Weak and Strong Forms of [omega] -Continuous Functions

Abstract

We introduce weak and strong forms of ω -continuous functions, namely, θ -ω -continuous and strongly θ -ω -continuous functions, and investigate their fundamental properties.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ahmad Al-Omari 1 and Takashi Noiri 2 and Mohd. Salmi Md. Noorani 3

Recommended by Naseer Shahzad

1, Department of Mathematics, Faculty of Science, Mu'tah University, P.O. Box 7, Karak 61710, Jordan
2, 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken 869-5142, Japan
3, School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Received 23 July 2009; Accepted 4 November 2009

1. Introduction

In 1943, Fomin [1] introduced the notion of θ -continuity. In 1968, the notions of θ -open subsets, θ -closed subsets, and θ -closure were introduced by Velic ko [2]. In 1989, Hdeib [3] introduced the notion of ω -continuity. The main purpose of the present paper is to introduce and investigate fundamental properties of weak and strong forms of ω -continuous functions. Throughout this paper, (X,τ) and (Y,σ) stand for topological spaces (called simply spaces) with no separation axioms assumed unless otherwise stated. For a subset A of X , the closure of A and the interior of A will be denoted by Cl(A) and Int(A) , respectively. Let (X,τ) be a space and A a subset of X . A point x∈X is called a condensation point of A if for each U∈τ with x∈U , the set U∩A is uncountable. However, A is said to be ω -closed [4] if it contains all its condensation points. The complement of an ω -closed set is said to be ω -open. It is well known that a subset W of a space (X,τ) is ω -open if and only if for each x∈W , there exists U∈τ such that x∈U and U-W is countable. The family of all ω -open subsets of a space (X,τ) , denoted by _τω or ωO(X) , forms a topology on X finer than τ . The family of all ω -open sets of X containing x∈X is denoted by ωO(X,x) . The ω -closure and the ω -interior, that can be defined in the same way as Cl(A) and Int(A) , respectively, will be denoted by ωCl(A) and ωInt(A) . Several characterizations of ω -closed subsets were provided in [5-8].

A point x of X is called a θ -cluster points of A if Cl(U)∩A≠[varphi] for every open set U of X containing x . The set of all θ -cluster points of A is called the θ -closure of A and is denoted by C_lθ (A) . A subset A is said to be θ -closed [2] if A=C_lθ (A) . The complement of a θ -closed set is said to be θ -open. A point x of X is called an ω -θ -cluster point of A if ωCl(U)∩A≠[varphi] for every ω -open set U of X containing x . The set of all ω -θ -cluster points of A is called the ω -θ -closure of A and is denoted by ωC_lθ (A) . A subset A is said to be ω -θ -closed if A=ωC_lθ (A) . The complement of a ω -θ -closed set is said to be ω -θ -open. The ω -θ -interior of A is defined by the union of all ω -θ -open sets contained in A and is denoted by ω_Intθ (A) .

2. θ -ω -Continuous Functions

We begin by recalling the following definition. Next, we introduce a relatively new notion.

Definition 2.1.

A function f:X[arrow right]Y is said to be ω -continuous (see [3]) (resp., almost weakly ω -continuous (see [9])) if for each x∈X and each open set V of Y containing f(x) , there exists U∈ωO(X,x) such that f(U)⊆V (resp., f(U)⊆Cl(V)) .

Definition 2.2.

A function f:X[arrow right]Y is said to be θ -ω -continuous if for each x∈X and each open set V of Y containing f(x) , there exists U∈ωO(X,x) such that f(ωCl(U))⊆Cl(V) .

Next, several characterizations of θ -ω -continuous functions are obtained.

Theorem 2.3.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is θ -ω -continuous;

(2) ω_Clθ (^f-1 (B))⊆^f-1 (_Clθ (B)) for every subset B of Y ;

(3) f(ω_Clθ (A))⊆_Clθ (f(A)) for every subset A of X .

Proof.

