Weak forms of open and closed functions via θgs-open sets. Md.Hanif.Page, Department of Mathematics BVB College of Engineering and Technology, Hubli-580031. Karnataka State, India. e-mail:
[email protected] Abstract: In this paper, we introduce and study two new classes of functions by using the notions of θgs-open sets and θgs-closure operator called weakly θgs-open and weakly θgsclosed functions.The connections between these functions and other existing well-known related functions are investigated. Mathematics Subject Classification 2000: 54A40,54C08, 54D10,54C10 Key words: θgs-open set, θgs-closed set,weakly θgs-closed functions.
1
Introduction
In 1987, Di Maio and Noiri [2] initiated a brief study of the concepts of semi-θ-open and semi-θ-closed sets which provide a formulation of semi-θ-closure of a set in a topological space. Recently in [5] the notion of of θ-generalized semi closed (briefly,θgs-closed)set was introduced using semi-θ-closure. The concepts of θgs-open and θgs-closed sets which provide a formulation of θgs-closure of a set in a topological space. The aim of this paper is to present the class of weakly θgs-openness(resp. θgs-closedness) as new genaralisation of θgs-openness(resp.θgs-closedness). we investigate some of the fundamental properties of this class of functions, with respect to theses notions.
2
Preliminaries
Throughout this paper (X, τ ), (Y, σ)(or simply X, Y )denote topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a space X the closure and interior of A with respect to τ are denoted by Cl(A) and Int(A) respectively. Definition 2.1 A subset A of a space X is called (1) a semi-open set [3] if A ⊂ Cl(Int(A)). (2) a semi-closed set [1] if Int(Cl(Int(A))) ⊂ A.
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M D Hanif Page, Journal of Global Research in Mathematical Archives, 1(12), December 2013, 55-60
Definition 2.2 [2] A point x ∈ X is called a semi-θ-cluster point of A if sCl(U )∩A 6= φ, for each semi-open set U containing x. The set of all semi-θ-cluster point of A is called semi-θ-closure of A and is denoted by sClθ (A). A subset A is called semi-θ-closed set if sClθ (A) = A. The complement of semi-θ-closed set is semi-θ-open set. Definition 2.3 [5] A subset A of X is θgeneralized semi-closed(briefly, θgs-closed)set if sClθ (A) ⊂ U whenever A ⊂ U and U is open in X. The complement of θgs-closed set is θgeneralized-semi open (briefly,θgs-open).The family of all θgs-closed sets of X is denoted by θGSC(X,τ ) and θgs-open sets by θGSO(X,τ ). Definition 2.4 [5] The intersection of all θgs-closed sets containing a set A is called θgs-closure of A and is denoted by θgsCl(A). A set A is θgs-closed if and only if θgsCl(A) = A. Definition 2.5 [5] The union of all θgs-open sets contained in A is called θgs-interior of A and is denoted by θgsInt(A). A set A is θgs-open if and only if θgsInt(A) = A. Definition 2.6 [7] A function f : X → Y is said to be θgs-open (resp., θgs-closed) if f (V ) is θgs-open (resp., θgs-closed) in Y for every open set (resp., closed) V in X. Definition 2.7 [9] A function f : X → Y is called (i) totally θgs-continuous function at a point x ∈ X if for each open subset V in Y containing f(x), there exists a θgs-clopen subset U in X conaining x such that f (U ) ⊂ V . (ii) totally θgs-continuous if it has this property at each point of X. Definition 2.8 [10] A function f : X → Y is said to be Almost contra θgs-continuous if f −1 (V ) is θgs-closed in X for each regular open set V in Y . Definition 2.9 [11] A function f : X → Y is said to be contra θgs-continuous if f −1 (V ) is θgs-closed in X for each open set V of Y
3
Weakly θgs-Open Functions
We define in this section the concept of weak θgs-openness Definition 3.1 A function f : X → Y is called Weakly θgs-Open if f(U)⊂ θgsInt(f(Cl(U))) for each open set U of X. Theorem 3.2 Every θgs-open function is also Weakly θgs-Open, but the converse is not generally true as illustrated by the following example. Example 3.3 Let X = {a, b, c} equipped with toplogy τ = {X, φ, {b}, {c}, {a, b}, {a, c}}. We have θGSO(X) = {X, φ, {b}, {c}, {b, c}, {a, c}}.Define a function f : X → X by f (a) = a,f (b) = b and f (c) = c. Then f is weakly θgs-open but not θgs-open function. Because for an open set {a, b} in X, f ({a, b}) = {a, b} is not θgs-open set in X.
