Takashi Noiri and Valeriu Popa - anubih

WEAKLY g-CLOSED SETS AND WEAKLY g-CONTINUOUS FUNCTIONS 139(4) f −1 (V ) is m-open in X for every θ-open set V of Y.Theorem 5.4. Let (Y, σ) be a regular space, (X, m X , n X ) a bi-m-space andwmnGO(X) a wmng-structure. For a function f : (X, m X , n X ) → (Y, σ),the following properties are equivalent:(1) f is wmng-continuous;(2) f −1 (Cl θ (B)) = wmnCl g (f −1 (Cl θ (B))) for every subset B of Y;(3) f −1 (K) = wmnCl g (f −1 (K)) for every θ-closed set K of Y;(4) f −1 (V ) = wmnInt g (f −1 (V )) for every θ-open set V of Y.Proof. The proof follows from Lemma 5.2.Corollary 5.3. Let (Y, σ) be a regular space, (X, m X , n X ) a bi-m-spaceand wmnGO(X) a wmng-structure with property B on X. For a functionf : (X, m X , n X ) → (Y, σ), the following properties are equivalent:(1) f is wmng-continuous;(2) f −1 (Cl θ (B)) is wmng-closed in X for every subset B of Y;(3) f −1 (K) is wmng-closed in X for every θ-closed set K of Y;(4) f −1 (V ) is wmng-open in X for every θ-open set V of Y.Proof. The proof follows from Corollary 5.2.Remark 5.2. Let (X, τ) be a topological space. If n X = τ and m X is an m-structure on X, then by Theorem 5.2 and Corollary 5.3 we obtain Theorem5.7 and Corollary 5.3 in [29], respectively.6. Some properties of wmng-continuous functionsDefinition 6.1. A function f : (X, m X ) → (Y, σ) is said to have a stronglym-closed graph (resp. m-closed graph) [33] if for each (x, y) ∈ (X×Y )−G(f),there exist U ∈ m X containing x and an open set V of Y containing y suchthat [U × Cl(V )] ∩ G(f) = ∅ (resp. [U × V ] ∩ G(f) = ∅).Definition 6.2. Let (X, m X , n X ) be a bi-m-space and wmnGO(X) a wmngstructureon X. A function f : (X, m X , n X ) → (Y, σ) is said to have astrongly wmng-closed graph (resp. wmng-closed graph) if a function f :(X, wmnGO(X)) → (Y, σ) has a strongly m-closed (resp. m-closed) graph,equivalently if for each (x, y) ∈ (X ×Y )−G(f), there exist U ∈ wmnGO(X)containing x and an open set V of Y containing y such that [U × Cl(V )] ∩G(f) = ∅ (resp. [U × V ] ∩ G(f) = ∅).Remark 6.1. Let (X, τ) be a topological space and m X , n X minimal structureson X. If n X = τ, then by Definition 6.2 we obtain the definition ofstrongly wmg ∗ -closed graphs and wmg ∗ -closed graphs in [29].□□