a unified theory of weak contra-continuity

Acta Math. Hungar., 132 (1–2) (2011), 63–77 DOI: 10.1007/s10474-010-0046-2 First published online December 11, 2010

A UNIFIED THEORY OF WEAK CONTRA-CONTINUITY T. NOIRI1 and V. POPA2 1

2

2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan e-mail: [email protected]

Department of Mathematics, Univ. Vasile Alecsandri of Bacˇ au, 600114 Bacˇ au, Rumania e-mail: [email protected] (Received May 21, 2010; accepted June 15, 2010)

Abstract. We introduce a new notion called weakly contra-m-continuous functions as functions from a set satisfying some minimal conditions into a topological space. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory of several modifications of weak contra-continuity due to Baker [9].

1. Introduction Semi-open sets, preopen sets, α-open sets, β-open sets and b-open sets play an important role in the researches of generalizations of continuity in topological spaces. By using these sets many authors introduced and studied various types of generalizations of continuity. Dontchev [19] introduced the notion of contra-continuous functions. Recently, new types of contracontinuous functions are introduced and studied: for example, contra-αcontinuity [28], contra-semi-continuity [20], contra-precontinuity [28], contra b-continuity [3], contra-β-continuity [13]. A unified form of these classes of functions is studied in [43]. Quite recently, Baker has introduced the notions of weak contra-continuity [9], weak contra-precontinuity [11], and weak contra-β-continuity [10] in topological spaces. On the other hand, the present authors introduced and investigated the notions of m-continuous functions [51], weakly m-continuous functions [53], slightly m-continuous functions [52], and contra-m-continuous functions [43]. In this paper, we introduce the notion of weakly contra-m-continuous functions as functions from a set X satisfying some minimal conditions into Key words and phrases: m-structure, m-space, contra-continuity, weak contra-continuity, weak m-continuity, weak contra-m-continuity. 2000 Mathematics Subject Classification: 54C08. c 2010 Akad´ 0236-5294/$ 20.00 emiai Kiad´ o, Budapest, Hungary

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a topological space and investigate their properties and the relationships between weak contra-m-continuity and other related generalized forms of continuity. It turns out that the weak contra-m-continuity is a unified form of several modifications of weak contra-continuity due to Baker [9]. 2. Preliminaries Let (X, τ ) be a topological space and A a subset of X. The closure of A and the interior of A are denoted by Cl(A) and Int (A), respectively. A subset A is said to be regular closed (resp. regular open) if Cl Int (A) = A (resp. Int Cl (A) = A). Definition 2.1. Let (X, τ ) be a topological space. A subset A of X is said to be semi-open [32] (resp. preopen [35], α-open [39], β-open [1] or semi-preopen [4], b-open [5]) if A ⊂ Cl Int (A) (resp. A ⊂Int Cl(A) , A ⊂ Int (Cl Int(A) ), A ⊂ Cl (Int Cl (A) )), A ⊂ Cl Int (A) ∪ Int Cl (A) ). The family of all semi-open (resp. preopen, α-open, β-open, b-open) sets in X is denoted by SO (X) (resp. PO (X), α(X), β(X) or SPO (X), BO (X)). Definition 2.2. The complement of a semi-open (resp. preopen, αopen, β-open, b-open) set is said to be semi-closed [15] (resp. preclosed [35], α-closed [36], β-closed [1] or semi-preclosed [4], b-closed [5]). Definition 2.3. The intersection of all semi-closed (resp. preclosed, αclosed, β-closed, b-closed) sets of X containing A is called the semi-closure [15] (resp. preclosure [22], α-closure [36], β-closure [2] or semi-preclosure [4], b-closure [5]) of A and is denoted by sCl (A) (resp. pCl (A), αCl (A), β Cl (A) or spCl (A), bCl (A)). Definition 2.4. The union of all semi-open (resp. preopen, α-open, β-open, b-open) sets of X contained in A is called the semi-interior (resp. preinterior, α-interior, β-interior or semi-preinterior, b-interior ) of A and is denoted by sInt (A) (resp. pInt (A), αInt (A), β Int (A) or spInt (A), bInt (A)). Throughout the present paper, (X, τ ) and (Y, σ) denote topological spaces and f : (X, τ ) → (Y, σ) presents a (single valued) function from a topological space (X, τ ) into a topological space (Y, σ). Definition 2.5. A function f : (X, τ ) → (Y, σ) is said to be semi-continuous [32] (resp. precontinuous [35], α-continuous [36], β-continuous [1], b-continuous [21]) if for each point x ∈ X and each open set V of Y containing f (x), there exists a semi-open (resp. preopen, α-open, β-open, b-open) set U of X containing x such that f (U ) ⊂ V . Acta Mathematica Hungarica 132, 2011

