On strongly faint e-continuous functions

A new class of functions, called strongly faint e-continuous function, has been defined and studied. Relationships among strongly faint e-continuous functions and econnected spaces, e-normal spaces and e-compact spaces are investigated. Furthermore, the relationships between strongly faint e-continuous functions and graphs are also investigated. 2000 Mathematics Subject Classification : 54B05, 54C08, 54D10.


Introduction
Recent progress in the study of characterizations and generalizations of continuity, compactness, connectedness, separation axioms etc. has been done by means of several generalized closed sets.The first step of generalizing closed set was done by Levine in 1970 [13].The notion of generalized closed sets has been studied extensively in recent years by many topologists.As a generalization of closed sets, e-closed sets and the related sets were introduced and studied by E. Ekici ([4], [5], [6], [7], [8], [9]).
Nasef and Noiri [18] introduce three classes of strong forms of faintly continuity namely: strongly faint semicontinuity, strongly faint precontinuity and strongly faint β-continuity.Recently Nasef [16] defined strong forms of faint continuity under the terminologies strongly faint α-continuity and strongly faint γ-continuity.In this paper using e-open sets, strongly faint e-continuity is introduced and studied.Moreover, basic properties and preservation theorems of strongly faint e-continuous functions are investigated and relationships between strongly faint e-continuous functions and graphs are investigated.

Preliminaries
Throughout the paper (X, τ ) and (Y, σ) (or simply X and Y ) represent topological spaces on which no separation axioms are assumed unless otherwise mentioned.For a subset A of a space (X, τ ), cl(A), int(A) and X\A denote the closure of A, the interior of A and the complement of A in X, respectively.A point x ∈ X is called a θ-cluster [23] (resp.δ-cluster [23]) The set of all θ-cluster (resp.δ-cluster) points of A is called the θ-closure (resp.δ-closure) of A and is denoted by cl θ (A) (resp.δcl(A)).If A = cl θ (A) (resp.A = δcl(A)), then A is said to be θ-closed (resp.δ-closed).The complement of a θ-closed (resp.δ-closed) set is said to be θ-open [23](resp.δ-open [23]).The union of all θ-open sets contained in a subset A is called the θ-interior of A and is denoted by int θ (A).It follows from [23] that the collection of θ-open sets in a topological space (X, τ ) forms a topology τ θ on X [14].The family of all θ-open (resp.θ-closed) subsets of X is denoted by θO(X) (resp.θC(X)).
Theorem 3.4.Let (X, τ ) be a regular space.Then for a function f : (X, τ ) → (Y, σ) the following properties are equivalent: (i) f is strongly e-continuous.(ii) f is strongly faint e-continuous.
Recall that, a topological space (Y, σ) is said to be a T e -space [2] if every e-open subset of (Y, σ) is open.
A space X is said to be submaximal if each dense subset of X is open in X and extremaly disconnected (briefly ED) if the closure of each open set of X is open in X.
Theorem 3.5.Let (Y, σ) be a submaximal ED, T e -space.Then the following are equivalent for a function f : (X, τ ) → (Y, σ): Proof.From [18], we have (i) Theorem 3.7.The following statements hold for functions f : X → Y and g : Y → Z: (i) If both f and g are strongly faint e-continuous, then the composition g • f : X → Z is strongly faint e-continuous.(ii) If f strongly faint e-continuous and g is an e-irresolute, then g • f is strongly e-continuous.(iii) If f strongly faint e-continuous and g is a e-continuous, then g • f is strongly θ-continuous.(iv) If f is quasi θ-continuous and g is strongly faint e-continuous, then g • f is strongly faint e-continuous.(v) If f is strongly θ-continuous and g is strongly faint e-continuous, then g • f is strongly faint e-continuous.
Theorem 3.8.Let (X, τ ) be a regular space.Then the set of all points x ∈ X in which a function f : (X, τ ) → (Y, σ) is not strongly faint econtinuous at x is identical with the union of the θ-frontier of the inverse images of e-open subsets of Y containing f (x).
Proof.Necessity.Suppose that f is not strongly faint e-continuous at x ∈ X.Then there exists a e-open set V of Y containing f (x) such that f (U ) is not a subset of V for each U ∈ τ θ containing x. Hence we have U ∩ (X\f −1 (V )) 6 = ∅ for each U ∈ τ θ containing x. Since X is regular, it follows that x ∈ cl θ (X\f −1 (V )).On the other hand we have that, ).This means that x ∈ θF r(f −1 (V )).Sufficiency.Suppose that x ∈ θF r(f −1 (V )) for some V ∈ eO(Y, f (x)).Now, we assume that f is strongly faint e-continuous at x ∈ X.Then there exists U ∈ τ θ containing x such that f (U ) ⊂ V .Therefore, we have . This is a contradiction.This means that f is not strongly faint e-continuous.Definition 6. (i) A space (X, τ ) is said to be e-connected [6,8] (resp.θ-connected [11]) if X cannot be written as the union of two nonempty disjoint e-open (resp.θ-open) sets.(ii) A subset K of a (X, τ ) space is said to be, e-compact [6,8] (resp.θcompact [11]) relative to (X, τ ), if for every cover of K by e-open (resp.θ-open) sets has a finite subcover.A topological space (X, τ ) is e-compact [6,8] It should be mentioned that θ-connected is equivalent with connected (see [11]).Theorem 3.9.If f : (X, τ ) → (Y, σ) is a strongly faint e-continuous surjection function and (X, τ ) is a θ-connected space, then Y is an e-connected space.
Proof.Assume that (Y, σ) is not e-connected.Then there exist nonempty e-open sets V 1 and Therefore (X, τ ) is not θ-connected.This is a contradiction and hence (Y, σ) is e-connected.Theorem 3.10.If f : (X, τ ) → (Y, σ) is a strongly faint e-continuous, then f (K) is e-compact relative to (Y, σ) for each subset K which is θcompact relative to (X, τ ).
Proof.Let {V i : i ∈ I} be any cover of cover of f (K) by e-open sets.For each x ∈ K, there exists i x ∈ I, such that f (x) ∈ V i x .Since f is strongly faint e-continuous, there exists Theorem 3.11.The surjective strongly faint e-continuous image of a θcompact space is e-compact.
Proof.Let f : (X, τ ) → (Y, σ) be a strongly faint e-continuous function from a θ-compact space X onto a space Y .Let {G α : α ∈ I} be any e-open cover of Y .Since f is strongly faint e-continuous,