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Theorem 3.4 For a function f : X → Y the following are equivalent (i) f is weaklly θgs-open. (ii) f (sIntθ ⊂ θgsInt(f (A)) for every subset A of X. (iii) f (sIntθ (f −1 (B)) ⊂ f −1 (θgsInt(B)) for every subset B of Y . Proof: (i)⇒(ii) Let A be an subset of X and let x ∈ sIntθ (A)). Then, there exists an open set U such that x ∈ U ⊂ Cl(U ) ⊂ A. Then, f (x) ∈ f (U ) ⊂ f (Cl(U )) ⊂ f (A). Since f is f is weaklly θgs-open, f (U ) ⊂ θgsInt(f (Cl(U ))) ⊂ θgsInt(f (A)). It implies that f (x) ∈ θgsInt(f (A)). This shows that x ∈ f −1 (θgsInt(f (A))). Thus sIntθ (A) ⊂ f −1 (θgsInt(f (A))) and so f (sIntθ (A)) ⊂ θgsInt(f (A)). (ii) ⇒(i). Let U be an open set in X. As U ⊂ θgsInt(Cl(U )) implies, f (U ) ⊂ f (θgsInt(Cl(U ))) ⊂ θgsInt(f (Cl(U ))). Hence f is weakly θgs-open. (ii) ⇒(iii). Let B be any subset of Y. Then by (ii), f (θgsInt(f −1 (B)) ⊂ θgsInt(B). Therefore θgsInt(f −1 (B)) ⊂ f −1 (θgsInt(B)). Theorem 3.5 If X be a regular space. Then f : X → Y is a weakly θgs-open if and only if for every open set W of X, f (W ) = θgsInt(f (W )) Proof: The sufficiency is clear. Necessisty. Let W be nonempty open subset of X. For each x in W . Let Ux be an open set such that x ∈ Ux ⊂ Cl(Ux ) ⊂ W . Hence we obtain that W = ∪ {Ux : x ∈ W } and f (W ) = ∪ {f (Ux ) : x ∈ W } ⊂ ∪ {θgsInt(f (Cl(Ux ))) : x ∈ W } ⊂ θgsInt(f (∪ {Cl(Ux ))) : x ∈ W }) = θgsInt(f (W )). Theorem 3.6 If f : X → Y is weakly θgs-open ifff for every subset A of X f (Intθ ) ⊂ θgsInt(f (A)). Proof: Let A ba a non empty subset of X and x ∈ Intθ . Then there exists an open set U such that x ∈ U ⊂ Cl(U ) ⊂ A. Then f (x) ∈ f (U ) ⊂ f (Cl(U )) ⊂ f (A). Since f is weakly θgs-open, so f (U ) ⊂ θgsInt(f (Cl(U ))) ⊂ θgsInt(f (A)). This shows that f (x) ∈ θgsInt(f (A)). Thus x ∈ f −1 (θgsInt(f (A))) ⇒ Intθ ⊂ f −1 (θgsInt(f (A))) ⇒ f (Intθ ) ⊂ θgsInt(f (A)). Conversely, let f (Intθ ) ⊂ θgsInt(f (A)) for every A ⊂ X. Let U be any open set in X. Now since U ⊂ Intθ (Cl(U )), therefore f (U ) ⊂ f (Intθ )(Cl(U ))) ⊂ θgsInt(f (Cl(U ))). Hence f is weakly θgs-open. Theorem 3.7 A function f : X → Y is weakly θgs-open if for each x ∈ X and each open set U of X containing x, there exists a θgs-open set V of Y containing f(x) such that V ⊂ f (Cl(U )). Proof: Let U ba a non empty open subset of X and x ∈ U .Then there exists a θgs-open set V containing f(x) such that V ⊂ f (Cl(U )).This implies that f (x) ∈ V = θgsInt(f (V )) ⊂ θgsInt(f (Cl(U ))).Thus f (U ) ⊂ θgsInt(f (Cl(U )))f is and hence weakly θgs-open. Theorem 3.8 For a function f : X → Y the following are equivalent (i) f is weaklly θgs-open. (ii) θgsCl(f (Int(F ))) ⊂ f (F ) for each closed F of X. (iii) θgsCl(f (Int(F ))) ⊂ f (Cl(F )) for each open set U of X.
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Proof:(i)⇒(ii) Let F be an closed set of X and we have Y − f (F ) = f (X − F ) ⊂ θgsInt(f (Cl(X − F ))) = θgsInt(f (X − Int(F ))) = θgsInt(Y − f (Int(F ))) = Y − θgsInt(f (Int(F ))). Hence θgsInt(f (Int(F ))) ⊂ f (F ). (ii)⇒(iii) Let U be any open set in X. Then U ⊂ Int(Cl(U )) and Cl(U )is a closed set in X.Therefore by (ii),we have θgsCl(f (U )) ⊂ θgsCl(f (Int(Cl(U )))) ⊂ f (Cl(U )). (iii)⇒(i) Let U be an open set in X. Y − θgsInt(f (Cl(U )))= θgsCl(Y − f (Cl(U )))= θgsCl(f (X − Cl(U )))=θgsCl(f (Int(X − U ))) ⊂ f (Cl(X − U ))=f (X − U )=Y − f (U ). Thus f (U ) ⊂ θgsInt(f (Cl(U ))) and hence f is weakly θgs-open. Theorem 3.9 If f : X → Y is weakly θgs-open and strongly continuous, then for each open set U of X, f (U )=θgsInt(f (U )). Proof:Let U be a non empty open subset of X. Since f is stongly continuous so f (Cl(U )) ⊂ f (U ). Also since f is weakly θgs-open, f (U ) ⊂ θgsInt(f (Cl(U ))) ⊂ θgsInt(f (U )) and thus f (U ) = θgsInt(f (U )). Definition 3.10 A functionf : X → Y is said to be contra θgs-closed if f(F) is θgs-open set for each closed set F of X. Theorem 3.11 If f : X → Y is contra θgs-closed then f is weakly θgs-open. Proof: Let U be a non empty open subset of X. As f is contra θgs-closed, therefore f (F ) = θgsInt(f (F )) for each closed set F of X. Hence we have f (U ) ⊂ f (Cl(U ))= θgsInt(f (Cl(U ))). Thus f is weakly θgs-open. Theorem 3.12 Let X be a hyperconnected space then a function f : X → Y is weakly θgs-open iff f (X) = θgsInt(f (X)) as a subset of Y. Proof: Let f(X)= θgsInt(f (X)) as a subset of Y. Let U be an open subset of X. Then f (U ) ⊂ f (X)= θgsInt(f (X))= θgsInt(f (Cl(U ))). The converse is obvious.
