ON COVERING PROPERTIES IN TERMS OF GENERALIZED FORM OF PREOPEN SETS

In this paper, we introduce the concept of a new class of sets viz. γ̂-preopen sets using an operation γ to preopen sets. This class of sets contains the class of preopen sets. Such sets are used to introduce and study some separation as well as covering axioms. AMS Subject Classification: 54A05, 54A10, 54C10, 54D20


Introduction and Preliminaries
Many classical topological notions such as continuity of functions, compactness and connectedness have been extended by using preopen sets replacing the role of open sets.From the literature, we see that the study of topological properties via preopenness has gained significant importance in General Topology.In this paper, we used operation approach on preopen sets in a different way to introduce a new kind sets known as γ-preopen sets, which contains the class of preopen [16] sets in a topological space.The concept of operation in a topological was first introduced by Kasahara [12] and it was further investigated by Jankovic [10].Then, Ogata [19] defined the concept of γ-operation in a topological space and consequently introduced γ-open sets.
By a space X we denote a topological space (X, τ ) without any separation axiom.For a subset A of X, we denote by int(A) and cl(A) the interior and clo-sure of A respectively.A subset A of X is called preopen [16] if A ⊂ int(cl(A)).Set of all preopen sets in X is denoted by P O(X) and that of containing a point x of X, by P O(X, x).Let A be a subset of space X, then pre-interior of A, denoted by pint(A) and defined as the union of all preopen sets contained in A and pre-closure of A denoted by pcl(A) and defined as the intersection of all preclosed sets containing A. A topological space (X, τ ) is called submaximal [2] if every dense subset of X is open equivalently, every preopen set is open.A topological space X is said to be strongly compact [3] if every cover of X by preopen sets admits a finite subcover.A topological space X is said to be p-closed [1] if for every preopen cover ) and every F ∈ F. A topological space (X, τ ) is said to be strongly p-regular [8] (resp.p-regular [6]) if for each point x ∈ X and each preclosed set (resp.closed set) F such that x ∈ F , there exist disjoint preopen sets U and V such that x ∈ U and F ⊂ V .A topological space (X, τ ) is called strongly normal [17] if for each pair of disjoint preclosed sets A and B of X, there exist disjoint preopen sets U and V containing them.A function f : (X, τ ) → (Y, τ ′ ) is called preirresolute [20] if f −1 (V ) is preopen in X for every preopen subset V of Y .
An operation γ [19] on a topology τ on X is a mapping γ : τ → P (X), such that V ⊂ V γ for each V ∈ τ , where P (X) is the power set of X and V γ denotes the value of γ at V .A subset A of X with an operation γ on τ is called γ-open [19] if for each x ∈ A, there exists an open set U such that x ∈ U and U γ ⊂ A. τ γ denotes the set of all γ-open sets in X. Clearly τ γ ⊂ τ .The τ γ -closure [19] of subset A of X is denoted by τ γ -cl(A) and is defined to be the intersection of all γ-closed sets containing S and τ γ -interior [13] of A is denoted by τ γ -int(A) and is defined as the union of all γ-open sets of X contained in A. A topological X with an operation γ on τ is said to be γ-regular [19] if for each x ∈ X and open neighborhood V of x, there exists an open neighborhood U of x such that U γ is contained in V .Let (X, τ ) and (Y, τ ′ ) be two topological spaces and γ be an operation on τ .Then a function f : (X, τ ) → (Y, τ ′ ) is said to be γ-continuous [5] at x if for each open set V containing f (x), there exists an γ-open set U in X containing the point x such that f (U ) ⊂ V .
In rest of this article by spaces X and Y we will denote topological spaces (X, τ ) and (Y, τ ′ ) with operations γ on τ and γ ′ on τ ′ respectively.By a function f : X → Y we mean a function f : (X, τ ) → (Y, τ ′ ) with operations γ on τ and γ ′ on Y respectively.
The following example shows that the above inclusions are proper in general.
Theorem 2.5.(a) Union of arbitrary family of γ-preopen sets is γpreopen.(b) If a topological space satisfies the property γ-p, then the intersection of any two γ-preopen sets is a γ-preopen set and γ-P O(X) forms a topology finer that τ .
Definition 2.6.A topological space X is said to be γ-submaximal every γ-preopen set in γ-open.
Theorem 2.7.If a topological space X is γ-submaximal then γ-P O(X) forms a topology coarser than τ .
Remark 2.8.The converse of the above theorem is not true.
We now state the following theorem without proof, which will be used in the consequent sections.
Theorem 2.11.For any subsets A and B of a topological space X, the following holds: Definition 2.12.A point x ∈ X is said to be a γ-pre-θ-accumulation point of a subset A of a topological space X if γ-pcl(U ) ∩ A = ∅ for every U ∈ γ-P O(X, x).
The set of all γ-pre-θ-accumulation points of a subset A of X is called γ-pre-θ-closure of A and is denoted by γ-pcl θ (A).
Proof.Let A be a γ-pre-θ-open and x ∈ A. Then (X − A) is γ-pre-θclosed set and so for each x ∈ A, there exists V ∈ γ-P O(X, x) such that γ-pcl(V ) ∩ (X − A) = ∅ and therefore γ-pcl(V ) ⊂ A.
Conversely, if the condition does not hold, there exists x ∈ A such that γ-pcl(V ) ⊂ A for all V ∈ γ-P O(X, x).This implies γ-pcl(V ) ∩ (X − A) = ∅ for all V ∈ γ-P O(X, x) and so x is a γ-pre-θ-accumulation point of (X − A).Hence (X − A) is not γ-pre-θ-closed.
Theorem 2.14.Let A and B by any two subsets a space X.Then the following properties hold: (a) Every θ-closed and pre-θ-closed sets are γ-pre-θ-closed. Proof.Straightforward.

