θ-GENERALIZED SEMI-OPEN AND θ-GENERALIZED SEMI-CLOSED FUNCTIONS

In this paper, we introduce and study the notions of θ-generalizedsemi-open function, θ-generalizedsemi-closed function,pre-θ-generalizedsemi-open function,pre-θ-generalized-semi-closed function, contra preθ-generalized-semi-open,contra pre-θ-generalized-semi-closed function and θ-generlized-sem-homeomorphism in topological spaces and study their properties. 2000 Mathematics Subject Classification : 54A05, 54C10, 54D10; Secondary: 54C08


INTRODUCTION
In 1970, Levine [10] first considered the concept of generalized closed (briefly, g-closed) sets were defined and investigated Arya and Nour [1] defined generalized semi-open (briefly, gs-open) sets using semi openness and obtained some characterization of s-normal space.The generalizations of generalized closed and generalized continuity were intensively studied in recent years by Balachandran, Devi ,Maki and Sundaram [2].Recently in [12] the notion of θ-generalized semi closed (briefly, θgs-closed) set was introduced.The aim of this paper is to introduce the notions of θ-generalized semi open (briefly, θgs-open) function, θ-generalized semi closed (briefly, θgs-closed) function, θ-generalized-semi-homeomorphisms and study their simple properties.

PRELIMINARIES
Through out this paper(X, τ ) and (Y, σ)(or simply X and Y)denote the spaces on which no separation axioms are assumed unless explicitly stated.If A is any subset of X, then Cl(A) and Int(A) denote the closure of A and the interior of A in X respectively.Definition 2.1 : A subset A of a topological space X is called (i) a semi-open set [9] if A ⊂ Cl(Int(A)), (ii) a semi-closed set [5] if A ⊂ Int(Cl(A)).

Definition 2.2 :
The semi-closure [5] of a subset of X is the intersection of all semiclosed sets that contain A and is denoted by sCl(A).

Definition 2.3 :
The θ-closure of a set A is denoted by Cl θ (A) [16] and is defined by (ii) gc-homeomorphism [11] if f and f −1 are gc-irresolute.Definition 2.9 : A subset A of a topological space X is called θ-generalizedsemi closed (briefly, θgs-closed) [12] We denote the family of θgs-closed sets of X by θGSC(X, τ ) and θgsopen sets by θGSO(X, τ ) .Definition 2.10 : For every set A ⊆ X, we define θgs-closure of A [12]to be the intersection of all θgs-closed sets containing A and is denoted by θgsCl (A).
Since every θ-g-closed set as well as semi-θ-closed set is θgs-closed set and hence we have, A ⊂ θgsCl(A) ⊂ Cl θ (A) and Definition 2.12 : A space X is called T θgs -space [13] if every θgs-closed set in it is closed set.Theorem 2.13 [13]: Intersection of arbitrary collection of θgs-closed sets is θgs-closed set.Definition 2.15 [15]: A topological space (X, τ ) is called (i)θgs−T 0 if for any pair of distinct points x and y of X, there exists a θgs-open set containing x but not y or a θgs-open set containing y but not x.
(ii)θ gs-T 1 if for every pair of distinct points xand y of X, there exists a θgs-open set containing x but not y and a θgs-open set containing y but not x.(iii)θ gs-T 2 if for each pair of distinct points x and y of X, there exist disjoint θgs-open sets, one containing x and the other containing y .

θgs-OPEN and θgs-CLOSED
Proof : Let y 1 and y 2 be two distinct points of Y .Since f is bijective, there exists distinct points x 1 and x 2 of X such that f (x 1 ) = y 1 and f (x 2 ) = y 2 .Since X is a θgs − T 1 space , there exists θgs-open sets G and H such that x 1 ∈ G and x 2 6 ∈ G and x 2 ∈ H and x 1 6 ∈ H. Again since X is T θgs -space, G and H are open sets in X.As f is θgs-open function, f (G) and f (H) are θgs-open sets such that Theorem 3.6 : The property of a space being θgs − T 0 space is preserved under one-one , onto , pre-θgs-open function and hence is a topological property.
Proof : Let X be a θgs − T 0 space and Y be any other topological space.Let f : X → Y be one-one, onto, pre-θgs-open function from X to Y. Let y 1 , y 2 ∈ Y with y 1 6 = y 2 and since f is one-one , onto,there exists distinct points and hence Y is θgs − T 0 space.Again as the property of being θgs − T 0 is preserved under one-one, onto mapping , it is also preserved under homeomorphism and hence is a topological property.Theorem 3.7 : Let f : X → Y be a θgs-closed and g: Y → Z pre-θgs-closed and continuous, then their composition gof: X → Z is θgs-closed.
Proof : Let A be a closed set of X.Then by hypothesis f (A) is θgsclosed set in Y .Since g is pre-θgs-closed and continuous, by Theorem 3.5 g(f (A) = (gof)(A) is θgs-closed in Z. Hence (gof ) is θgs-closed.