(1)[implies] (2) Let B be any subset of Y . Suppose that x∉^f-1 (C_lθ (B)) . Then f(x)∉C_lθ (B) and there exists an open set V containing f(x) such that Cl(V)∩B=[varphi] . Since f is θ -ω -continuous, there exists U∈ωO(X,x) such that f(ωCl(U))⊆Cl(V) . Therefore, we have f(ωCl(U))∩B=[varphi] and ωCl(U)∩^f-1 (B)=[varphi] . This shows that x∉ωC_lθ (^f-1 (B)) . Thus, we obtain ωC_lθ (^f-1 (B))⊆^f-1 (C_lθ (B)) .

(2)[implies] (1) Let x∈X and V be an open set of Y containing f(x) . Then we have Cl(V)∩(Y-Cl(V))=[varphi] and f(x)∉C_lθ (Y-Cl(V)) . Hence, x∉^f-1 (C_lθ (Y-Cl(V))) and x∉ωC_lθ (^f-1 (Y-Cl(V))) . There exists U∈ωO(X,x) such that ωCl(U)∩^f-1 (Y-Cl(V))=[varphi] and hence f(ωCl(U))⊆Cl(V) . Therefore, f is θ -ω -continuous.

(2)[implies] (3) Let A be any subset of X . Then we have ωC_lθ (A)⊆ωC_lθ (^f-1 (f(A)))⊆^f-1 (C_lθ (f(A))) and hence f(ωC_lθ (A))⊆C_lθ (f(A)) .

(3)[implies] (2) Let B be a subset of Y . We have f(ωC_lθ (^f-1 (B)))⊆C_lθ (f(^f-1 (B)))⊆C_lθ (B) and hence ωC_lθ (^f-1 (B))⊆^f-1 (C_lθ (B)) .

Proposition 2.4.

A subset U of a space X is ω -θ -open in X if and only if for each x∈U , there exists an ω -open set V with x∈V such that ωCl(V)⊆U .

Proof.

Suppose that U is ω -θ -open in X . Then X-U is ω -θ -closed. Let x∈U . Then x∉ωC_lθ (X-U)=X-U , and so there exists an ω -open set V with x∈V such that ωCl(V)∩(X-U)=[varphi] . Thus ωCl(V)⊆U . Conversely, assume that U is not ω -θ -open. Then X-U is not ω -θ -closed, and so there exists x∈ωC_lθ (X-U) such that x∉X-U . Since x∈U , by hypothesis, there exists an ω -open set V with x∈V such that ωCl(V)⊆U . Thus ωCl(V)∩(X-U)=[varphi] . This is a contradiction since x∈ωC_lθ (X-U) .

Theorem 2.5.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is θ -ω -continuous;

(2) ^f-1 (V)⊆ω_Intθ (^f-1 (Cl(V))) for every open set V of Y ;

(3) ω_Clθ (^f-1 (V))⊆^f-1 (Cl(V)) for every open set V of Y .

Proof.

(1)[implies] (2) Suppose that V is any open set of Y and x∈^f-1 (V) . Then f(x)∈V and there exists U∈ωO(X,x) such that f(ωCl(U))⊆Cl(V) . Therefore, x∈U⊆ωCl(U)⊆^f-1 (Cl(V)) . This shows that x∈ω_Intθ (^f-1 (Cl(V))) . Therefore, we obtain ^f-1 (V)⊆ω_Intθ (^f-1 (Cl(V))) .

(2)[implies] (3) Suppose that V is any open set of Y and x∉^f-1 (Cl(V)) . Then f(x)∉Cl(V) and there exists an open set W containing f(x) such that W∩V=[varphi] ; hence Cl(W)∩V=[varphi] . Therefore, we have ^f-1 (Cl(W))∩^f-1 (V)=[varphi] . Since x∈^f-1 (W) , by (2) x∈ω_Intθ (^f-1 (Cl(W))) , there exists U∈ωO(X,x) such that ωCl(U)⊆^f-1 (Cl(W)) . Thus we have ωCl(U)∩^f-1 (V)=[varphi] and hence x∉ωC_lθ (^f-1 (V)) . This shows that ωC_lθ (^f-1 (V))⊆^f-1 (Cl(V)) .

