1_xxx_noiri.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 4, 2009, (473-493) ISSN 1307-5543 – www.ejpam.com A Generalization of Some Forms of g-Irresolute Functions Takashi Noiri1∗ and Valeriu Popa2 1 2949-1 Shiokita-Cho, Hinagu, Yatsushiro-Shi, Kumamoto-Ken, 869-5142 Japan 2 Department Of Mathematics, University Of Bacǎu, 600 114 Bacǎu, Romania Abstract. In this paper, by using gm-closed sets [27], we obtain the unified definitions and properties for g-continuity, gs-continuity, gp-continuity, αg-continuity, γg-continuity and gsp- continuity. 2000 Mathematics Subject Classifications: 54A05, 54C08, 54C10. Key Words and Phrases: m-structure, g-closed, gm-closed, gm-continuous. 1. Introduction The concept of generalized closed (briefly g-closed) sets in topological spaces was introduced by Levine [20] in 1970. These sets were also considered by Dunham [15] and Dunham and Levine [16]. The notion of αg-closed [12] (resp. gs-closed [11], ∗Corresponding author. Email addresses: t.noiri�nifty. om (T. Noiri), vpopa�ub.ro (V. Popa) http://www.ejpam.com 473 c© 2009 EJPAM All rights reserved. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 474 gp-closed [6], g b-closed or γg-closed [18], gsp-closed or gβ -closed [14]) sets is in- troduced and investigated. In 1981, Munshy and Bassan [25] introduced the notion of generalized continuous (briefly g-continuous) functions which are called in [7] as g-irresolute functions. Furthermore, the notion of gs-irresolute [11] (resp. gp- irresolute [6], αg-irresolute [12], g b-irresolute [3], gsp-irresolute [32]) functions is introduced. Recently, the present authors [29], [30] have introduced the notions of m-structures, m-spaces and M -continuity. In [27], the first author introduced the notion of gener- alized m-closed (briefly gm-closed) sets and tried to unify certain types of modifica- tions of g-closed sets such as stated above. In this paper, by using gm-closed sets, we obtain the unified definitions and properties for g-irresoluteness, gs-irresoluteness, gp-irresoluteness, αg-irresoluteness, g b-irresoluteness and gsp-irresoluteness. 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. We recall some generalized open sets in topological spaces. Definition 1. Let (X ,τ) be a topological space. A subset A of X is said to be (1) α-open [26] if A⊂ Int(Cl(Int(A))), (2) semi-open [19] if A⊂ Cl(Int(A)), (3) preopen [22] if A⊂ Int(Cl(A)), (4) β -open [1] or semi-preopen [4] if A⊂ Cl(Int(Cl(A))), (5) γ-open [18] or b-open [5] if A⊂ Int(Cl(A))∪Cl(Int(A)). The family of all α-open (resp. semi-open, preopen, β -open, γ-open) sets in (X ,τ) is denoted by α(X ) (resp. SO(X ), PO(X ), β(X ) or SPO(X ), γ(X ) or BO(X )). T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 475 Definition 2. Let (X ,τ) be a topological space. A subset A of X is said to be α- closed [23] (resp. semi-closed [10], preclosed [22], β -closed [1] or semi-preclosed [4], γ-closed [18] or b-closed [5]) if the complement of A is α-open (resp. semi-open, preopen, β -open, γ-open). Definition 3. Let (X ,τ) be a topological space and A a subset of X . The intersection of all α-closed (resp. semi-closed, preclosed, β -closed, γ-closed) sets of X containing A is called the α-closure [23] (resp. semi-closure [10], preclosure [17], β -closure [2] or semi-preclosure [4], γ-closure [18] or b-closure [5]) of A and is denoted by αCl(A) (resp. sCl(A), pCl(A), βCl(A) or spCl(A)), Clγ(A) or bCl(A)). Definition 4. Let (X ,τ) be a topological space and A a subset of X . The union of all α-open (resp. semi-open, preopen, β -open, γ-open) sets of X contained in A is called the α-interior [23] (resp. semi-interior [10], preinterior [17], β -interior [2] or semi-preinterior [4], γ-interior [18] or b-interior [5]) of A and is denoted by αInt(A) (resp. sInt(A), pInt(A), β Int(A) or spInt(A)), Intγ(A) or bInt(A)). 