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Definition 2.6. A function f : (X, τ ) → (Y, σ) is said to be contracontinuous [19] (resp. contra-semi-continuous [20], contra-precontinuous [28], contra-α-continuous [27], contra-β-continuous [13], contra-γ-continuous [38]) if f −1 (V ) is closed (resp. semi-closed, preclosed, α-closed, β-closed, b-closed) for every open set V of Y . Definition 2.7. A function f : (X, τ ) → (Y, σ) is said to be weakly continuous [31] (resp. weakly quasicontinuous [54] or weakly semi-continuous [7], [14], [30], almost weakly continuous [29] or quasi precontinuous [48], weakly α-continuous [40], weakly β-continuous [49], weakly b-continuous [56]) if for each x ∈ X and each open set V of Y containing f (x), there exists an open (resp. semi-open, preopen, α-open, β-open, b-open) set U of X containing x such that f (U ) ⊂ Cl (V ). A unified theory of weakly continuous functions is investigated in [53]. Definition 2.8. A function f : (X, τ ) → (Y, σ) is said to be slightly continuous [26] (resp. slightly semi-continuous [46], slightly precontinuous or faintly semi-continuous [47], slightly β-continuous [41], slightly bcontinuous [21]) if for each point x ∈ X and each clopen set V of Y containing f (x), there exists an open (resp. semi-open, preopen, β-open, b-open) set U of X containing x such that f (U ) ⊂ V . A unified theory of slightly continuous functions is investigated in [52]. Definition 2.9. A function f : (X, τ ) → (Y, σ) is said to be weakly contra-continuous [9] (resp. weakly contra-precontinuous [11], weakly contraβ-continuous [10]) if for set V of Y and each each open closed set A of Y such that A ⊂ V , Cl f −1 (A) ⊂ f −1 (V ) (resp. pCl f −1 (A) ⊂ f −1 (V ), spCl f −1 (A) ⊂ f −1 (V )). Similarly, we can obtain the following definitions: Definition 2.10. A function f : (X, τ ) → (Y, σ) is said to be weakly contra-semi-continuous (resp. weakly contra-α-continuous, weakly contraγ-continuous or weakly contra-b-continuous) if for each open set V of Y −1 and each closed set A of Y such that A ⊂ V , sCl f (A) ⊂ f −1 (V ) (resp. −1 −1 −1 αCl f (A) ⊂ f (V ), bCl f (A) ⊂ f −1 (V )). 3. Contra-m-continuous functions Definition 3.1. A subfamily mX of the power set P(X) of a nonempty set X is called a minimal structure (briefly m-structure) on X [50], [51] if ∅ ∈ mX and X ∈ mX . Acta Mathematica Hungarica 132, 2011

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By (X, mX ), we denote a nonempty subset X with a minimal structure mX on X and call it an m-space. Each member of mX is said to be mX open (briefly m-open) and the complement of an mX -open set is said to be mX -closed (briefly m-closed). Remark 3.1. Let (X, τ ) be a topological space. Then the families τ , SO (X), PO (X), α(X), β(X) or SPO (X), BO (X) are all m-structures on X. Definition 3.2. Let X be a nonempty set and mX an m-structure on X. For a subset A of X, the mX -closure of A and the mX -interior of A are defined in [34] as follows: (1) mCl (A) = {F : A ⊂ F, X \ F ∈ mX }, (2) mInt (A) = {U : U ⊂ A, U ∈ mX }. Remark 3.2. Let (X, τ ) be a topological space and A a subset of X. If mX = τ (resp. SO (X), PO (X), α(X), β(X), SPO (X), BO (X)), then we have (1) mCl (A) = Cl (A) (resp. sCl (A), pCl (A), αCl (A), β Cl (A), spCl (A), bCl (A)), (2) mInt (A) = Int (A) (resp. sInt (A), pInt (A), αInt (A), β Int (A), spInt (A), bInt (A)). Lemma 3.1 (Maki et al. [34]). Let X be a nonempty set and mX an m-structure on X . For subsets A and B of X , the following properties hold: (1) mCl (X \ A) = X \ mInt (A) and mInt (X \ A) = X \ mCl (A), (2) If (X \ A) ∈ mX , then mCl (A) = A and if A ∈ mX , then mInt (A) = A, (3) mCl (∅) = ∅, mCl (X) = X , mInt (∅) = ∅ and mInt (X) = X , (4) If A ⊂ B , then mCl (A) ⊂ mCl (B) and mInt (A) ⊂ mInt (B), (5) A ⊂ mCl (A) and mInt (A) ⊂ A, (6) mCl mCl (A) = mCl (A) and mInt mInt (A) = mInt (A). Definition 3.3. An m-structure mX on a nonempty set X is said to have property B [34] if the union of any family of subsets belonging to mX belongs to mX . Remark 3.3. Let (X, τ ) be a topological space. Then the families τ , SO (X), PO (X), α(X), β(X), SPO (X), BO (X) have all property B. Lemma 3.2 (Popa and Noiri [43]). Let X be a nonempty set and mX an m-structure on X satisfying property B . For a subset A of X , the following properties hold: (1) A ∈ mX if and only if mInt (A) = A, (2) A is mX -closed if and only if mCl (A) = A, (3) mInt (A) ∈ mX and mCl (A) is mX -closed. Acta Mathematica Hungarica 132, 2011


