Takashi Noiri and Valeriu Popa - anubih

138 TAKASHI NOIRI AND VALERIU POPACorollary 5.1. Let (X, m X , n X ) be a bi-m-space and wmnGO(X) a wmngstructureon X with property B. For a function f : (X, m X , n X ) → (Y, σ),the following properties are equivalent:(1) f is wmng-continuous;(2) f −1 (V ) is wmng-open in X for every open set V of Y;(3) f −1 (F ) is wmng-closed in X for every closed set F of Y.Proof. The proof follows from Corollary 3.1.Definition 5.2. Let (X, m X ) be an m-space and A a subset of X. The m X -frontier of A, mFr(A), [33] is defined by mFr(A) = mCl(A) ∩ mCl(X − A) =mCl(A) − mInt(A).If (X, m X , n X ) is a bi-m-space and wmnGO(X) a wmng-structure, thenwmnFr g (A) = wmnCl g (A)∩wmnCl g (X −A) = wmnCl g (A)−wmnInt g (A).Lemma 5.1. (Popa and Noiri [33]). The set of all points of X at whicha function f : (X, m X ) → (Y, σ) is not m-continuous is identical with theunion of the m-frontiers of the inverse images of open sets containing f(x).Theorem 5.3. Let (X, m X , n X ) is a bi-m-space and wmnGO(X) a wmngstructure.Then, the set of all points of X at which a function f :(X, m X , n X )→ (Y, σ) is not wmng-continuous is identical with the union of the wmngfrontiersof the inverse images of open sets containing f(x).Proof. The proof follows from Lemma 5.1.Let (X, τ) be a topological space and A a subset of X. A point x ∈ Xis called a θ-cluster point of A if Cl(V ) ∩ A ≠ ∅ for every open set Vcontaining x. The set of all θ-cluster points of A is called the θ-closure of Aand is denoted by Cl θ (A) [44]. If A = Cl θ (A), then A is said to be θ-closed.The complement of a θ-closed set is said to be θ-open.Lemma 5.2. (Noiri and Popa [28]). Let (Y, σ) be a regular space. For afunction f : (X, m X ) → (Y, σ), the following properties are equivalent:(1) f is m-continuous;(2) f −1 (Cl θ (B)) = mCl(f −1 (Cl θ (B))) for every subset B of Y;(3) f −1 (K) = mCl(f −1 (K)) for every θ-closed set K of Y;(4) f −1 (V ) = mInt(f −1 (V )) for every θ-open set V of Y.Corollary 5.2. (Noiri and Popa [28]). Let (Y, σ) be a regular space and m Xan m-structure on X with property B. For a function f : (X, m X ) → (Y, σ),the following properties are equivalent:(1) f is m-continuous;(2) f −1 (Cl θ (B)) is m-closed for every subset B of Y;(3) f −1 (K) is m-closed in X for every θ-closed set K of Y;□□