Separation Axioms
Recall that a topological space (X, τ ) is said to be: (i) e-T 1 [6,8] (resp.θ-T 1 ) if for each pair of distinct points x and y of X, there exists e-open (resp.θ-open) sets U and V containing x and y, respectively such that y / ∈ U and x / ∈ V .(ii) e-T 2 [6,8] (resp.θ-T 2 [22]) if for each pair of distinct points x and y in X, there exists disjoint e-eopen (resp.θ-open) sets U and V in X such that x ∈ U and y ∈ V .
) is strongly faint e-continuous injection and Y is an e-T 1 space, then X is a θ-T 1 (or Hausdorff) space.
Proof.Suppose that Y is e-T 2 .For any pair of distinct points x and y in X, there exist disjoint e-open sets U and V in Y such that f (x) ∈ U and f (y) ∈ V .Since f is strongly faint e-continuous, f −1 (U ) and f −1 (V ) are θ-open in X containing x and y, respectively.Therefore,   Proof.Let F 1 and F 2 be disjoint e-closed subsets of Y .Since f is strongly faint e-continuous, f −1 (F 1 ) and f −1 (F 2 ) are θ-closed sets.Take U = f −1 (F 1 ) and V = f −1 (F 2 ).We have U ∩ V = ∅.Since X is θ-normal, there exist disjoint θ-open sets A and B such that U ⊂ A and V ⊂ B. We obtain that F 1 = f (U ) ⊂ f (A) and F 2 = f (V ) ⊂ f (B) such that f (A) and f (B) are disjoint e-open sets.Thus, Y is e-normal.
(i)⇔(ii): Clear.The relationships between this new class of functions and other corresponding types of functions are shown in the following diagram.s. θ-continuous ← s. f aintly e-continuous -↓ s. f aintly α-continuous % s. f aintly precontinuous s. f aintly semicontinuous -% s. faintly γ-continuous ↑ s. f aintly β-continuous However, none of these implications is reversible as shown by the following examples and well-known facts.Example 3.2.(i) In ([18], Examples 3.2), it is shown that a strong faint semicontinuity which is not a strong faint precontinuity.(ii) In ([16], Examples 4.3, (resp.Examples: 4.4 and 4.5)), it is shown a strong faint semicontinuity which is not a strong faint γ-continuity (resp.a strong faint precontinuity which is not a strong faint γ-continuity and a strong faint γ-continuity which is not a strong faint β-continuity).

Theorem 4 . 4 .
If f, g : X → Y are strongly faint e-continuous functions and Y is e-T 2 , thenE = {x ∈ X: f (x) = g(x)} is closed in X. Proof.Suppose that x / ∈ E. Then f (x) 6 = g(x).Since Y is e-T 2 , there exist V ∈ eO(Y, f(x)) and W ∈ eO(Y, g(x)) such that V ∩ W = ∅.Since f and g are strongly faint e-continuous, there exist a θ-open set U of X containing x and a θ-open set G of X containing x such that f (U ) ⊂ V and g(G) ⊂ W . Set D = U ∩ G. then D ∩ E = ∅ with D a θ-open subset and hence open such that x ∈ D. Then x / ∈ cl(E) and thus E is closed in X. Definition 7. A space (X, τ ) is said to be: (i) θ-regular (resp.e-regular [5]) if for each θ-closed (resp.e-closed) set F and each point x / ∈ F , there exist disjoint θ-open (resp.e-open) sets U and V such that F ⊂ U and x ∈ V .(ii) θ-normal (resp.e-normal [5]) if for any pair of disjoint θ-closed (resp.e-closed) subsets F 1 and F 2 of X, there exist disjoint θ-open (resp.e-open) sets U and V such that F 1 ⊂ U and F 2 ⊂ V .

Theorem 4 . 5 .
If f is strongly faint e-continuous θe-open injection from a θ-regular space (X, τ ) onto a space (Y, σ), then (Y, σ) is e-regular.Proof.Let F be an e-closed subset of Y and y / ∈ F .Take y

Theorem 4 . 6 .
and f (V ) are disjoint e-open sets.This shows that Y is e-regular.If f is strongly faint e-continuous θe-open injection from a θ-normal space (X, τ ) onto a space (Y, σ), then Y is e-normal.

Definition 8 .
A graph G(f ) of a function f : (X, τ ) → (Y, σ) is said to be (e, θ)-closed if for each (x, y) ∈ (X × Y )\G(f ), there exist a θ-open U set of X containing x and an e-open set V of Y containing y such that (U × V ) ∩ G(f ) = ∅.Lemma 4.7.A graph G(f ) of a function f : (X, τ ) → (Y, σ) is (e, θ)closed in X × Y if and only if for each (x, y) ∈ (X × Y ) \ G(f ), there exist a θ-open set U of X containing x and an e-open set V of Y containing y such that f (U ) ∩ V = ∅.Proof.It is an immediate consequence of Definition 8.