Applications of some operators on supra topological spaces

Abstract In this paper, the notion of an operator γ \gamma on a supra topological space ( X , μ ) (X,\mu ) is studied and then utilized to analyze supra γ \gamma -open sets. The notions of μ γ {\mu }_{\gamma } -g.closed sets on the subspace are introduced and investigated. Furthermore, some new μ γ {\mu }_{\gamma } -separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator γ \gamma on μ \mu are presented and investigated. Finally, we give some properties of S ( γ , β ) {S}_{(\gamma ,\beta )} -closed graph and strongly S ( γ , β ) {S}_{(\gamma ,\beta )} -closed graph.


Introduction
Kasahara [1] defined an operator associated with a topology, namely, an α operator. He initiated some concepts that are equivalent to those given in topological spaces when the operator is the identity operator. Also, he studied α-closed graphs of α-continuous functions and α-compact spaces. Then, Jankovic [2] used α operator to introduce α-closure of a set and give some characterizations on α-closed graph of functions. Later, Ogata [3] defined the notion of γ-open sets to study operator-functions and operator-separation. Rosas and Vielma [4] investigated some features of operator-compact spaces and defined the concept of operator-connected spaces. Kalaivani and Krishnan [5] formulated the concept ofα γ-open sets in a topological space and studied their corresponding closure and interior operators. Quite recently, many notions of operators have been investigated on different classes of open sets and generalizations of open sets; see [6][7][8][9][10][11][12][13][14][15].
In 1983, Mashhour et al. [16] introduced supra topological spaces (STSs) by neglecting an intersection condition of topology. This makes supra topology (ST) more flexible to describe some real-life problems (see, [17]) and construct easily some examples that show the relationships between certain topological concepts. Al-shami [18] investigated the classical topological notions such as limit points of a set, compactness, and separation axioms on the STSs. Investigation of several types of compactness and Lindelöfness was the goal of some papers such as [19][20][21][22]. Al-shami [23] introduced the concept of paracompactness on STSs and explored main properties. Recently, the authors of [24][25][26][27] have employed some generalizations of supra open sets given in [28][29][30][31] to study limit points and separation axioms on STSs. They have provided various interesting examples to explain the given relationships and results. In [32][33][34], some new concepts and notions were introduced using supra b-open sets and supra D-open sets.
This paper is organized as follows: after this introduction, we recall some basic definitions that are necessary to understand this work. In Section 3, an operator γ depending on supra open sets is studied and then employed to analyze supra γ-open sets. In Section 4, we introduce and discuss μ γ -g.closed sets on subspace ST. In Section 5, some new μ γ -separation axioms are formulated using the operator γ on μ and the relationships between them are elucidated. In Section 6, some new classes of functions are defined and some characterizations of these functions are given. In Section 7, two new classes of closed graphs are studied and some relations and properties are obtained. Section 8 concludes the paper with summary.

Preliminaries
Let X be a non-empty set and ( ) P X be the power set of X.
Definition 2.1. [16] A subfamily μ of ( ) P X is called an ST if it is closed under arbitrary union and X is a member of μ.
Then the pair ( ) X μ , is called an STS. Terminologically, a member of μ is called a supra open set and its complement is called a supra closed set. , is said to be: (i) supra T 0 (briefly, S-T 0 ) if ∀ ≠ ∈ x y X, ∃ ∈ U μ such that either ∈ x U and ∉ y U or ∈ y U and ∉ x U; (ii) supra T 1 (briefly, S-T 1 ) if ∀ ≠ ∈ x y X, ∃ ∈ U V μ , with ∈ x U but ∉ y U and ∈ y V but ∉ x V; (iii) supra T 2 (briefly, S-T 2 ) if ∀ ≠ ∈ x y X, ∃ ∈ U V μ , with ∈ x U, ∈ y V and ∩ = U V ϕ; (iv) supra regular if for every supra closed set F and every ∉ a F, there exist disjoint supra open sets U and V containing F and a, respectively; (v) supra normal if for every disjoint supra closed sets F and H, there exist disjoint supra open sets U and V containing F and H, respectively; (vi) S-T 3 (resp. S-T 4 ) if it is both supra regular (resp. supra normal) and S-T 1 .

