ON GENERALIZED ω-CLOSED SETS

The class of ω-closed subsets of a space 
(X,τ) was defined to introduce ω-closed functions. The aim of this paper is to introduce and study the class of 
gω-closed sets. This class of sets is finer than g-closed 
sets and ω-closed sets. We study the fundamental properties 
of this class of sets. In the space (X,τω), the concepts closed set, g-closed set, and gω-closed set 
coincide. Further, we introduce and study gω-continuous and 
gω-irresolute functions.


Introduction
Throughout this work, a space will always mean a topological space on which no separation axiom is assumed unless explicitly stated.Let (X,τ) be a space and let A be a subset of X.A point x ∈ X is called a condensation point of A if for each U ∈ τ with x ∈ U, the set U ∩ A is uncountable.A is called ω-closed [10] if it contains all its condensation points.The complement of an ω-closed set is called ω-open.It is well known that a subset W of a space (X,τ) is ω-open if and only if for each x ∈ W, there exists U ∈ τ such that x ∈ U and U − W is countable.The family of all ω-open subsets of a space (X,τ), denoted by τ ω , forms a topology on X finer than τ.
In 1970, Levine [13] introduced the notion of generalized closed sets.He defined a subset A of a space (X,τ) to be generalized and closed (briefly g-closed) if cl τ (A) ⊆ U whenever U ∈ τ and A ⊆ U.
In Section 2 of the present work, we follow a similar line to introduce generalized ωclosed sets by utilizing the ω-closure operator.We study g-closed sets and gω-closed sets in the spaces (X,τ) and (X,τ ω ).In particular, we show that a subset A of a space (X,τ) is closed in (X,τ ω ) if and only if it is g-closed in (X,τ ω ) if and only if it is gω-closed in (X,τ ω ).
In Section 3, we introduce gω-continuity and gω-irresoluteness by using gω-closed sets and study some of their fundamental properties.Now we begin to recall some known notions, definitions, and results which will be used in the work.
Let (X,τ) be a space and let A be a subset of X.The closure of A, the interior of A, and the relative topology on A will be denoted by cl τ (A), int τ (A), and τ A , respectively.The ω-interior (ω-closure) of a subset A of a space (X,τ) is the interior (closure) of A in the space (X,τ ω ), and is denoted by int τω (A)(cl τω (A)).[13] if every g-closed set is closed (equivalently if every singleton is open or closed, see [8]).
It is clear that if (X,τ) is a countable space, then GωC(X,τ) = ᏼ(X), where ᏼ(X) is the power set of X. Proposition 2.2.Every g-closed set is gω-closed.
The proof follows immediately from the definitions and the fact that τ ω is finer than τ for any space (X,τ).However, the converse is not true in general as the following example shows.
Proof.We need to prove that cl τ (A) ⊆ cl τω (A).Suppose that there exists x ∈ cl τ (A) − cl τω (A).Then, x / ∈ cl τω (A), and so there exists , which is a contradiction and the result follows.
Theorem 2.6.Let (X,τ) be any space and A ⊆ X.Then the following are equivalent.
Proof.(a)⇒(b).It follows from the fact that every closed set is g-closed.
(b)⇒(c).It is obvious by using Proposition 2.2.(c)⇒(a).We show that cl τω (A ,τ).In the same way, it can be shown that a subset A of a space (X,τ) is closed if and only The proof is obvious.
Example 2.8.Let X = R be the set of all real numbers with the topology τ = {φ, X,{1}} and put
In Example 2.8, A ∈ GC(X,τ) − GC(X,τ ω ).In the following, we give an example of a space (X,τ) and a subset A of X such that A ∈ GC(X,τ ω ) − GC(X,τ).In other words, for a space (X,τ), the collections GC(X,τ) and GC(X,τ ω ) are independent from each other.
Example 2.9.Consider X = R with the usual topology τ u .Put A = (0,1) Note that in Example 2.9, (R,τ u ) is anti-locally countable and R,τ u ).Thus the condition that (A,τ A ) is anti-locally countable in Corollary 2.5 cannot be replaced by the condition that (X,τ) is anti-locally countable.
Proposition 2.10.Let A be a gω-closed subset of a space (X,τ) and B ⊆ X.Then the following hold.
(a) cl τω (A) − A contains no nonempty closed set.
Proof.(a) Suppose by contrary that cl τω (A) − A contains a nonempty closed set C. Then Proof.Let A be a gω-closed subset of (X,τ).By Theorem 2.6, we show that A is ωclosed in (X,τ).Suppose, to the contrary, that there exists x ∈ cl τω (A) − A. Then, by Proposition 2.10(a), {x} is not closed.Since (X,τ) is a T 1/2 -space, {x} is open in (X,τ), and thus it is ω-open.Therefore, {x} ∩ A = ∅, a contradiction.
In the space X from Example 2.3, every gω-closed set is ω-closed while (X,τ) is not a T 1/2 -space.Thus, the converse of Theorem 2.11 is not true in general.
Theorem 2.12.Let (X,τ) be an anti-locally countable space.Then (X,τ) is a T 1 -space if and only if every gω-closed set is ω-closed.
Proof.We need to show the sufficiency part only.Let x ∈ X and suppose that {x} is not closed.Then A = X − {x} is not open, and thus A is gω-closed (the only open set containing A is X).Therefore, by assumption, A is ω-closed, and thus {x} is ω-open.So there exists U ∈ τ such that x ∈ U and U − {x} is countable.It follows that U is a nonempty countable open subset of (X,τ), a contradiction.Proposition 2.13.If Ꮽ = {A α : α ∈ I} is a locally finite collection of gω-closed sets of a space (X,τ), then A = α∈I A α is gω-closed (in particular, a finite union of gω-closed sets is gω-closed).
The following two examples show that a countable union of gω-closed sets and a finite intersection of gω-closed sets need not be gω-closed.
Example 2.14.(a) Consider X = R with the usual topology τ u .For each n ∈ N, put (b) Let X be an uncountable set and let A be a subset of X such that A and X − A are uncountable.Let τ = {∅, A,X}.Choose x 0 ,x 1 / ∈ A and In [11], Hdeib shows that if A is an ω-open subset of a space (X,τ) and B is an ωopen subset of a space (Y ,σ), then To prove that the other inclusion always holds, we need the following lemma.
Theorem 2.17.Let (X,τ) and (Y ,σ) be two topological spaces.Then Proof.Let F A be a closed subset of (X,τ) and let F B be a closed subset of (Y ,σ) such that by using Theorem 2.17.Therefore, F A ⊆ int τω (A) and F B ⊆ int σω (A), and the result follows.
The converse of the above theorem need not be true in general.
Theorem 2.21.Let (Y ,τ Y ) be a subspace of a space (X,τ) and A ⊆ Y .Then the following hold. (a Proof.(a) Let V be an open set of (Y ,τ Y ) such that A ⊆ V .By using Lemma 1.3(b), there exists an open set X,τ).Therefore, the condition that Y is ω-closed in Theorem 2.21 (b) cannot be dropped.The proof follows from the definitions and Propositions 2.2 and 2.7.