(3)[implies] (1) Suppose that x∈X and V are any open set of Y containing f(x) . Then V∩(Y-Cl(V))=[varphi] and f(x)∉Cl(Y-Cl(V)) . Therefore x∉^f-1 (Cl(Y-Cl(V))) and by (3) x∉ωC_lθ (^f-1 (Y-Cl(V))) . There exists U∈ωO(X,x) such that ωCl(U)∩^f-1 (Y-Cl(V))=[varphi] . Therefore, we obtain f(ωCl(U))⊆Cl(V) . This shows that f is θ -ω -continuous.

A subset A of X is said to be regular open (resp., regular closed) (see [10]) if A=Int(Cl(A)) (resp., A=Cl(Int(A)) ). Also, the family of all regular open (resp., regular closed) sets of X is denoted by RO(X) (resp., RC(X) ).

Theorem 2.6.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is θ -ω -continuous;

(2) ω_Clθ [^f-1 (Int(_Clθ (B)))]⊆^f-1 (_Clθ (B)) for every subset B of Y ;

(3) ω_Clθ [^f-1 (Int(Cl(V)))]⊆^f-1 (Cl(V)) for every open set V of Y ;

(4) ω_Clθ [^f-1 (Int(K))]⊆^f-1 (K) for every closed set K of Y ;

(5) ω_Clθ [^f-1 (Int(R))]⊆^f-1 (R) for every regular closed set R of Y .

Proof.

(1)[implies] (2) This follows immediately from Theorem 2.5(3) with V=Int(C_lθ (B)) .

(2)[implies] (3) This is obvious since C_lθ (V)=Cl(V) for every open set V of Y.

(3)[implies] (4) For any closed set K of Y , Int(K)=Int(Cl(Int(K))) and by (3) [figure omitted; refer to PDF]

(4)[implies] (5) This is obvious.

(5)[implies] (1) Let V be any open set of Y . Since Cl(V) is regular closed, by (5) ωC_lθ (^f-1 (V)))⊂ωC_lθ (^f-1 (Int(Cl(V)))))⊂^f-1 (Cl(V)) . It follows from Theorem 2.5 that f is θ -ω -continuous.

Definition 2.7.

A subset A of a space X is said to be semi-open (see [11]) (resp., preopen (see [12]), β -open (see [13])) if A⊆Cl(Int(A)) (resp., A⊆Int(Cl(A)) , A⊆Cl(Int(Cl(A))) ).

Theorem 2.8.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is θ -ω -continuous;

(2) ω_Clθ [^f-1 (Int(Cl(G)))]⊆^f-1 (Cl(G)) for every β -open set G of Y ;

(3) ω_Clθ [^f-1 (Int(Cl(G)))]⊆^f-1 (Cl(G)) for every semi-open set G of Y .

Proof.

(1)[implies] (2) This is obvious by Theorem 2.6(5) since Cl(G) is regular closed for every β -open set set G .

(2)[implies] (3) This is obvious since every semi-open set is β -open.

(3)[implies] (1) This follows immediately from Theorem 2.5(3) and since every open set is semi-open.

Theorem 2.9.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is θ -ω -continuous;

(2) ω_Clθ [^f-1 (Int(Cl(G)))]⊆^f-1 (Cl(G)) for every preopen set G of Y ;

(3) ω_Clθ [^f-1 (G)]⊆^f-1 (Cl(G)) for every preopen set G of Y ;

(4) ^f-1 (G)⊂ω_Intθ (^f-1 (Cl(G))) for every preopen set G of Y .

Proof.

(1)[implies] (2) The proof follows from Theorem 2.8 (2) since every preopen set is β -open.

(2)[implies] (3) This is obvious by the definition of a preopen set.

(3)[implies] (4) Let G be any preopen set of Y . Then, by (3) we have [figure omitted; refer to PDF] Therefore, we obtain ^f-1 (G)⊂ω_Intθ (^f-1 (Cl(G))) .

(4)[implies] (1) This is obvious by Theorem 2.5 since every open set is preopen.

Definition 2.10.

A function f:X[arrow right]Y is said to be almost ω -continuous if for each x∈X and each regular open set V of Y containing f(x) , there exists U∈ωO(X,x) such that f(U)⊆V .