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Definition 3.4. A function f : (X, mX ) → (Y, σ) is said to be weakly m-continuous [53] (resp. m-continuous [51]) if for each point x ∈ X and each open set V of Y containing f (x), there exists U ∈ mX containing x such that f (U ) ⊂ Cl (V ) (resp. f (U ) ⊂ V ). Remark 3.4. Let f : (X, τ ) → (Y, σ) be a function and mX = τ (resp. SO (X), PO (X), α(X), β(X), BO (X)). (1) If f : (X, mX ) → (Y, σ) is m-continuous, then f is continuous (resp. semi-continuous, precontinuous, α-continuous, β-continuous, b-continuous), (2) If f : (X, mX ) → (Y, σ) is weakly m-continuous, then f is weakly continuous (resp. weakly semi-continuous, almost weakly continuous, weakly α-continuous, weakly β-continuous, weakly b-continuous). Definition 3.5. A function f : (X, mX ) → (Y, σ) is said to be contram-continuous [43] if f −1 (V ) = mCl f −1 (V ) for every open set V of Y . Remark 3.5. (1) Let f : (X, τ ) → (Y, σ) be a function and mX = τ (resp. SO (X), PO (X), α(X), β(X), BO (X)). If f : (X, mX ) → (Y, σ) is contra-m-continuous, then f is contra-continuous (resp. contra-semicontinuous, contra-precontinuous, contra-α-continuous, contra-β-continuous, contra-b-continuous), (2) The notions of m-continuity and contra-m-continuity are independent by Examples 2.1 and 2.2 of [13] and Examples 2.1 and 2.2 of [28]. Definition 3.6. A function f : (X, mX ) → (Y, σ) is said to be weakly contra-m-continuous iffor eachopen set V of Y and each closed set A of Y such that A ⊂ V , mCl f −1 (A) ⊂ f −1 (V ). Remark 3.6. Let f : (X, τ ) → (Y, σ) be a function and mX = τ (resp. SO (X), PO (X), α(X), β(X), BO (X)). If f : (X, mX ) → (Y, σ) is weakly contra-m-continuous, then f is weakly contra-continuous (resp. weakly contra-semi-continuous, weakly contra-precontinuous, weakly contra-α-continuous, weakly contra-β-continuous, weakly contra-b-continuous). Theorem 3.1. If a function f : (X, mX ) → (Y, σ) is contra-m-continuous, then f is weakly contra-m-continuous. Proof. Let V be any open set of Y and A a closed set of Y such that (V ) = mCl f −1 (V ) . ThereA ⊂ V . Since f is contra-m-continuous, f −1 −1 −1 fore, we have mCl f (A) ⊂ mCl f (V ) = f −1 (V ). This shows that f is weakly contra-m-continuous. Remark 3.7. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 3.1 we obtain the results established in Theorem 3.1 of [9] (resp. Theorem 3.3 of [11], Theorem 3.3 of [10]). Acta Mathematica Hungarica 132, 2011