gω-continuous functions
Example 3.3.(a) Let X be an uncountable set and let A be a proper uncountable subset of X.Let f : (X,τ indis ) → (X,τ dis ) be the identity function.Then f is gω-continuous (GC(X,τ indis ) = ᏼ(X)).However, f is not ω-continuous since A is closed in (X,τ dis ) and (b) Let (X,τ) be as in Example 2.3.Then, the identity function f : (X,τ) → (X,τ dis ) is gω-continuous but not g-continuous.
(a) f ω is gω-continuous if and only if it is g-continuous.
In Example 3.3(a), f is both gω-continuous and g-irresolute.However, f ω is neither g-continuous nor gω-irresolute ((τ indis ) ω = τ coc is the cocountable topology).Therefore, the converses of parts (b) and (c) of Theorem 3.5 are not true in general.Example 3.8 shows that also the converse of part (d) is not true.Proposition 3.6.Every gω-irresolute function is gω-continuous but not conversely.
The proof follows immediately from the definitions.For the converse, see Example 3.8.
The following example shows that the condition that f ω is gω-continuous in Theorem 3.7 cannot be weakened to f being gω-continuous.The converse of the above proposition is not true in general as the following example shows.

Lemma 2 .
16. (a) If A is an ω-open subset of a space (X,τ), then A − C is ω-open for every countable subset C of X.(b) The open image of an ω-open set is ω-open.Proof.Part (a) is clear.To prove part (b), let f : (X,τ) → (Y ,σ) be an open function and let W be an ω-open subset of (X,τ).Let y ∈ f (W).There exists x ∈ W such that y