Lemma 2.11.

For a function f:X[arrow right]Y , the following assertions are equivalent:

(1) f is almost ω -continuous;

(2) for each x∈X and each open set V of Y containing f(x) , there exists U∈ωO(X,x) such that f(U)⊆Int(Cl(V)) ;

(3) ^f-1 (F)∈ωC(X) for every F∈RC(Y) ;

(4) ^f-1 (V)∈ωO(X) for every V∈RO(Y) .

Proposition 2.12.

For a function f:X[arrow right]Y , the following properties hold:

(1) if f is almost ω -continuous, then it is θ -ω -continuous;

(2) if f is θ -ω -continuous, then it is almost weakly ω -continuous.

Proof.

(1) Suppose that x∈X and V is any open set of Y containing f(x) . Since f is almost ω -continuous, ^f-1 (Int(Cl(V))) is ω -open and ^f-1 (Cl(V)) is ω -closed in X by Lemma 2.11. Now, set U=^f-1 (Int(Cl(V))) . Then we have U∈ωO(X,x) and ωCl(U)⊆^f-1 (Cl(V)) . Therefore, we obtain f(ωCl(U))⊆Cl(V) . This shows that f is θ -ω -continuous.

(2) The proof follows immediately from the definition.

Example 2.13.

Let X be an uncountable set and let A,B, and C be subsets of X such that each of them is uncountable and the family {A,B,C} is a partition of X . We define the topology τ={[varphi],X,{A},{B},{A,B}} . Then, the function f:(X,τ)[arrow right](X,τ) defined by f(A)=A, f(B)=C, and f(C)=A is θ -ω -continuous (and almost weakly ω -continuous) but is not almost ω -continuous since for _xc ∈C⊆X , A is regular open and f(_xc )∈A but there is not open set _Uxc containing _xc such that f(_Uxc )⊆A.

Question 2.

Is the converse of Proposition 2.12(2) true?

It is clear that, for a subset A of a space X , x∈ωCl(A) if and only if for any ω -open set U containing x , U∩A≠[varphi] .

Lemma 2.14.

For an ω -open set U in a space X , ωCl(U)=ω_Clθ (U) .

Proof.

By definition, ωCl(U)⊆ωC_lθ (U) . Let x∈ωC_lθ (U) . Then for any ω -open set V containing x , ωCl(V)∩U≠[varphi] . Let z∈ωCl(V)∩U . Then U∩V≠[varphi] and x∈ωCl(U) . Thus ωC_lθ (U)⊆ωCl(U) .

Definition 2.15.

A topological space X is said to be ω -regular (resp., ^ω* -regular) if for each ω -closed (resp., closed) set F and each point x∈X-F , there exist disjoint ω -open sets U and V such that x∈U and F⊆V .

Lemma 2.16.

A topological space X is ω -regular (resp., ^ω* -regular) if and only if for each U∈ωO(X) (resp., U∈O(X) ) and each point x∈U , there exists V∈ωO(X,x) such that x∈V⊆ωCl(V)⊆U .

Proposition 2.17.

Let X be an ω -regular space. Then f:X[arrow right]Y is θ -ω -continuous if and only if it is almost weakly ω -continuous.

Proof.

Suppose that f is almost weakly ω -continuous. Let x∈X and V be any open set of Y containing f(x) . Then, there exists U∈ωO(X,x) such that f(U)⊆Cl(V) . Since X is ω -regular, by Lemma 2.16 there exists W∈ωO(X,x) such that x∈W⊆ωCl(W)⊆U . Therefore, we obtain f(ωCl(W))⊆Cl(V) . This shows that f is θ -ω -continuous.

Theorem 2.18.

Let f:X[arrow right]Y be a function and g:X[arrow right]X×Y the graph function of f defined by g(x)=(x,f(x)) for each x∈X . Then g is θ -ω -continuous if and only if f is θ -ω -continuous.

Proof

Necessity.