4
Weakly θgs-Closed Functions
We define in this section the concept of weakly θgs-closed function. Definition 4.1 A function f : X → Y is said to be Weakly θgs-Closed if θgsCl(f (Int(F ))) ⊂ f (F ) for each closed set F of X. Remark 4.2 Clearly every θgs-Closed funation is weakly θgs-Closed function, but converse is not true. Example 4.3 In the example 3.3, f is weakly θgs-closed but not θgs-closed function because for closed set {c} , f ({c}) = {c} is not θgs-closed set in X. Theorem 4.4 For a function f : X → Y the following are equivalent (i) f is weaklly θgs-closed. (ii) θgsCl(f (U ))) ⊂ f (Cl(U )) for each open set U of X.
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Proof:(i)⇒(ii) Let U be an open subset of X. Then θgsCl(f (U )) = θgsCl(f (Int(U ))) ⊂ θgsCl(f (Int(Cl(U )))) ⊂ (f (Cl(U ))). (ii)⇒(i) Let F be closed subset of X. Then θgsCl(f (Int(F ))) ⊂ f (Cl(Int(F ))) ⊂ (f (Cl(F )))=f (F ). Theorem 4.5 For a function f : X → Y if θgsCl(f (U )) ⊂ f (Cl(U )) for each regular open subset U of X then for each subset F in Y and each open set U in X with f −1 (F ) ⊂ U , there exist a θgs-open set A in Y with f ⊂ A and f −1 (F ) ⊂ Cl(U ). Proof: Let F be a subset in Y and U be open in X with f −1 (F ) ⊂ U . Then f −1 (F ) ⊂ U ∩ Cl(X − Cl(U )) = φ and consequently F ∩ f (Cl(X − Cl(U ))) = φ. Since X − Cl(U ) is regular open, F ∩ θgsCl(f (X − Cl(U ))) = φ. Let A = Y − θgsCl(f (X − Cl(U ))). Then A is θgs-open with F ⊂ A and f −1 (A) ⊂ X − f −1 (θgsCl(f (X − Cl(U )))) ⊂ X − f −1 (f (X − Cl(U ))) ⊂ Cl(U ). Theorem 4.6 If f : X → Y is injective weakly θgs-closed function, then for every subset F in Y and every open set U in X with f −1 (F ) ⊂ U ,there exists a θgs-closed set B in Y such that and f −1 (B) Cl(U ). Proof: Let F be subset of Y and U ba an open subset of X with f −1 (F ) ⊂ U .Put B = θgsCl(f (Int(Cl(U )))), then B is θgs-closed set of Y such that F ⊂ B. Since F ⊂ f (U ) ⊂ f (Int(Cl(U ))) ⊂ θgsCl(f (Int(Cl(U )))) = B. By weakly θgs-closedness of f, it follows thatf −1 (B) ⊂ Cl(U ).
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[5] Govindappa Navavalgi and Md. Hanif Page,On θgs-Neighbourhoods, Indian Journal of Mathematics and Mathematical Sciences, Vol.4, No.1,(June-2008), 21-31. [6] Govindappa Navavalgi and Md. Hanif Page,On θgs-Open and θgs-Closed functions,Proycecciones Journal of Mathematics, Vol.28,(April-2009), 111-123. [7] Govindappa Navavalgi and Md. Hanif Page,On Some separation axioms via θgs-open sets,Bulletin of Allahabad Mathematical Society, Vol.25,Part 1,(2010), 13-22.
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[8] Md. Hanif Page,On Some more properties of θgs-Neighbourhodds,American Journal of Applied Mathematics and Mathematical Analysis, Vol.1, No.2,(2012). [9] Md. Hanif Page,On Totally θgs-continuous Functions,InternationalJournal of Pure and Applied Mathematical Sciences Volume 6, No. 2 (2013), pp. 177-184 [10] Md. Hanif Page,On Almost contra θgs-continuous Functions,Genaral Mathematics Notes,Vol. 15,No. 2, April, 2013, pp 45-54. [11] Md. Hanif Page,On Contra θgs-continuous Functions,International Journal of Mathematics Trends and Technology Volume 5, January- 2014, 16-21.
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