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Theorem 3.2. If a function f : (X, mX ) → (Y, σ) is m-continuous, then f is weakly contra-m-continuous. Proof. Let V be any open set of Y and A a closed set of Y such that A ⊂ V . Since f is m-continuous, by Theorem 3.1 of [51] we have mCl f −1 (A) = f −1 (A) ⊂ f −1 (V ). Therefore f is weakly contra-m-continuous. Remark 3.8. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 3.2 we obtain the results established in Theorem 3.4 of [9] (resp. Theorem 3.2 of [11], Theorem 3.2 of [10]). Definition 3.7. A function f : (X, mX ) → (Y, σ) is said to be slightly m-continuous [52] if for each point x ∈ X and each clopen set V of Y containing f (x), there exists U ∈ mX containing x such that f (U ) ⊂ V . Theorem 3.3. If a function f : (X, mX ) → (Y, σ) is weakly contra-mcontinuous, then f is slightly m-continuous. Proof. Let V be a clopen set of Y . If we put A = V , then by the weak contra-m-continuity we have mCl f −1 (V ) ⊂ f −1 (V ) and hence −1 −1 mCl f (V ) = f (V ) by Lemma 3.1. It follows from Theorem 3.1 of [52] that f is slightly m-continuous. Remark 3.9. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 3.3 we obtain the results established in Theorem 3.7 of [9] (resp. Theorem 3.6 of [11], Theorem 3.6 of [10]). Lemma 3.3 (Popa and Noiri [52]). If a function f : (X, mX ) → (Y, σ) is weakly m-continuous, then f is slightly m-continuous. Remark 3.10. (1) By Theorems 3.1–3.3, Lemma 3.3 and Theorem 4.1 of [43], for a function f : (X, mX ) → (Y, σ) the following implications hold: ⎧ ⎪ ⎪ ⎨

(∗)

⎪ ⎪ ⎩ m-continuity

contra-m-continuity

weak m-continuity

weak contra-m-continuity

slightly m-continuity

(2) It follows from [9], [10], [11] and [52] that the implications in (∗) are strict. (3) Let f : (X, τ ) → (Y, σ) be a function and mX an m-structure on X. If mX = τ (resp. PO (X), β(X)), then by (∗) we obtain the diagram constructed in [9] (resp. [11], [10]). Acta Mathematica Hungarica 132, 2011


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Definition 3.8. A topological space (Y, σ) is said to be 0-dimensional [58] if each point of Y has a neighborhood base consisting of clopen sets. Let (Y, σ) be a topological space. Since the intersection of two clopen sets of (Y, σ) is clopen, the family of the clopen sets of (Y, σ) may be used as a base for a topology on Y . The topology is called the ultra-regularization [41] of σ and is denoted by σU . A topological space (Y, σ) is said to be ultraregular [23] if σ = σU . Theorem 3.4. Let (Y, σ) be 0-dimensional (resp. ultra-regular). Then for a function f : (X, mX ) → (Y, σ), the following properties are equivalent: (1) f is m-continuous; (2) f is weakly contra-m-continuous; (3) f is slightly m-continuous. Proof. The proof follows from (∗) and Theorem 4.3 (resp. Theorem 4.4) of [52]. Definition 3.9. A topological space (Y, σ) is said to be extremally disconnected [58] (briefly E.D.) if the closure of each open set of Y is open in Y . Theorem 3.5. If f : (X, mX ) → (Y, σ) is weakly contra-m-continuous and (Y, σ) is E.D., then f is weakly m-continuous. Proof. Let V be an open set of Y . Since (Y, σ) is E.D., Cl(V ) is open. Since f is weakly contra-m-continuous, mCl f −1 (V ) ⊂ mCl (f −1 Cl (V ) ) ⊂ f −1 Cl (V ) . Hence mCl f −1 (V ) ⊂ f −1 Cl (V ) for every open set V of Y . It follows from Theorem 3.1 of [53] that f is weakly m-continuous. Remark 3.11. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 3.5 we obtain the results established in Corollary 3.9 of [9] (resp. Corollary 3.10 of [11], Corollary 3.9 of [10]). 4. Weak contra-m-continuity and gm∗ -continuity Definition 4.1. Let (X, τ ) be a topological space. A subset A of X is said to be (1) g-closed [33] if Cl (A) ⊂ U whenever A ⊂ U and U ∈ τ , (2) αg-closed [17] if αCl (A) ⊂ U whenever A ⊂ U and U ∈ τ , (3) gs-closed [16] if sCl (A) ⊂ U whenever A ⊂ U and U ∈ τ , (4) gp-closed [6] if pCl (A) ⊂ U whenever A ⊂ U and U ∈ τ , (5) gsp-closed [18] if spCl (A) ⊂ U whenever A ⊂ U and U ∈ τ , (6) γg-closed [24] if bCl (A) ⊂ U whenever A ⊂ U and U ∈ τ . Acta Mathematica Hungarica 132, 2011