Supra γ -open sets and operators
In this section, we introduce and study the concept of γ operator on an ST. Then, we define supra γ-regular and supra open operators and investigate main properties. We construct some examples to show the obtained results.
, be an STS. An operator γ on an ST μ is a mapping from μ to ( ) P X such that ⊆ ( ) U γ U ∀ ∈ U μ, where ( ) γ U denotes the value of γ at U. This operator will be denoted by → ( ) γ μ P X : .
Applications of some operators on supra topological spaces  293 , be an STS and → ( ) γ μ P X : be an operator on μ.
Suppose that the empty set ϕ is also supra γ-open set for any operator → ( ) γ μ P X : . We denote the class of all supra γ-open subsets of an STS ( ) X μ , by μ γ .
The identity operator id on μ is a mapping But the converse of this relation is not true as illustrated in the following example.
, is an STS. Define an operator → ( ) γ μ P X : as follows: Remark 3.7. Lemma 3.5 demonstrates that μ γ is an ST on X, and Example 3.6 shows that μ γ is not always a topology.
, be any STS. An operator γ on μ is said to be supra regular if ∀ ∈ x X and ∀ ∈ be the mapping defined by: .
Thus, it can easily check that → ( ) γ μ P X : is a supra regular operator.
be supra regular operator on μ.
is a supra regular operator on μ, then ∃ ∈ W μ containing x such that Remark 3.11. If γ is a supra regular operator on μ, then μ γ is a topology on X. be the mapping defined by: Clearly, γ is not a supra regular operator on μ. Thus, is not a topology on X.
Theorem 3.14. Let ( ) X μ , be an STS and → ( ) γ μ P X : be an operator on μ. Then the following statements are equivalent: Again, by using (2) for the set W, we obtain ∈ W μ γ such that ∈ x W and ⊆ W U .
Definition 3.16. Let A be any subset of an STS ( ) X μ , and γ be an operator on μ. Then 2. The supra γ-closure of A is denoted by -( ) μ cl A γ μ and is defined as A F : is supra closed set in and . γ μ Theorem 3.17. Let A be any subset of an STS ( ) X μ , and γ be an operator on μ. Then ∈ -( ) This is a contradiction. Hence, the proof is complete.
Conversely, let ∉ -( ) . This is a contradiction. Therefore, ∈ -( ) , be an STS and γ be an operator on μ. Then the following statements are true for any subsets ⊆ A B X , , and γ be a supra regular operator on μ. Then Proof.
(1) It follows directly from Lemma 3.18 Then it is enough to Since γ is a supra regular operator on μ, then by Theorem 3.10, This means that ∉ -( ∪ ) Since γ is a supra regular operator on μ, The disjoint of ( ∪ ) , be an STS and γ be a supra regular operator on μ.
, and γ is an operator on μ, then the next four properties are equivalent: Theorem 3.23. Let A be any subset of an STS ( ) X μ , . If γ is a supra open operator on μ, then 1.

closed sets and operator on subspace ST
Through this section, we present the concept of μ γ -generalized closed and give some characterizations.
be an STS and γ be an operator on μ.
Then by hypothesis, is supra open.
is a supra open operator, then by Theorem 3.23 By using the assumption of the converse of Theorem 4.3, , and let γ be an operator on μ. Then A is supra is supra γ-closed and A is μ γ -g.closed. It follows from Theorem 4.3 that , be an STS and γ be an operator on μ. If A is μ γ -g.closed and supra γ-open subset of X, then A is supra γ-closed.
X μ , and γ be an operator on μ. Then the μ γ -kernel of A, denoted by μ γ -ker(A), is defined as follows: Conversely, let Now we define an operator on subspace ST as follows: be an operator on μ. We define the restriction of γ to μ A , denoted by γ A , to be the mapping from μ A into ( ) x U A and , A be a subspace of an STS ( ) X μ , and μ A be the restriction of γ to μ A . If the mapping γ is supra open and the set B is supra This completes the proof. □ , and μ A be the restriction of γ to μ A . If the mapping γ is supra open, the set B is μ γ A -g closed in A and A is μ γ -g closed in X, then B is μ γ -g closed in X. . Therefore, B is μ γ -g.closed in X. □ Corollary 4.16. If γ is supra open, A is μ γ -g.closed in X and F is supra γ-closed in X, then ∩ A F is μ γ -g.closed in X.