3. Minimal structures and m-continuity Definition 5. Let X be a nonempty set and P (X ) the power set of X . A subfamily mX of P (X ) is called a minimal structure (briefly m-structure) on X [29], [30] if ; ∈ mX and X ∈ mX . By (X , mX ), we denote a nonempty set X with an m-structure mX on X and call it an m-space. Each member of mX is said to be mX -open and the complement of an mX -open set is said to be mX -closed. Remark 1. Let (X ,τ) be a topological space. Then the family α(X ) is a topology finer than τ. The families SO(X ), PO(X ), β(X ), and γ(X ) are all m-structures on X . T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 476 Definition 6. 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 [21] as follows: (1) mCl(A) = ∩{F : A⊂ F, X − F ∈ mX}, (2) mInt(A) = ∪{U : U ⊂ A, U ∈ mX }. Remark 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 ), γ(X )), then we have (1) mCl(A) = Cl(A) (resp. sCl(A), pCl(A), αCl(A), βCl(A), Clγ(A)), (2) mInt(A) = Int(A) (resp. sInt(A), pInt(A), αInt(A), β Int(A), Intγ(A)). Lemma 1. (Maki et al. [21]). Let X be a nonempty set and mX a minimal 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). Lemma 2. (Popa and Noiri [29]). Let X be a nonempty set with a minimal structure mX and A a subset of X. Then x ∈ mCl(A) if and only if U ∩ A 6= ; for every U ∈ mX containing x. Definition 7. An m-structure mX on a nonempty set X is said to have propertyB [21] if the union of any family of subsets belong to mX belongs to mX . Remark 3. If (X ,τ) is a topological space, then SO(X ), PO(X ), α(X ), β(X ) and γ(X ) have propertyB , Lemma 3. (Popa and Noiri [30]). Let X be a nonempty set and mX an m-structure on X satisfying propertyB . For a subset A of X, the following properties hold: T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 477 (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. Definition 8. A function f : (X , mX )→ (Y, mY ) is said to be M-continuous at a point x ∈ X [30] if for each x ∈ X and each V ∈ mY containing f (x), there exists U ∈ mX containing x such that f (U) ⊂ V . A function f : (X , mX ) → (Y, mY ) is said to be M-continuous if it has this property at each point x ∈ X . Theorem 1. For a function f : (X , mX )→ (Y, mY ), the following properties are equiva- lent: (1) f is M-continuous at x ∈ X; (2) x ∈mInt( f −1(V )) for every V ∈ mY containing f(x); (3) x ∈ f −1(mCl( f (A))) for every subset A of X with x ∈mCl(A); (4) x ∈ f −1(mCl(B)) for every subset B of Y with x ∈mCl( f −1(B)); (5) x ∈mInt( f −1(B)) for every subset B of Y with x ∈ f −1(mInt(B)); (6) x ∈ f −1(K) for every mY -closed set K of Y such that x ∈mCl( f −1(K)). Proof. (1) ⇒ (2): Let V ∈ mY containing f (x). Then, there exists U ∈ mX con- taining x such that f (U) ⊂ V . Thus x ∈ U ⊂ f −1(V ). Since U ∈ mX , we have x ∈mInt( f −1(V )). (2) ⇒ (3): Let A be any subset of X . Let x ∈ mCl(A) and V ∈ mY containing