SLIGHTLY β-CONTINUOUS FUNCTIONS

We define a function f : X → Y to be slightly β-continuous if for every clopen set V of Y , f−1(V)⊂ Cl(Int(Cl(f−1(V)))). We obtain several properties of such a function. Especially, we define the notion of ultra-regularizations of a topology and obtain interesting characterizations of slightly β-continuous functions by using it. 2000 Mathematics Subject Classification. 54C08.


Introduction.
Semi-open sets, preopen sets, α-sets, and β-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.In 1980 Jain [15] introduced the notion of slightly continuous functions.Recently, Nour [24] defined slightly semi-continuous functions as a weak form of slight continuity and investigated the functions.Quite recently, Noiri and Chae [23] have further investigated slightly semi-continuous functions.On the other hand, Pal and Bhattacharyya [7] defined a function to be faintly precontinuous if the preimages of each clopen set of the codomain is preopen and obtained many properties of such functions.Slight continuity implies both slight semi-continuity and faint precontinuity but not conversely.
In this paper, we introduce the notion of slight β-continuity which is implied by both slight semi-continuity and faint precontinuity.We establish several properties of such functions.Especially, we define the notion of ultra-regularization of a topology and obtain interesting characterizations of slight β-continuity, slight semi-continuity, faint precontinuity and slight continuity.Moreover, we investigate the relationships between slight β-continuity, contra-β-continuity [13], and β-continuity [1].
The following basic properties of the semi-preclosure are useful in the sequel.

Characterizations
Definition 3.1.A function f : (X, τ) → (Y , σ ) is said to be slightly β-continuous (briefly sl.β.c.) if for each point x ∈ X and each clopen set V containing f (x) there exists a β-open set U of X containing x such that f (U) ⊂ V .Theorem 3.2.For a function f : (X, τ) → (Y , σ ), the following statements are equivalent: (a) f is slightly β-continuous; Proof.The proof is easily obtained by using Lemma 2.2.
Let (X, τ) be a topological space.Since the intersection of two clopen sets of (X, τ) is clopen, the clopen subsets of (X, τ) may be used as a base for a topology on X.The topology is called the ultra-regularization of τ and is denoted by τ u .A topological space (X, τ) is said to be ultra regular [12] if τ = τ u .Each element of τ u is said to be δ * -open [29].Note that ultra-regular spaces are known as 0-dimensional spaces.Definition 3.3.A function f : (X, τ) → (Y , σ ) is said to be clopen-continuous [28] if for each point x of X and each open set V containing f (x), there exists a clopen set U containing x such that f (U) ⊂ V .Remark 3.4.A space (X, τ) is ultra-regular if and only if every continuous function f : (X, τ) → (Y , σ ) is clopen-continuous.Theorem 3.5.For a function f : (X, τ) → (Y , σ ), the following statements are equivalent: Proof.The proof is similar to that of Theorem 3.5 and is thus omitted.

Comparisons.
In this section, we investigate the relationships between slightly β-continuous functions and other related functions.For this purpose, we will recall some definitions of functions.Definition 4.1.A function f : X → Y is said to be weakly β-continuous [27] (resp., weakly semi-continuous [6], almost weakly continuous [16], or quasi precontinuous [25]) if for each point x ∈ X and each open set A function is said to be β-irresolute [18] We give an interesting characterization of β-quasi-irresolute functions and make clear the fact that β-irresolute functions are From the above definitions we obtain the following diagram: and f is slightly continuous.Remark 4.12.We may define a function f :

Properties.
The composition of two slightly β-continuous functions need not be slightly β-continuous as shown by the following example due to Pal and Bhattacharyya [7].
This shows that g is sl.β.c.Lemma 5.3 (see Abd El-Monsef et al. [1]).Let X be a topological space and A, U subsets of X.

Proof
Necessity.Let γ be an arbitrarily fixed index and Proof.Suppose that g is sl.β.c.Let F be a clopen set of Y .Then X ×F is a clopen set of X × Y .Since g is sl.β.c., g −1 (X × F) = f −1 (F ) ∈ SPO(X).Therefore, f is sl.β.c.
Theorem 5.9.Let f : X → Y be a sl.β.c.injection.Then Proof.(a) Let x 1 , x 2 be two distinct points of X.Then since f is injective and (c) Let F 1 , F 2 be disjoint closed subsets of X.Since f is closed and injective, f (F 1 ) and f (F 2 ) are disjoint closed subsets of Y .Since Y is ultra normal, f (F 1 ) and f (F 2 ) are separated by disjoint clopen sets V 1 and V 2 .Therefore, we obtain A subset A of a topological space X is said to be semi pre β-closed if for each x ∈ X − A there exists a β-clopen set U containing x such that U ∩ A = ∅.Proof.(a) Let (x, y) ∈ (X × Y )− G(f ).Then y ≠ f (x) and there exist clopen sets V and W such that y ∈ V , f (x) ∈ W , and V ∩ W = ∅.Since f is sl.β.c., there exists a β-clopen set U containing x such that f (U) ⊂ W .Therefore, we obtain V ∩ f (U) = ∅ and hence (U × V ) ∩ G(f ) = ∅ and U × V is a β-clopen set of X × Y .This shows that G(f ) is semi pre β-closed in X × Y .
A topological space X is said to be β-connected [27] if X cannot be expressed as the union of two disjoint nonempty β-open sets.Proof.Assume that Y is not connected.Then there exist nonempty open sets V 1 and V 2 such that V 1 ∩ V 2 = ∅ and V 1 ∪ V 2 = Y .Therefore, V 1 and V 2 are clopen sets of Y .Since f is sl.β.c., f −1 (V 1 ) and f −1 (V 2 ) are β-open sets in X.Moreover, we have ) and f −1 (V 2 ) are nonempty.Therefore, X is not β-connected.This is a contradiction and hence Y is connected.
Corollary 5.12 (see Popa and Noiri [27]).If f : X → Y is a weakly β-continuous surjection and X is β-connected, then Y is connected.Corollary 5.13.If f : X → Y is a contra β-continuous surjection and X is βconnected, then Y is connected.