Proof. By Lemma 4.12, ∩
A F is supra γ A -closed in A and hence ∩ A F is μ γ -g.closed in A. Thus, by Theorem 4.15, . Therefore, B is μ γ A -g closed in A. □ In this section, we investigate some types of μ γ -separation axioms. Some results and examples of these spaces are studied.
, with an operator γ on μ is said to be: with ≠ x y, ∃ ∈ U μ such that either ∈ x U and ∉ ( ) y γ U or ∈ y U and ∉ ( ) with ≠ x y, ∃ ∈ U μ γ such that either ∈ x U and ∉ y U or ∈ y U and ∉ x U.
if every μ γ -g.closed set in X is supra γ-closed.
Conversely, let F be any μ γ -g.closed set in the STS ( ) X μ , . We have to show that F is supra γ-closed (i.e., x X. So there are two cases: , be a - * μ T γ 0 space. Thus, ∃ ∈ U μ γ with ∈ x U and ∉ ( ) y γ U . Since γ is a supra open operator on μ, then ∃ ∈ W μ γ with ∈ x W and ⊆ ( ) , .Therefore, we obtain that Proof. (Necessity) Let X be aμ T γ 0 space and ∈ x y X , with ≠ x y. Then ∃ ∈ U μ γ (say ∈ x U, but ∉ y U). So X U \ is a supra γ-closed set, which does not contain x, but contains y.
Theorem 5.6. For an STS ( ) X μ , with an operator γ on μ. Then the following conditions are true: , be an STS and γ be an operator on μ. Then the following statements are equivalent: , and any operator γ on μ, the following properties hold.
Proof. The proofs are obvious from their definitions and hence they are omitted.

□
Observe that the converse of each part of Theorem 5.8 is not true as shown by the following examples.
be an operator on μ defined as follows: or or , or , ; otherwise.
Example 5.13. Consider = { } X a b c , , and = ( ) μ P X . Define an operator γ on μ as follows: if or , ; otherwise. X a b a b a c b c  , , , , , , , , , . Define an operator → ( ) γ μ P X : Remark 5.15. By Theorem 5.6 and Theorem 5.8, we obtain the following diagram of implications.
(1) Let ∈ ( ( )) y f cl A γ μ and ∈ V ν in Y with ∈ y V. Then by hypothesis, ∃ ∈ x X and ∃ ∈ U μ in X with ∈ x U and (2) Let F be any supra β-closed set of ( ) Y ν , . So, . Then by using (1), we have , is supra β-regular, we prove the implication: , is a supra β-regular space, then by Theorem 3.14, ∈ V ν β in Y. By This implies that ( ( )) ⊆ ⊆ ( ) f γ U V β V . Therefore, f is ( ) S γ β , -continuous. Now, when β is a supra open operator, we show the implication: , is said to be 1. ν β -closed if the image of each supra γ-closed set of X is supra β-closed in Y.

( )
S id β , -closed if the image of each supra closed set of X is supra β-closed in Y.

If
, has an ( ) S γ β , -closed graph, then it has an S-closed graph. Proof.
Proof. We have to show that is supra associated operator with γ and γ, and it is supra regular with γ and γ. An STS ( ) X μ , is - * μ T γ 2 iff the diagonal set = {( ) ∈ } x x x X Δ , : is supra ρ-closed of ( × × ) X X μ μ , .