f (x). Then x ∈mInt( f −1(V )). There exists U ∈ mX such that x ∈ U ⊂ f −1(V ). Since x ∈mCl(A), by Lemma 2, U ∩A 6= ; and ; 6= f (U ∩A) ⊂ f (U)∩ f (A)⊂ V ∩ f (A). Since V ∈ mY containing f (x), f (x) ∈mCl( f (A)) and hence x ∈ f −1(mCl( f (A)). (3) ⇒ (4): Let B be any subset of Y and x ∈ mCl( f −1(B)), then by (3) x ∈ f −1(mCl( f ( f −1(B)))) ⊂ f −1(mCl(B)). Hence, we have x ∈ f −1(mCl(B)). (4) ⇒ (5): Let B be any subset of Y such that x /∈ mInt( f −1(B)). Then x ∈ X − mInt( f −1(B)) = mCl(X − f −1(B)) = mCl( f −1(Y − B)). By (4), we have x ∈ T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 478 f −1(mCl(Y −B)) = f −1(Y −mInt(B)) = X − f −1(mInt(B)). Hence, x /∈ f −1(mInt(B)). (5) ⇒ (6): Let K be any mY -closed set of Y such that x /∈ f −1(K). Then x ∈ X − f −1(K) = f −1(Y −K) = f −1(mInt(Y −K)) because Y −K is mY -open. By (5), x ∈ mInt( f −1(Y −K)) =mInt(X − f −1(K)) = X −mCl( f −1(K)). Hence x /∈mCl( f −1(K)). (6)⇒ (2): Let x ∈ X and V ∈ mY containing f (x). Suppose that x /∈mInt( f −1(V )). Then x ∈ X −mInt( f −1(V )) = mCl(X − f −1(V )) = mCl( f −1(Y − V )). By (6), x ∈ f −1(Y − V ) = X − f −1(V ). Hence x /∈ f −1(V ). This contraries to the hypothesis. (2) ⇒ (1): Let V ∈ mY containing f (x). By (2), x ∈ mInt( f −1(V )) and hence there exists U ∈ mX containing x such that x ∈ U ⊂ f −1(V ). Therefore, f (U) ⊂ V and f is M -continuous at x . For a function f : (X , mX )→ (Y, mY ), we define DM( f ) as follows: DM( f ) = {x ∈ X : f is not M -continuous at x}. Theorem 2. For a function f : (X , mX )→ (Y, mY ), the following properties hold: DM ( f ) = ⋃ G∈mY { f −1(G)−mInt( f −1(G))} = ⋃ B∈P (Y ) { f −1(Int(B))−mInt( f −1(B))} = ⋃ B∈P (Y ) {mCl( f −1(B))− f −1(mCl(B))} = ⋃ A∈P (X ) {mCl(A)− f −1(mCl( f (A)))} = ⋃ K∈F {mCl( f −1(K))− f −1(K)}, where F is the family of mY -closed sets of Y . Proof. We show only the first equality because the proofs of the others are similar to the first one. Let x ∈ DM ( f ). By Theorem 1, there exists V ∈ mY such that f (x) ∈ V and x /∈ mInt( f −1(V )). Therefore, we have x ∈ f −1(V )−mInt( f −1(V )) ⊂ ⋃ G∈mY { f −1(G)−mInt( f −1(G))}. Conversely, let x ∈ ⋃ G∈mY { f −1(G)−mInt( f −1(G))}. There exists V ∈ mY such that x ∈ f −1(V )−mInt( f −1(V )). By Theorem 1, x ∈ DM( f ). Theorem 3. (Popa and Noiri [29]). For a function f : (X , mX )→ (Y, mY ), the following properties are equivalent: T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 479 (1) f is M-continuous; (2) f −1(V ) =mInt( f −1(V )) for every V ∈ mY ; (3) f (mCl(A)) ⊂ Cl( f (A)) for every subset A of X; (4) mCl( f −1(B)) ⊂ f −1(mCl(B)) for every subset B of Y; (5) f −1(Int(B)) ⊂mInt( f −1(B)) for every subset B of Y; (6) mCl( f −1(K)) = f −1(K) for every mY -closed set K of Y. Corollary 1. (Popa and Noiri [29]). For a function f : (X , mX )→ (Y, mY ), where mX has propertyB , the following properties are equivalent: (1) f is M-continuous; (2) f −1(V ) is mX -open for every V ∈ mY ; (3) f −1(F) is mX -closed in X for every mY -closed set F of Y. Definition 9. A function f : (X , mX ) → (Y, mY ) is said to be M ∗-continuous [24] if f −1(V ) is mX -open for each mY -open set V of Y . Remark 4. (1) If f : (X , mX ) → (Y, mY ) is M ∗-continuous, then it is M -continuous. By Example 3.4 of [24], an M -continuous function may not be M ∗-continuous. (2) If mX has propertyB , then M -continuity and M ∗-continuity are equivalent. 