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Definition 4.2. Let (X, τ ) be a topological space and mX an mstructure on X. A subset A of X is said to be generalized m-closed (briefly gm-closed ) [42] if mCl (A) ⊂ U whenever A ⊂ U and U ∈ τ . Remark 4.1. Let (X, τ ) be a topological space and mX an m-structure on X. We put mX = τ (resp. SO (X), PO (X), α(X), β(X), BO (X)). Then, a gm-closed set is a g-closed (resp. gs-closed, gp-closed, αg-closed, gsp-closed, γg-closed) set. Definition 4.3. A function f : (X, τ ) → (Y, σ) is said to be g-continuous [12] or weakly g-continuous [37] (resp. gs-continuous [16], gp-continuous [6], αg-continuous [17], gsp-continuous [18], γg-continuous [24]) if f −1 (K) is a g-closed (resp. gs-closed, gp-closed, αg-closed, gsp-closed, γg-closed) set in X for every closed set K of Y . Definition 4.4. A function f : (X, mX ) → (Y, σ) is said to be gm∗ continuous [45] if f −1 (K) is gm-closed for every closed set K of Y . Remark 4.2. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. SO (X), PO (X), α(X), SPO (X), BO (X)) and f : (X, mX ) → (Y, σ) is gm∗ -continuous, then f is g-continuous (resp. gs-continuous, gp-continuous, αg-continuous, gsp-continuous, γgcontinuous). Definition 4.5. A function f : (X, τ ) → (Y, σ) is said to be approximately continuous [8] (resp. approximately semi-continuous, approximately precontinuous [11], approximately α-continuous, approximately β-continuous [10], approximately b-continuous) if Cl (A) ⊂ f −1 (V ) (resp. sCl(A) ⊂ f −1 (V ),

pCl(A) ⊂ f −1 (V ),

spCl (A) ⊂ f −1 (V ),

αCl (A) ⊂ f −1 (V ),

bCl (A) ⊂ f −1 (V ))

whenever V is open in Y and A is g-closed (resp. gs-closed, gp-closed, αgclosed, gsp-closed, γg-closed) in X such that A ⊂ f −1 (V ). Definition 4.6. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. A function f : (X, mX ) → (Y, σ) is said to be approximately m-continuous if mCl (A) ⊂ f −1 (V ) whenever V is open in Y and A is gmclosed in X such that A ⊂ f −1 (V ). Remark 4.3. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. SO (X), PO (X), α(X), SPO (X), BO (X)) and f : (X, mX ) → (Y, σ) is approximately m-continuous, then f is approximately continuous (resp. approximately semi-continuous, approximately precontinuous, approximately α-continuous, approximately β-continuous, approximately b-continuous). Acta Mathematica Hungarica 132, 2011


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Theorem 4.1. If f : (X, mX ) → (Y, σ) is a gm∗ -continuous and approximately m-continuous function, then f is weakly contra-m-continuous. Proof. Let V be an open set of Y and A a closed set of Y such that A ⊂ V . Since f is gm∗ -continuous, f −1 (A) is gm-closed. f −1 (A) −1 Since −1 ⊂ f (V ) and f is approximately m-continuous, mCl f (A) ⊂ f −1 (V ). This shows that f is weakly contra-m-continuous. Remark 4.4. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 4.1 we obtain the results established in Theorem 3.11 of [9] (resp. Theorem 3.11 of [11], Theorem 3.10 of [10]). Theorem 4.2. If f : (X, mX ) → (Y, σ) is weakly contra-m-continuous and f (A) is closed in Y for every gm-closed set A of X , then f is approximately m-continuous. Proof. Let V be any open set of Y and A a gm-closed set of X such that A ⊂ f −1 (V ). Then f (A) is closed and f (A) ⊂ V . Since f is weakly contra-m-continuous, mCl (f −1 f (A) ) ⊂ f −1 (V ) and hence mCl (A) ⊂ mCl (f −1 f (A) ) ⊂ f −1 (V ). This shows that f is approximately m-continuous. Remark 4.5. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 4.2 we obtain the results established in Theorem 3.12 of [9] (resp. Theorem 3.12 of [11], Theorem 3.11 of [10]). 5. Properties of weak contra-m-continuity Definition 5.1. A topological space (X, τ ) is said to be strongly Sclosed [19] (resp. strongly S-semi-closed, strongly S-preclosed [11], strongly S-α-closed, strongly Sβ -closed [10], strongly S-b-closed ) if every cover of X by closed (resp. semi-closed, preclosed, α-closed, β-closed, b-closed) sets of (X, τ ) has a finite subcover. Definition 5.2. An m-space (X, mX ) is said to be strongly S-m-closed if every cover of X by m-closed sets of (X, mX ) has a finite subcover. Remark 5.1. Let (X, τ ) be a topological space and mX = τ (resp. SO (X), PO (X), α(X), SPO (X), BO (X)). If (X, mX ) is strongly S-mclosed, (X, τ ) is strongly S-closed (resp. strongly S-semi-closed, strongly S-preclosed, strongly S-α-closed, strongly Sβ -closed, strongly S-b-closed). Acta Mathematica Hungarica 132, 2011