Example 4 . 6 .Remark 4 . 7 .Example 4 . 8 .Example 4 . 9 .Theorem 4 . 11 .
Let X = {a, b, c}, τ = {∅, {a}, {b}, {a, b},X}, and σ = {∅, {a}, {b, c}, X}.Then the identity f : (X, τ) → (X, σ ) is slightly semi-continuous by[23,  Example 2.1] but not faintly precontinuous as f −1 ({a}) is not preclosed in (X, τ).Contra-β-continuity and β-continuity are independent of each other as Examples 4.8 and 4.9 show.The identity function on the real line with the usual topology is continuous and hence β-continuous.But it is not contra-β-continuous since the preimage of any singleton is not β-open.Let X = {a, b} be the Sierpinski space by setting τ = {∅, {a},X} and σ = {∅, {b},X}.The identity function f : (X, τ) → (Y , σ ) is contra-continuous by[10, Example 2.5] and hence contra-β-continuous but not β-continuous.Definition 4.10.A topological space X is said to be (a) extremally disconnected (briefly E.D.) if the closure of each open set of X is open in X, (b) a PS-space [4] if every preopen set of X is semi-open in X, (c) locally indiscrete[20] if every open set of X is closed in X.For a function f : X → Y , the following properties hold:(a) If f is sl.β.c. and X is E.D., then f is faintly precontinuous.(b) If f is sl.β.c. and X is a PS-space, then f is slightly semi-continuous.(c) If f is sl.β.c. and X is an E.D. and PS-space, then f is slightly continuous.Proof.(a) Let x ∈ X and V ∈ CO(Y , f (x)).Now, put U = f −1 (V ).Since X is E.D., we have U ∈ PO(X, x) by [4, Theorem 5.1] and f (U) ⊂ V .Therefore, f is faintly precontinuous.(b) Since X is a PS-space, every β-open set of X is semi-open by [4, Theorem 2.1] and the result follows easily.(c) Let V ∈ CO(Y ).Then by (a) and (b), f −1 (V ) is semi-regular and pre-clopen in X.Since f −1 (V ) is semi-closed and preopen, we have Int(Cl(f −1

Theorem 4 . 13 .Theorem 4 . 14 .
and only if it is semi-open and preopen.Therefore, by the proof for Theorem 4.11(c) each α-open and α-closed set is clopen.Hence, slight α-continuity is equivalent to slight continuity.For a function f : (X, τ) → (Y , σ ), the following properties hold:(a) If f is sl.β.c. and (Y , σ ) is E.D., then f is β-quasi-irresolute.(b) If f is sl.β.c. and (Y , σ ) is ultra regular, then f is β-continuous.(c) If f is sl.β.c. and (X, τ) is a PS-space and (Y , σ ) is E.D., then f is weakly semicontinuous.(d) If f is sl.β.c. and (Y , σ ) is locally indiscrete, then f is β-continuous and contra β-continuous.Proof.(a) Let x ∈ X and V ∈ SO(Y ) containing f (x).Then we have Cl(V ) = Cl(Int(V )) and hence Cl(V ) is clopen in (Y , σ ) since (Y , σ ) is E.D. Since f is sl.β.c., there exists U ∈ SPO(X, x) such that f (U) ⊂ Cl(V ).Therefore, f is β-quasi-irresolute.(b) Since (Y ,σ ) is ultra regular, σ u = σ and by Theorem 3.7 the proof is obvious.(c) Let x ∈ X and V any open set containing f (x).Then we have Cl(V ) ∈ CO(Y ) since (Y , σ ) is E.D. Since f is sl.β.c., there exists U ∈ SPO(X, x) such that f (U) ⊂ Cl(V ).Since (X, τ) is a PS-space, U ∈ SO(X) by [4, Theorem 2.1], hence f is weakly semi-continuous.(d) Let V be any open set of (Y , σ ).Since (Y , σ ) is locally indiscrete, V is clopen and hence f −1 (V ) is β-open and β-closed in (X, τ).Therefore, f is β-continuous and contra β-continuous.For a function f : X → Y , the following properties hold: (a) If f is sl.β.c., X is E.D. and Y is locally indiscrete, then f is contra-precontinuous.(b) If f is sl.β.c. and X and Y are E.D., then f is almost weakly continuous.Proof.(a) Let F be any closed set of Y .By Theorem 4.13(d), f is contra-β-continuous and f −1 hence f is almost weakly continuous by [26, Theorem 3.1].

Theorem 5 . 10 .
If f : X → Y is sl.β.c. and Y is ultra Hausdorff, then (a) the graph G(f ) of f is semi pre β-closed in the product space X × Y , (b) the set {(x 1 ,x 2 ) : f (x 1 ) = f (x 2 )} is semi pre β-closed in the product space X ×X.

Theorem 5 . 11 .
If f : X → Y is a sl.β.c.surjection and X is β-connected, then Y is connected.