4. gm-closed sets and gM -continuity Definition 10. Let (X ,τ) be a topological space. A subset A of X is said to be (1) g-closed [20] if Cl(A) ⊂ U whenever A⊂ U and U ∈ τ, (2) αg-closed [12] if αCl(A)⊂ U whenever A⊂ U and U ∈ τ, (3) gs-closed [11] 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) gb-closed or γg-closed [18] if bCl(A) ⊂ U whenever A⊂ U and U ∈ τ, (6) gsp-closed [14] or gβ -closed if spCl(A)⊂ U whenever A⊂ U and U ∈ τ, T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 480 Definition 11. A subset A of a topological space is said to be g-open (resp. gs-open, gp-open, αg-open, g b-open, gsp-open) if X − A is g-closed (resp. gs-closed, gp- closed αg-closed, g b-closed, gsp-closed). The family of all g-open (resp. gs-open, gp-open, αg-open, g b-open, gsp-open) sets of X is denoted by GO(X ) (resp. GSO(X ), GPO(X ), αGO(X ), GBO(X ), GSPO(X )). Definition 12. Let (X ,τ) be a topological space and A a subset of X . The intersection of all g-closed (resp. αg-closed, gs-closed, gp-closed, gsp-closed, g b-closed) sets of X containing A is called the g-closure [15] (resp. αg-closure, gs-closure, gp-closure, gsp-closure, gb-closure) of A and is denoted by Clg(A) (resp. αClg(A), sClg(A), pClg(A), spClg(A), bClg(A)). Definition 13. Let (X ,τ) be a topological space and A a subset of X . The union of all g-open (resp. αg-open, gs-open, gp-open, gsp-open, g b-open) sets of X contained in A is called the g-interior [9] (resp. αg-interior, gs-interior, gp-interior, gsp-interior, gb- interior) of A and is denoted by Intg(A) (resp. αIntg(A), sIntg(A), pIntg(A), spIntg(A), bIntg(A)). Remark 5. Let (X ,τ) be a topological space and A a subset of X . (1) Then, GO(X ), GSO(X ), GPO(X ), αGO(X ) and GSPO(X ) are all m-structures on X . Hence, if we put mX = GO(X ) (resp. αGO(X ), GSO(X ), GPO(X ), GSPO(X )), then we have (i) mCl(A) = Clg(A) (resp. αClg(A), sClg(A), pClg(A), spClg(A)), (ii) mInt(A) = Intg(A) (resp. αIntg(A)), sIntg(A), pIntg(A), spIntg(A)). (2) If mX = GO(X ), then by Lemma 1 we obtain the results established in Theorem 2.1 (4), (5) and Theorem 2.8 (2), (3), (5), (6) in [9]. By Lemma 2, we obtain the result established in Theorem 2.1 (4) in [9]. (3) The m-structures GO(X ), GSO(X ), GPO(X ), αGO(X ), GSPO(X ) and GBO(X ) do not have propertyB , in general. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 481 Definition 14. Let (X ,τ) be a topological space and mX an m-structure on X . A subset A of X is said to be generalized m-closed (briefly gm-closed) [27] if mCl(A)⊂ U whenever A⊂ U and U ∈ τ. The complement of a gm-closed set is said to be gm-open. The family of all gm- open sets of a topological space (X ,τ) is denoted by GMO(X ). Obviously, GMO(X ) is an m-structure on X and is called a gm-structure on X . Remark 6. 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 ), SPO(X ), BO(X )). Then, a gm-closed set is a g-closed (resp. gs-closed, gp-closed, αg-closed, gsp-closed, g b-closed) set. Definition 15. A function f : (X ,τ) → (Y,σ) is said to be g-irresolute [7] or g- continuous [25] (resp. gs-irresolute [11], gp-irresolute [6], αg-irresolute [12], gsp- irresolute [32], gb-irresolute [3]) if f −1(K) is a g-closed (resp. gs-closed, gp-closed, αg-closed, gsp-closed, g b-closed) in X for every g-closed (resp. gs-closed, gp-closed, αg-closed, gsp-closed, g b-closed) set K of Y . Definition 