-closed sets as a branch of them and the possible application in computer [15][16][17] and quantum.
Data Availability e data are included only in references of manuscript.

Conflicts of Interest
e author declares that there are no conflicts of interest.