Separation Axioms
In this section we focus our attention to the introduction of certain separation axioms utilizing the concepts developed in the earlier section and discuss some of their characterizations.Definition 3.1.A space X is called (a) γ-pre-T 0 if and only if for each pair of distinct points x, y ∈ X, there exists an U ∈ γ-P O(X, x) such that either x ∈ U and y / ∈ U or x / ∈ X and y ∈ U .(b) γ-pre-T 1 if and only if each pair of distinct points x, y ∈ X, there exists two γ-preopen sets U, V such that x ∈ U but y / ∈ U and y ∈ V but x / ∈ V .(c) γ-pre-T 2 if for each pair of distinct points x, y ∈ X, there exists U ∈ γ-P O(X, x) and V ∈ γ-P O(X, y) such that U ∩ V = ∅.(d) γ-pre-Urysohn if for each pair of distinct points x, y ∈ X, there exists U ∈ γ-P O(X, x) and V ∈ γ-P O(X, y) such that γ-pcl(U ) ∩ γ-pcl(V ) = ∅.Definition 3.2.A space X is called (e) γ-strongly pre-regular if for each point x ∈ X and each γ-preclosed set F not containing x, there exist disjoint γ-preopen sets U and V such that x ∈ U and F ⊂ V .(f) γ-strongly pre-normal if for any pair of disjoint γ-preclosed sets A, B of X, there exists disjoint γ-preopen sets U and V such that A ⊂ U and B ⊂ V .Remark 3.3.(a) The family of pre-T 0 [11] (resp.pre-T 1 [11], pre-T 2 [11], strongly pre-regular [6], strongly pre-normal [17]) spaces are contained in the family of γ-T 0 (resp.γ-pre-T 1 , γ-pre-T 2 , γ-strongly pre-regular, γ-strongly pre-normal) spaces.(b) The family of Urysohn [22] and pre-Urysohn spaces [15] are contained in γ-pre-Urysohn spaces.(c) The family of γ-pre-Urysohn spaces is contained in the family of γ-pre-T 2 spaces and the family of γ-pre-T 2 spaces is contained in the family of γ-pre-T 1 spaces.

Covering Axioms
Definition 4.1.A subset S of a space X is called γ-strongly compact (resp.γ-p-closed) relative to X if every cover of S by γ-preopen sets of X has a finite subfamily whose members (resp.γ-preclosures) cover S.
Proof.Let S be an arbitrary subset of X and {V i : i ∈ I} be a cover of S by γ-preopen sets of X.Then the family {V i : i ∈ I} ia γ-preopen cover of the γ-preopen set ∪{V i : i ∈ I}.Hence by hypothesis there is finite subfamily {V i j : j ∈ N 0 } which covers ∪{V i : i ∈ I}, where N 0 is finite subset of the naturals N.This subfamily is also a cover of the set S. Hence the theorem.Definition 4.4.A filter base F on a topological space X is said to be γ-pre-converge (resp.γ-pre-θ-converge) to a point x ∈ X if for each V ∈ γ-P O(X, x), there exists an A filter base F on a topological space X is said to be γ-pre-accumulate (resp.γ-pre-θ-accumulate) at x ∈ X if V ∩ F = ∅ (resp.γ-pcl(V ) ∩ F = ∅) for every V ∈ γ-P O(X, x) and every F ∈ F.
Theorem 4.5.The following conditions are equivalent for a topological space X: (a) X is γ-p-closed; (b) every ultrafilter base γ-pre-θ-converges at some point of X; (c) every filter base γ-pre-θ-accumulates at some point of X; (d) for every family {V α : α ∈ Λ} of γ-preclosed subsets such that Proof.(a) ⇒ (b): Suppose X is γ-p-closed and Ω is an ultrafilter base on X which does not γ-pre-θ-converges to any point of X.Now, Ω being an ultrafilter base on X, it can not γ-pre-θ-accumulate at any point of X.Then for each x ∈ X, there is an F x ∈ Ω and a V x ∈ γ-P O(X, x) such that γ-pcl(V x ) ∩ F x = ∅ and so the family {V x : x ∈ X} forms a cover of X by γ-preopen sets.Now, X being γ-p-closed, there exist finite number of points x 1 , x 2 , ..., x n ∈ X such that X = ∪ n i=1 γ-pcl(V x i ).Again since Ω is a filter base on X, there exists an F 0 ∈ Ω such that F 0 ⊂ ∩ n i=1 F x i and thus F 0 = ∅ -a contradiction.(b) ⇒ (c): Let F be any filter base on X.Then there is an ultrafilter base Ω containing F γ-pre-θ-converging to some point of x ∈ X.We take any V ∈ γ-P O(X, x) and F ∈ F, then there exists an F ′ ∈ Ω such that F ′ ⊂ γ-pcl(V ) and F ∩ F ′ = ∅.Therefore, ∅ = F ∩ F ′ ⊂ γ-pcl(V ) ∩ F and so the filter base F, γ-pre-θ-accumulates at x ∈ X.