16. A function f : (X ,τ)→ (Y,σ) is said to be (1) gM-continuous at a point x ∈ X if f : (X , GMO(X )) → (Y, GMO(Y )) is M - continuous at a point x ∈ X . The function f : (X ,τ) → (Y,σ) is said to be gM- continuous if it is gM -continuous at each point x ∈ X . (2) gM-irresolute if f : (X , GMO(X ))→ (Y, GMO(Y )) is M ∗-continuous. Remark 7. (1) Every gM -irresolute function is gM -continuous. (2)If mX = GO(X ) (resp. GSO(X ), GPO(X ), αGO(X ), GSPO(X ), BO(X )), mY = GO(Y ) (resp. GSO(Y ), GPO(Y ), αGO(Y ), GSPO(Y ), BO(Y )) and f : (X ,τ) → (Y,σ) is gM -irresolute, then f is g-irresolute (resp. gs-irresolute, gp-irresolute, αg- irresolute, gsp-irresolute, g b-irresolute). T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 482 Definition 17. Let (X ,τ) be a topological space and GMO(X ) a gm-structure on X . For a subset A of X , the gm-closure of A and the gm-interior of A are defined as follows: (1) mClg(A) = ∩{F : A⊂ F, X − F ∈ GMO(X )}, (2) mIntg(A) = ∪{U : U ⊂ A, U ∈ GMO(X )}. By Definition 16 and Theorem 3, we obtain the following theorem and corollary. Theorem 4. For a function f : (X ,τ)→ (Y,σ), the following properties are equivalent: (1) f is gM-continuous; (2) f −1(V ) =mIntg( f −1(V )) for every gm-open set V of Y; (3) mClg( f −1(F)) = f −1(F) for every gm-closed set F of Y; (4) mClg( f −1(B)) ⊂ f −1(mClg(B)) for every subset B of Y; (5) f (mClg(A)) ⊂mClg( f (A)) for every subset A of X; (6) f −1(mIntg(B)) ⊂mIntg( f −1(B)) for every subset B of Y. Corollary 2. For a function f : (X ,τ) → (Y,σ), where GMO(X) has property B , the following properties are equivalent: (1) f is gM-continuous; (2) f −1(V ) is gm-open for every gm-open set V of Y; (3) f −1(F) is gm-closed for every gm-closed set F of Y. Let (X ,τ) be a topological space and GMO(X ) a gm-structure on X . For a function f : (X ,τ) → (Y,σ), we denote by DgM( f ) the set of all points of X at which the function f is not gM -continuous. Then by Definition 16 and Theorem 4, we obtain the following theorem. Theorem 5. For a function f : (X ,τ)→ (Y,σ), the following properties hold: DgM( f ) = ⋃ G∈GMO(Y ){ f −1(G)−mIntg( f −1(G))} = ⋃ B∈P (Y ) { f −1(mIntg(B))−mIntg( f −1(B))} = ⋃ B∈P (Y ) {mClg( f −1(B))− f −1(mClg(B))} T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 483 = ⋃ A∈P (X ) {mClg(A)− f −1(mClg( f (A)))} = ⋃ K∈F g {mClg( f −1(K))− f −1(K)}, where F g is the family of gm-closed sets of Y . Definition 18. Let (X , mX ) be an m-space and A a subset of X . The mX -frontier of A, mFr(A), [30] is defined by mFr(A) = mCl(A)∩mCl(X − A) =mCl(A)−mInt(A). If (X ,τ) is a topological space and GMO(X ) is a gm-structure on X , then gmFr(A) = mClg(A)∩mClg(X − A) =mClg(A)−mIntg(A). Theorem 6. The set of all points of X at which a function f : (X , mX ) → (Y, mY ) is not M-continuous is identical with the union of the m-frontiers of the inverse images of mY -open sets containing f(x). Proof. Suppose that f is not M -continuous at x ∈ X . There exists an mY -open set V of Y containing f (x) such that U ∩ (X − f −1(V )) 6= ; for every mX -open set U containing x . By Lemma 2, we have x ∈ mCl(X − f −1(V )). On the other hand, we have x ∈ f −1(V ) and hence x ∈mFr( f −1(V )). Conversely, suppose that f is M -continuous at x ∈ X . Then, for any mY -open set V of Y containing f (x), there exists U ∈ mX containing x such that f (U) ⊂ V ; hence U ⊂ f −1(V ). Therefore, we have x ∈ U ⊂ mInt( f −1(V )). This contradicts to the fact that x ∈mFr( f −1(V )). Corollary 3. Let (X ,τ) (resp. (Y,σ)) be a topological space and GOM(X) (resp. GOM(Y)) a gm-structure on X (resp. Y). Then, the set of all points at x ∈ X which a function f : (X ,τ)→ (Y,σ) is not gM-continuous is identical with the union of the gm-frontiers of the inverse images of gm-open sets containing f(x). Proof. This follows immediately from Theorem 6. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 484 5. Some properties of gM -continuity In this section, we use gm-open sets and gm-closed sets in order to obtain some properties of gm-T2 spaces and the preservation theorems of gm-compact spaces and gm-connected spaces. Furthermore, we investigate some properties of strongly m- closed graphs. Definition 19. An m-space (X , mX ) is said to be m-T2 [29] if for any distinct points x , y, there exist U , V ∈ mX such that x ∈ U , y ∈ V , and U ∩ V = ;. Remark 8. (1) Let (X ,τ) be a topological space, then (X ,τ) is said to be gm-T2 if the m-space (X , GMO(X )) is m-T2. (2) If GMO(X ) = GO(X ) (resp. GSO(X ), GPO(X ), αGO(X ) GBO(X ), GSPO(X )) and (X ,τ) is mg-T2, then (X ,τ) is said to be g-T2 [8] (resp. gs-T2, gp-T2, αg-T2, g b-T2, gsp-T2). Lemma 4. (Popa and Noiri [29]). If f : (X , mX )→ (Y, mY ) is an M-continuous injec- tion and (Y, mY ) is m-T2, then (X , mX ) is m-T2. Theorem 7. If f : (X ,τ)→ (Y,σ) is a gM-continuous injection and (Y,σ) is a gm-T2- space, then (X ,τ) is gm-T2. Proof. The proof follows from Remark 8 and Lemma 4. Corollary 4. If f : (X ,τ)→ (Y,σ) is a gM-irresolute injection and (Y,σ) is a gm-T2- space, then (X ,τ) is gm-T2. Definition 20. An m-space (X , mX ) is said to be m-compact [29] if every cover of X by sets of mX has a finite subcover. A subset K of an m-space (X , mX ) is said to be m-compact [29] if every cover of K by subsets of mX has a finite subcover. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 485 Remark 9. (1) If (X ,τ) is a topological space and (X , GMO(X )) is m-compact, then (X ,τ) is said to be gm-compact. (2) If GMO(X ) = GO(X ) (resp. GSO(X ), GPO(X ), αGO(X )), then we obtain the definition of GO-compactness [7] (resp. GSO-compactness [11], GPO-compactness [6], αGO-compactness [12]). Lemma 5. (Popa and Noiri [29]). If a function f : (X , mX )→ (Y, mY ) is M-continuous and K is an m-compact set of X, then f(K) is m-compact. Theorem 8. If f : (X ,τ)→ (Y,σ) is a gM-continuous function and K is a gm-compact set of X, then f(K) is gm-compact. Proof. The proof follows from Definition 20 and Lemma 5. Corollary 5. If f : (X ,τ)→ (Y,σ) is a gM-irresolute function and K is a gm-compact set of X, then f(K) is gm-compact. Remark 10. If GMO(X ) = GO(X ) (resp. GSO(X ), GPO(X ), αGO(X )) and GMO(Y ) = GO(Y ) (resp. GSO(Y ), GPO(Y ), αGO(Y )), then by Corollary 5 we obtain the result established in Proposition 9(ii) of [7] (resp. Proposition 5.5(iii) of [11], Theorem 5.5(iii) of [6], Proposition 4.3(iii) [12]). Definition 21. An m-space (X , mX ) is said to be m-connected [29] if X cannot be written as the union of two nonempty disjoint mX -open sets. Remark 11. Let (X ,τ) be a topological space and GMO(X ) a gm-structure on X , then (1) (X ,τ) is said to be gm-connected if X cannot be written as the union of two nonempty disjoint gm-open sets. (2) If GMO(X ) = GO(X ) (resp. αGO(X )), then we obtain the definition of GO- connected spaces [7] (resp. αGO-connected spaces [12]). T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 486 Lemma 6. If f : (X , mX ) → (Y, mY ) is an M ∗-continuous surjection and (X , mX ) is m-connected, then (Y, mY ) is m-connected. Proof. Suppose that (Y, mY ) is not m-connected. Then there exist nonempty mY -open sets V1 and V2 such that V1 ∩ V2 = ; and V1 ∪ V2 = Y . Hence we have f −1(V1) ∩ f −1(V2) = ; and f −1(V1) ∪ f −1(V2) = X . Since f is an M ∗-continuous sur- jection, f −1(V1) and f −1(V2) are nonempty mX -open sets. Therefore, (X , mX ) is not m-connected. This is a contradiction and hence (Y, mY ) is m-connected. Theorem 9. If f : (X ,τ) → (Y,σ) is a gM-irresolute surjection and (X ,τ) is gm- connected, then (Y,σ) is gm-connected. Proof. The proof follows from Definition 21, Remark 11 and Lemma 6. Remark 12. If GMO(X )= GO(X ), then we obtain the result established in Proposition 13 of [7]. Definition 22. A function f : (X , mX ) → (Y, mY ) is said to have a strongly m-closed graph (resp. m-closed graph) [29] if for each (x , y) ∈ (X × Y ) − G( f ), there exist U ∈ mX containing x and V ∈ mY containing y such that [U ×mCl(V )] ∩ G( f ) = ; (resp. [U × V ]∩G( f ) = ;). Remark 13. Let (X ,τ) (resp. (Y,σ)) be a topological space and GMO(X ) (resp. GMO(Y )) a gm-structure on X (resp. Y ). A function f : (X ,τ) → (Y,σ) is said to have a strongly gm-closed graph (resp. gm-closed graph) if for each (x , y) ∈ (X × Y )− G( f ), there exist U ∈ GMO(X ) containing x and V ∈ GMO(Y ) containing y such that [U ×mClg(V )]∩G( f ) = ; (resp. [U × V ]∩G( f ) = ;). Lemma 7. (Popa and Noiri [29]). A function f : (X , mX )→ (Y, mY ) is M-continuous and (Y, mY ) is m-T2, then f has a strongly m-closed graph. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 487 Theorem 10. Let (X ,τ) (resp. (Y,σ)) be a topological space and GMO(X) (resp. GMO(Y )) a gm-structure on X (resp. Y ). If a function f : (X ,τ) → (Y,σ) is gM- continuous and (Y,σ) is gm-T2, then f has a strongly gm-closed graph. Proof. The proof follows from Definition 22, Remark 13 and Lemma 7. Corollary 6. If a function f : (X ,τ)→ (Y,σ) is gM-irresolute and (Y,σ) is gm-T2, then f has a strongly gm-closed graph. Remark 14. If (Y,σ) is g-T2 (resp. gs-T2, gp-T2, αg-T2, g b-T2, gsp-T2) and f : (X ,τ) → (Y,σ) is a g-irresolute (resp. gs-irresolute, gp-irresolute, αg-irresolute, g b-irresolute, gsp-irresolute) function, then G( f ) is strongly g-closed (resp. strongly gs-closed, strongly gp-closed, strongly αg-closed, strongly g b-closed, strongly gsp- closed). Lemma 8. (Popa and Noiri [29]). If f : (X , mX )→ (Y, mY ) is a surjective function with a strongly m-closed graph, then (Y, mY ) is m-T2. Theorem 11. Let (X ,τ) (resp. (Y,σ)) be a topological space and GMO(X) (resp. GMO(Y )) a gm-structure on X (resp. Y ). If f : (X ,τ)→ (Y,σ) is a surjective function with a strongly gm-closed graph, then (Y,σ) is gm-T2. Proof. The proof follows from Definition 22 and Lemma 8. Remark 15. If f : (X ,τ) → (Y,σ) is a surjective function with a strongly g-closed (resp. strongly gs-closed, strongly gp-closed, strongly αg-closed, strongly g b-closed, strongly gsp-closed), then Y is g-T2 (resp. gs-T2, gp-T2, αg-T2, g b-T2, gsp-T2). Lemma 9. (Popa and Noiri [29]). Let f : (X , mX )→ (Y, mY ) be a function, where mX has propertyB . If f is