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Definition 5.3. A topological space (X, τ ) is called a PΣ -space [9] or C-space [10], [11] if for every open set U and each x ∈ U , there exists a closed set A such that x ∈ A ⊂ U . Theorem 5.1. Let f : (X, mX ) → (Y, σ) be a weakly contra-m-continuous function, let mX have property B and let (Y, σ) be a C -space. If (X, mX ) is strongly S -m-closed, then f (X) is compact. Proof. Let (X, mX ) be strongly S-m-closed and {Vα : α ∈ Δ} any cover of f (X) by open sets of (Y, σ). For each x ∈ X, there exists α(x) ∈ Δ such that f (x) ∈ Vα(x) . Since Y is a C-space, there exists a closed set Aα(x) such that f (x) ∈ Aα(x) ⊂ Vα(x) . Since f is weakly contra-m-continuous, mCl (f −1 (Aα(x) )) ⊂ f −1 (Vα(x) ). Since mX has property B, the family { mCl (f −1 (Aα(x) )) : x ∈ X } is an m-closed cover of X. Since X is strongly S-m-closed, there exists a finite number of points, say, x1 , x2 , . . . , xn in X such that X = { mCl (f −1 (Aα(xk ) )) : xk ∈ X, 1 k n}. Therefore, we obtain f (X) =

f mCl (f −1 (Aα(xk ) )) : xk ∈ X, 1 k n

⊂

{Vα(x

k

)

: xk ∈ X, 1 k n}.

This shows that f (X) is compact. Remark 5.2. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X), β(X)), then by Theorem 5.1 we obtain the results established in Theorem 4.1 of [9] (resp. Theorem 4.1 of [11], Theorem 4.11 of [10]). Definition 5.4. An m-space (X, mX ) is said to be m-T1 [44] if for any pair of distinct points x, y of X, there exist an m-open set of X containing x but not y and an m-open set of X containing y and not x. Remark 5.3. Let (X, τ ) be a topological space and mX = τ (resp. SO (X), PO (X), α(X), SPO (X), BO (X)). If (X, mX ) is m-T1 , then (X, τ ) is T1 (resp. semi-T1 , pre-T1 , αT1 , βT1 , bT1 ). Theorem 5.2. Let f : (X, mX ) → (Y, σ) be a weakly contra-m-continuous injection and let mX have property B . If (Y, σ) is T1 , then (X, mX ) is m - T1 . Proof. Let x1 and x2 be any distinct points in X. Then, since f is injective, f (x1 ) = f (x2 ). Moreover, since (Y,σ) is T1 , there exists an open set V such that f (x1 ) ∈ V , f (x2 ) ∈ V . Since f (x1 ) is closed, by the weak Acta Mathematica Hungarica 132, 2011