CONTRA PRE-θgs-CLOSED and CONTRA PRE-θgs-OPEN FUNCTIONS
The following example shows that contra pre-θgs-closedness and contra pre-θgs-openness are independent.
There are two cases to be considered: Let (1) ,( 2) and (3) be properties of a function f : X → Y as follows.
Proof : (i)(1) → (2).Let B be the subset of Y and O be a θgs-open subset of X with f (2) → (1).Let E be a θgs-closed subset of X.

Remark 2. 14 :
(i) Any intersection of θgs-closed sets is θgs-closed set.Hence, by complement, any union of θgs-open sets is θgs-open.(ii) Union of θgs-closed sets may fail to be θgs-closed set.

Example 4 . 3 :Remark 4 . 4 :Theorem 4 . 5 :
Let X = Y = {a, b, c},τ = {X, φ, {a} , {b} , {a, b} , {a, c} , {b, c}} and σ = {Y, {a} , {b} , {a, b}}.We have θgs-open sets in X are {X, {a} , {b, c}} and θgs-open sets in Y are {Y, {a} , {b} , {a, c} , {b, c}}.Letf : X → Y is defined by f (a) = f (c) = c, f (b) = b and g: X → Y is by g(a) = g(b) = a,g(c) = b.Then f is contra pre-θgs-open but not contra pre-θgs-closed and g is contra pre-θgs-closed but not contra pre-θgs-open.Contra pre-θgs-closedness and contra pre-θgs-openness are equivalent if the function is bijective.For a function f : X → Y the following properties are equivalent.(i) f is contra pre-θgs-open.(ii) For every subset B of Y and every θgs-closed subset F of X with f −1 (B) ⊆ F , there exists a θgs-open subset O of Y with B ⊆ O and f −1 (O) ⊆ F .(iii) For every point y ∈ Y and every θgs For the case where f −1 (y) = φ , we have f −1 (y) = φ ⊆ F .For both cases , we can use (iii) and get the following: there exists a θgs-openset O y ⊆ Y such that y ∈ O y and f −1 (O y ) ⊆ F = A C .Namely, (*) f −1 (O y )∩A = φ holds for each y ∈ [f (A)] C .Finally we claim that [f (A)] C = ∪ n O y : y ∈ [f (A)] C o .Obviously, we have that [f (A)] C ⊆ ∪ n O y : y ∈ [f (A)] C o .Conversely, let z ∈ ∪ n O y : y ∈ [f (A)] C o .Then there exists a point w ∈ [f (A)] C such that z ∈ O w .Suppose that z / ∈ [f (A)] C .Then z ∈ f (A) and there exists a point b ∈ A such that f (b) = z.Thus we have that f (b) ∈ O w and so b ∈ f −1 (O w ).We have a contradiction to (*) above , that is , b ∈ f −1 (O w )∩A.Hence , we show that ∪ n O y : y ∈ [f (A)] C o ⊆ [f (A)] C and so [f (A)] C = ∪ n O y : y ∈ [f (A)] Co .Consequently, by Remark 2.12 (i), f(A) is a θgs-closed subset of Y .Theorem 4.6 :

2 . 3 .
For every set B of Y and every θgs-open subset O of X with f −1 (B) ⊆, there exists a θgs-closed subset O of Y with B ⊆ O and f −1 (O) ⊆ F .For every point y ∈ Y and every θgs-open subset O of X with f −1 (y) ⊆ O, there exists a θgs-closed subset F of Y with y ∈ F and f −1 (F ) ⊆ O. Then (i) The implications (1) → (2) → (3)hold.

Theorem 4 . 7 :Theorem 4 . 9 :
Let f : X → Y and g : Y → Z are θgs-closed functions and Y be T θgs − space.Then their composition gof is θgs-closed.Proof : Let A be a closed set of X.Then by hypothesis f(A) is a θgsclosed set in Y .Since Y is T θgs − space, f (A) is closed in Y. Since g is θgs-closed , g(f (A)) is θgs-closed in Z.But g(f (A)) = (gof )(A).Hence gof is θgs-closed.Theorem 4.8 : The composition of a closed functions f : X → Y and g : Y → Z is θgs-closed function from X to Z. Proof : Trivial.Let f : X → Y and g : Y → Z be two functions such that their composition gof : X → Z is θgs-closed function.Then the following statements holds; i) If f is continuous and surjective , then g is θgs-closed.