Noiri T., Weakly λ-continuous functions

It is the objective of this paper to introduce a new class of generalizations of continuous functions via ‚-open sets called weakly ‚- continuous functions. Moreover, we study some of its fundamental prop- erties. It turns out that weak ‚-continuity is weaker than ‚-continuity (1).


Introduction
Maki [13] offered a new and useful notion in the field of topology which he called a Λ-set.A Λ-set is a set A which is equal to its kernel (= saturated set), i.e. to the intersection of all open supersets of A. Arenas et al. [1] introduced and investigated the notion of λ-closed sets by involving Λ-sets and closed sets.By utilizing λ-closed sets, they introduced and to some extent investigated the notion of λ-continuity.Quite recently, several authors investigated some new maps and notions via λ-open and λ-closed sets (see for example [2], [3], [4], [10], [5] and [7]).
In this paper, we establish a new class of functions called weakly λ-continuous functions which is weaker than λ-continuous functions.We also investigate some of the fundamental properties of this type of functions.
Throughout the paper a space will always mean a topological space on which no separation axioms are assumed unless explicitly stated.Definition 1.A subset A of a space (X, τ ) is called (1) a Λ-set [13] if it is equal to its kernel (= saturated set), i.e. to the intersection of all open supersets of A. ( The family of all λ-open subsets of a space (X, τ ) shall be denoted by λO(X).
The set of all λ-cluster points is called the λ-closure of A [3] and is denoted by Cl λ (A).A point x ∈ X is said to be a λ-interior point of a subset A ⊂ X [2] if there exists a λ-open set B containing x such that B ⊂ A. The set of all λ-interior points of A is said to be λ-interior of A and is denoted by Int λ (A).
(3) regular open [17] (resp.regular closed A space X is called locally indiscrete [15] if every open set is closed.Recall that a space is rim-compact if it has a basis of open sets with compact boundaries.The graph of a function f : X → Y , denoted by G(f ), is the subset {(x, f (x)) : x ∈ X} of the product space X × Y .A subset A of a space X is said to be N -closed relative to X [6] if for each cover {B i : i ∈ I} of A by open sets of X, there exists a finite subfamily I 0 ⊂ I such that A ⊂ ∪ i∈I0 Int(Cl(B i )).

Weakly λ-continuous functions
Theorem 2.2.For a function f : X → Y the following are equivalent: (1) f is weakly continuous, (2) f is weakly g-continuous and weakly λ-continuous.
Proof.It follows directly from Theorem 2.4 of [1].
(2) ⇒ (3) : Let F be any regular closed set in Y .Then ( (1) ⇒ (6) : Let U be any preopen set of Y and x ∈ X\f −1 (Cl(U )).There exists an open set G containing f (x) such that G∩U = ∅.We have Recall that a point x ∈ X is said to be in the θ-closure [18] of a subset A of X, denoted by Theorem 2.5.For a function f : X → Y , the following equivalent: (1) f is weakly λ-continuous, Proof.Follows from Theorem 2.5.

Theorem 4 . 4 . 2 Theorem 4 . 5 . 2 Corollary 4 . 6 .
Let f, g : X → Y be weakly λ-continuous functions and Y be Urysohn.If λO(X) is closed under the finite intersections, then the set {x ∈ X :f (x) = g(x)} is λ-closed in X.Proof.Obvious.Let f : X → Y be a weakly λ-continuous function and K be a θ-closed subset of X × Y .Suppose that λO(X) is closed under the finite intersections.Thenp(K ∩ G(f )) is λ-closed in X, where p is the projection of X × Y onto X. Proof.Let x ∈ Cl λ (p(K ∩ G(f ))), G be an open subset of X containing x and H be an open subset of Y containing f (x).Since f is weakly λ-continuous, then x ∈ f −1 (H) ⊂ Int λ (f −1 (Cl(H))).This implies that x ∈ G∩Int λ (f −1 (Cl(H))).Since x ∈ Cl λ (p(K ∩G(f ))), then (G∩Int λ (f −1 (Cl(H))))∩p(K ∩G(f )) contains a point x 0 ∈ X.We have (x 0 , f (x 0 )) ∈ K and f (x 0 ) ∈ Cl(H).Then ∅ = (G × Cl(H)) ∩ K ⊂ Cl(G × H) ∩ K and (x, f (x)) ∈ θ-Cl(K).Since K is θclosed, (x, f (x)) ∈ K ∩ G(f ) and x ∈ p(K ∩ G(f )).This shows that p(K ∩ G(f )) is λ-closed in X.Let f : X → Ybe a function with the θ-closed graph and g : X → Y be a weakly λ-continuous function.Suppose that λO(X) is closed then there exist open sets V 1 and V 2 containing y 1 and y 2 , respectively, such that Cl(V 1 ) ∩ Cl(V 2 ) = ∅.Since f is weakly λ-continuous at x 1 and x 2 , then there exist λ-open setsU i for i = 1, 2 containing x i such that f (U i ) ⊂ Cl(V i ).This shows that U 1 ∩ U 2 = ∅ and hence X is λ-T 2 . 2 Theorem 4.2.If f : X → Y is weakly λ-continuous and g : Y → Z is continuous, then the composition gof : X → Z is weakly λ-continuous.Proof.Let x ∈ X and A be an open set of Z containing g(f (x)).We haveg −1 (A) is an open set of Y containing f (x).Then there exists a λ-open set B containing x such that f (B) ⊂ Cl(g −1 (A)).Since g is continuous, then (gof )(B) ⊂ g(Cl(g −1 (A))) ⊂ Cl(A).Thus, gof is weakly λ-continuous.2 Remark 4.3.Here we have an observation concerning λ-connectedness.By definition, if a space X can not be written as the union of two nonempty disjoint λ-open sets, then X is said to be λ-connected.It is obvious that every λconnected space is indiscrete.Because we know that if a space is not indiscrete, then there is a nontrivial open set.This set and its complement provide a decomposition of the space into nonempty disjoint λ-open sets.Hence every λconnected space must be indiscrete and therefore the notion is not interesting.