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contra-m-continuity of f mCl (f f (x1 ) ) ⊂ f −1 (V ). Since x2 ∈ f −1 (V ), x2 ∈ mCl (f −1 f (x1 ) ). Since mX has property B, by Lemma 3.2 −1 f (x1 ) ) is m-closed and hence X \ mCl (f −1 f (x1 ) ) is an mmCl (f open set of X containing x2 but not x1 . This shows that X is m-T1 . Remark 5.4. Let f : (X, τ ) → (Y, σ) be a function. If mX = PO (X), then by Theorem 5.2 we obtain the results established in Theorem 4.2 of [11]. Definition 5.5. A function f : (X, mX ) → (Y, σ) is said to have a (regular )contra-m-closed graph if for each (x, y) ∈ (X × Y ) \ G(f ), there exist an mX -closed set A containing x and a (regular) closed set B of Y containing y such that (A × B) ∩ G(f ) = ∅. Remark 5.5. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X)) and f has a (regular) contra-mclosed graph, then f has a (regular) contra-closed graph [9] (resp. a (regular) contra-preclosed graph [11]). Theorem 5.3. If f : (X, mX ) → (Y, σ) is a weakly contra-m-continuous function, where mX has property B and (Y, σ) is T1 , then G(f ) is contra-mclosed. Proof. Suppose that (x, y) ∈ (X × Y ) \ G(f ). Then y = f (x). Since Y is T1 , there exists an open set V of Y suchthat y ∈ V and f (x) ∈ V . Sincef is weakly contra-m-continuous and f (x) is closed in Y , mCl (f −1 f (x) ) ⊂ f −1 (V ) and hence we have (x, y) ∈ mCl (f −1 f (x) ) × (Y \ V ) ⊂ (X × Y ) \ G(f ). This shows that G(f ) is contra-m-closed. Remark 5.6. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X)), then by Theorem 5.3 we obtain the results established in Theorem 4.3 of [9] (resp. Theorem 4.3 of [11]). Theorem 5.4. If f : (X, mX ) → (Y, σ) is a weakly contra-m-continuous function, where mX has property B and (Y, σ) is Urysohn, then G(f ) is regular contra-m-closed. Proof. Let (x, y) ∈ (X × Y ) \ G(f ). Then y = f (x). Since Y is Urysohn, there exist open sets U and V in Y containing y and f (x), respectively, such that Cl (U ) ∩ Cl (V ) = ∅; hence Cl(V ) ⊂ Y \ Cl (U ). Since f is weakly contra-m-continuous, mCl (f −1 Cl (V ) ) ⊂ f −1 Y \ Cl (U ) . There fore, (x, y) ∈ mCl (f −1 Cl (V ) ) × Cl (U ) ⊂ (X × Y ) \ G(f ). Hence G(f ) is regular contra-m-closed. Remark 5.7. Let f : (X, τ ) → (Y, σ) be a function and mX an mstructure on X. If mX = τ (resp. PO (X)), then by Theorem 5.4 we obtain the results established in Theorem 4.4 of [9] (resp. Theorem 4.4 of [11]). Acta Mathematica Hungarica 132, 2011

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6. New varieties of contra-continuity First, we recall the θ-closure and the δ-closure of a subset in a topological space. Let (X, τ ) be a topological space and A a subset of X. A point x ∈X is called a θ-cluster (resp. δ-cluster) point of A if Cl (V ) ∩ A = ∅ (resp. Int Cl (V ) ∩ A = ∅) for every open set V containing x. The set of all θcluster (resp. δ-cluster) points of A is called the θ-closure (resp. δ-closure) of A and is denoted by Clθ (A) (resp. Clδ (A)) [57]. Definition 6.1. A subset A of a topological space (X, τ ) is said to be (1) δ-preopen [55] (resp. θ-preopen [42]) if A ⊂ Int Clδ (A) (resp. A ⊂ Int Clθ (A) ), (2) δ-β-open [25] (resp. θ-β-open [42]) if A ⊂ Cl (Int Clδ (A) ) (resp. A ⊂ Cl (Int Clθ (A) )). By δPO (X) (resp. δβ(X), θPO (X), θβ(X)), we denote the collection of all δ-preopen (resp. δ-β-open, θ-preopen, θ-β-open) sets of a topological space (X, τ ). These four collections are all m-structures with property B and they contain τ . Definition 6.2. Let (X, τ ) be a topological space. A subset A of X is said to be δ-preclosed [55] (resp. θ-preclosed [42], δ-β-closed [25], θ-βclosed [42]) if the complement of A is δ-preopen (resp. θ-preopen, δ-β-open, θ-β-open). Definition 6.3. Let (X, τ ) be a topological space and A a subset of X. The intersection of all δ-preclosed (resp. θ-preclosed, δ-β-closed, θ-β-closed) sets of (X, τ ) containing A is called the δ-preclosure [55] (resp. θ-preclosure [42], δ-β-closure [25], θ-β-closure [42]) of A and is denoted by pClδ (A) (resp. pClθ (A), β Clδ (A), β Clθ (A)). Let f : (X, τ ) → (Y, σ) be a function and mX = δPO (X) (resp. δβ(X), θPO (X), θβ(X)). Then, for a function f : (X, mX ) → (Y, σ), we can define m-continuity, weak m-continuity, contra-m-continuity, weak contra-mcontinuity, and slight m-continuity. For example, the definitions of weakly contra-m-continuous functions are given as follows: Definition 6.4. A function f : (X, τ ) → (Y, σ) is said to be weakly contra-δ-precontinuous (resp. weakly contra-θ-precontinuous, weakly contraδ-β-continuous, weakly contra-θ-β-continuous) if for each open set V of Y −1 f (A) ⊂ f −1 (V ) and each closed set A of Y such that A ⊂ V , pCl δ −1 −1 −1 −1 −1 (resp. pClθ f (A) ⊂ f (V ), β Clδ f (A) ⊂ f (V ), β Clθ f (A) ⊂ f −1 (V )). Definition 6.5. A subset A of a topological space (X, τ ) is said to be gδp -closed (resp. gθp -closed, gδβ -closed, gθβ -closed ) [42] if pClδ (A) Acta Mathematica Hungarica 132, 2011