an M-continuous surjection with an m-closed graph, then (X , mX ) is m-T2. T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 488 Theorem 12. Let (X ,τ) (resp. (Y,σ)) be a topological space and GMO(X) (resp. GMO(Y )) a gm-structure on X (resp. Y ) and GMO(X ) a gm-structure satisfying prop- erty B . If f : (X ,τ) → (Y,σ) is a gM-continuous surjection with a gm-closed graph, then X is gm-T2. Proof. The proof follows from Definition 22 and Lemma 9. Corollary 7. If a function f : (X ,τ) → (Y,σ) is a gM-irresolute surjection with a gm- closed graph and GMO(X ) has propertyB , then (X ,τ) is gm-T2. Definition 23. Let A a subset of an m-space (X , mX ). A point x ∈ X is called an mθ -adherent point of A [31] if mCl(U) ∩ A 6= ; for every mX -open set U containing x . The set of all mθ -adherent points of A is called the mθ -closure of A and is denoted by mClθ (A). If A = mClθ (A), then A is said to be mθ -closed. The complement of a mθ -closed set is said to be mθ -open. The union of all mθ -open sets contained in A is called the mθ -interior of A and is denoted by mIntθ (A). Remark 16. Let A be a subset of a topological space (X ,τ) and mX an m-structure on X . If mX = τ (resp. SO(X ), PO(X )), then mClθ(A) = Clθ (A) [33] (resp. sClθ(A) [13], pClθ (A) [28]). Lemma 10. (Popa and Noiri [31]). Let A be a subset of an m-space (X , mX ). Then the following properties hold: (1) If A is mX -open in X, then mClθ (A) =mCl(A), (2) If mX has propertyB , then mClθ(A) is mX -closed in X for every subset A of X. Definition 24. An m-space (X , mX ) is said to be m-regular [31] if for each mX -closed set F of X and each point x /∈ F , there exist disjoint mX -open sets U and V such that x ∈ U and F ⊂ V . Lemma 11. (Popa and Noiri [31]). Let (X , mX ) be an m-regular m-space. Then the following properties hold: T. Noiri and V. Popa / Eur. J. Pure Appl. Math, 2 (2009), (473-493) 489 (1) mClθ (A) =mCl(A) for every subset A of X, (2) Every mX -open set is mθ -open. Theorem 13. Let (Y, mY ) be an m-regular m-space and mY have property B . For a function f : (X , mX )→ (Y, mY ), the following properties are equivalent: (1) f is M-continuous; (2) f −1(mClθ(B)) =mCl( f −1(mClθ (B))) for every subset B of Y; (3) f −1(K) =mCl( f −1(K)) for every mθ -closed set K of Y; (4) f −1(V ) =mInt( f −1(V )) for every mθ -open set V of Y. Proof. (1) ⇒ (2): Let B be any subset of Y . Then, by Lemma 10 mClθ(B) is mY - closed in Y . By Theorem 3, we obtain f −1(mClθ (B)) =mCl( f −1(mClθ(B))). (2) ⇒ (3): Let K be an mθ -closed set of Y . Then mClθ (K) = K . Then by (2) we obtain f −1(K) =mCl( f −1(K)). (3) ⇒ (4): Let V be an mθ -open set of Y . Then Y − V is mθ -closed and f −1(Y − V ) =mCl( f −1(Y −V )). Therefore, X − f −1(V ) = X −mInt( f −1(V )). Hence we obtain f −1(V ) =mInt( f −1(V )). (4) ⇒ (1): Let V be any mY -open set of Y . Since Y is m-regular, by Lemma 11 V is mθ -open and by (4) we have f −1(V ) = mInt( f −1(V )). By Theorem 1, f is M -continuous. Theorem 14. Let (Y, mY ) be m-regular and let mX and mY have property B . For a function f : (X , mX )→ (Y, mY ), the following properties are equivalent: (1) f is M-continuous; (2) f −1(mClθ(B)) is mX -closed for every subset B of Y; (3) f −1(K) is mX -closed for every mθ -closed set K of Y; (4) f −1(V ) is mX -open for every mθ -open set V of Y. Proof. 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