Alpha Weakly Semi Closed Sets in Topological Spaces

Accepted 12/Aug/2018, Online 30/Aug/2018 Abstract – N. Levine introduced the concept of generalized closed (briefly g-closed) sets in topology. Researches in topology studied several versions of generalized closed sets and they characterized that sets. In this paper, we introduce a new class of closed sets which is called Alpha weakly semi closed sets in topological spaces and we study the relationships of this set with some other generalized closed sets. Also we study some of its basic properties.


I. INTRODUCTION
In 1970, Levine [1] introduced generalized closed (briefly g-closed) sets in topology. Researches in topology studied several versions of generalized closed sets. In 2000, M. Sheik John [2] introduced and investigated wclosed sets in topology. In 2017, Veeresha A Sajjanar [3] introduced weakly semi closed sets and investigated some of their properties. In this paper, Section I contains the concept of Alpha Weakly semi-closed (briefly αws-closed) set is introduced and their properties are investigated. Section II contains the Certain preliminary concepts, Section III contain the concept of αwsclosed set is studied and a diagram also included which states the relationships among the generalized closed sets in topological spaces and Section IV contains the conclusions and Section V contains the references.

II. PRELIMINARIES
Throughout this paper X and Y represents the topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a topological space X, clA and intA denote the closure of A and the interior of A respectively. X -A denotes the complement of A in X. We recall the following definitions.   [32] if every gs-closed set is closed.

Lemma 2.: [6]
For any subset A of X, the following results hold:

III. ALPHA WEAKLY SEMI CLOSED SETS
In this section, we introduce a new type of closed sets namely αws-closed sets in topological spaces and study some of their properties.

Definition 3.1:
A subset A of a space X is called Alpha Weakly Semi closed (briefly αws-closed) if αclA ⊆ U whenever A ⊆ U and U is ws-open.
(v) Every gsp)*-closed set is αws-closed. Proof: Therefore αclA  U. Hence A is αws-closed in X. (iii) Let A be a π -closed subset of X. Since every π-closed set is closed [19] and by (i), we have A is αws-closed. (iv) Let A be a regular-closed subset of X. Since every regular-closed set is closed [8] and By (i), we have A is αws-closed.
The concept " -closed" is independent from the concepts "g-closed", "gr-closed", "g-closed", "rg-closed", "g # p # -closed", "g-closed", " g-closed" as seen in the following Examples 3.11 & 3.12 Example 3.11: Let X = {a, b, c} with topology τ ={ϕ,{a},{a, b},X}. {b} is -closed but not g-closed and {a, c} is g-closed but not -closed. {b} is -closed but not gr -closed and {a, c} is gr -closed but not -closed. {b} is -closed but not g -closed and {a, c} is g -closed but not -closed. {c} is -closed but not rg-closed and {a, b} is rg-closed but not -closed. {c} is -closed but not g # p # -closed and {b, d} is g # p # -closed but not -closed. {c} is -closed but not g-closed and {a, b, d} is g-closed but not -closed. {c} is -closed but not g-closed and {b, d} is g-closed but not -closed.
Thus the above discussions lead to the following diagram. In this diagram, "A→B" means A implies B but not conversely and "A B" means A and B are independent of each other. The union of two αws-closed subsets of X is αws-closed set.

Proof:
Let A and B be any two αws -closed sets in X. Let [35].

Theorem: 3.14
If a subset A of X is αws-closed in X, then αcl A A does not contain any non-empty ws-closed set in X.

Proof:
Let A be a αws-closed set in X and F be a ws-closed subset of αclA -A.
Since A is αwsclosed set and X F is ws-open, then αclA  X -F ie then F  X -αcl A We have F  αclA. Therefore, F  (X -αclA)  αclA = ϕ. Thus F ϕ.
Hence αclA -A does not contain any non-empty ws-closed set in X.

Theorem: 3.15
If a subset A is αws-closed set in X and, A  B  αclA, then B is also αws-closed set.

Proof:
Let A be a αws-closed set in X such that A  B αcl (A). To prove B is also αws-closed set in X. It is enough to prove αcl (B) Therefore αcl (X -{x}) is either X -{x} or X. Therefore Take αcl (X {x}) = X -{x}, then X {x} is α-closed. By Preposition 3.2 (i) every α-closed set is αws-closed, X-{x} is αws closed. This is contradiction to our assumption. Therefore Let X and Y are topological spaces and A Y X. Suppose that A is αws-closed set in X then A is αws-closed relative to Y.