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⊂ U (resp. pClθ (A) ⊂ U , β Clδ (A) ⊂ U , β Clθ (A) ⊂ U ) whenever A ⊂ U and U ∈ τ. Let f : (X, τ ) → (Y, σ) be a function and mX = δPO (X) (resp. δβ(X), θPO (X), θβ(X)). Then, for a function f : (X, mX ) → (Y, σ), we can define gm∗ -continuity and approximate m-continuity. Conclusion. If we give some necessary definitions, then we can obtain the corresponding results to δPO (X), δβ(X), θPO (X), and θβ(X) from the results in Sections 3, 4 and 5. References [1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud, β-open sets and β-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12 (1983), 77–90. [2] M. E. Abd El-Monsef, R. A. Mahmoud and E. R. Lashin, β-closure and β-interior, J. Fac. Ed. Ain Shams Univ., 10 (1986), 235–245. [3] A. Al-Omari and Mohd Salmi Md. Noorani, Some properties of contra-b-continuous and almost contra-b-continuous functions, Europ. J. Pure Appl. Math., 2 (2009), 213–230. [4] D. Andrijević, Semi-preopen sets, Mat. Vesnik, 38 (1986), 24–32. [5] D. Andrijević, On b-open sets, Mat. Vesnik, 48 (1996), 59–64. [6] I. Arokiarani, K. Barachandran and J. Dontchev, Some characterization of gpirresolute and gp-continuous maps between topological spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 20 (1999), 93–104. [7] S. P. Arya and M. P. Bhamini, Some weaker forms of semi-continuous functions, Ganita, 33 (1982), 124–134. [8] C. W. Baker, On preserving g-closed sets, Kyungpook Math. J., 36 (1998), 195–199. [9] C. W. Baker, Weakly contra continuous functions, Internat. J. Pure Appl. Math., 40 (2007), 265–271. [10] C. W. Baker, Weakly contra β-continuous functions and Sβ -closed sets, J. Pure Math., 24 (2007), 31–38. [11] C. W. Baker, On a weak form of contra-continuity (submitted). [12] K. Balachandran, P. Sundarm and H. Maki, On generalized continuous maps in topological spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 12 (1991), 5–13. [13] M. Caldas and S. Jafari, Some properties of contra-β-continuous functions, Mem. Fac. Kochi Univ. Ser. Math., 22 (2001), 19–28. [14] Gh. Costovici, Other elementary properties of topological spaces, Bull. Inst. Politehnic Ia¸si, Sect. I, 26(30) (1980), 19–21. [15] S. G. Crossley and S. K. Hildebrand, Semi-closure, Texas J. Sci., 22 (1971), 99–112. [16] R. Devi, K. Balachandran and H. Maki, Semi-generalized homeomorphisms and generalized semi-homeomorphisms in topological spaces, Indian J. Pure Appl. Math., 26 (1995), 271–284. [17] R. Devi, K. Balachandran and H. Maki, On generalized α-continuous maps and αgeneralized continuous maps, Far East J. Math. Sci., Special Volume (1997), Part I, 1–15. [18] J. Dontchev, On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., 16 (1995), 35–48. [19] J. Dontchev, Contra-continuous functions and strongly S-closed spaces, Internat. J. Math. Math. Sci., 19 (1996), 303–310. Acta Mathematica Hungarica 132, 2011

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