Suppose that g is θ -ω -continuous. Let x∈X and V be an open set of Y containing f(x) . Then X×V is an open set of X×Y containing g(x) . Since g is θ -ω -continuous, there exists U∈ωO(X,x) such that g(ωCl(U))⊆Cl(X×V)=X×Cl(V) . Therefore, we obtain f(ωCl(U))⊆Cl(V) . This shows that f is θ -ω -continuous.

Sufficiency.

Let x∈X and W be any open set of X×Y containing g(x) . There exist open sets _U1 ⊆X and V⊆Y such that g(x)=(x,f(x))∈_U1 ×V⊆W . Since f is θ -ω -continuous, there exists _U2 ∈ωO(X,x) such that f(ωCl(_U2 ))⊆Cl(V) . Let U=_U1 ∩_U2 , then U∈ωO(X,x) . Therefore, we obtain g(ωCl(U))⊆Cl(_U1 )×f(ωCl(_U2 ))⊆Cl(_U1 )×Cl(V)⊆Cl(W) . This shows that g is θ -ω -continuous.

3. Strongly θ -ω -Continuous Functions

We introduce the following relatively new definition.

Definition 3.1 (see [14]).

A function f:X[arrow right]Y is said to be strongly θ -continuous if for each x∈X and each open set V of Y containing f(x) , there exists an open neighborhood U of x such that f(Cl(U))⊆V .

Definition 3.2.

A function f:X[arrow right]Y is said to be strongly θ -ω -continuous if for each x∈X and each open set V of Y containing f(x) , there exists U∈ωO(X,x) such that f(ωCl(U))⊆V .

Clearly, the following holds and none of its implications is reversible:

Remark 3.3.

Strong θ -ω -continuity is stronger than ω -continuity and is weaker than strong θ -continuity. Strong θ -ω -continuity and continuity are independent of each other as the following examples show.

Example 3.4.

Let X={a,b,c} , τ={[varphi],X,{a,b}} , and σ={[varphi],X,{c}} . Define a function f:(X,τ)[arrow right](X,σ) as follows: f(a)=a, f(b)=f(c)=c. Then f is strongly θ -ω -continuous but it is not continuous.

Example 3.5.

Let X be an uncountable set and let A,B, and C be subsets of X such that each of them is uncountable and the family {A,B,C} is a partition of X. We defined the topology τ={[varphi],X,{A},{B},{A,B}} and σ={[varphi],X,{A},{A,B}}. Then, the identity function f:(X,τ)[arrow right](X,σ) is continuous (and ω -continuous) but is not strongly θ -ω -continuous.

Next, several characterizations of strongly θ -ω -continuous functions are obtained.

Theorem 3.6.

For a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is strongly θ -ω -continuous;

(2) ^f-1 (V) is ω -θ -open in X for every open set V of Y ;

(3) ^f-1 (F) is ω -θ -closed in X for every closed set F of Y ;

(4) f(ω_Clθ (A))⊆Cl(f(A)) for every subset A of X ;

(5) ω_Clθ (^f-1 (B))⊆^f-1 (Cl(B)) for every subset B of Y .

Proof.

(1)[implies] (2) Let V be any open set of Y . Suppose that x∈^f-1 (V) . Since f is strongly θ -ω -continuous, there exists U∈ωO(X,x) such that f(ωCl(U))⊆V . Therefore, we have x∈U⊆ωCl(U)⊆^f-1 (V) . This shows that ^f-1 (V) is ω -θ -open in X .

(2)[implies] (3) This is obvious.

(3)[implies] (4) Let A be any subset of X . Since Cl(f(A)) is closed in Y , by (3) ^f-1 (Cl(f(A))) is ω -θ -closed, and we have ωC_lθ (A)⊆ωC_lθ (^f-1 (f(A)))⊆ωC_lθ (^f-1 (Cl(f(A))))=^f-1 (Cl(f(A))) . Therefore, we obtain f(ωC_lθ (A))⊆Cl(f(A)) .

(4)[implies] (5) Let B be any subset of Y . By (4), we obtain f(ωC_lθ (^f-1 (B)))⊆Cl(f(^f-1 (B)))⊆Cl(B) and hence ωC_lθ (^f-1 (B))⊆^f-1 (Cl(B)) .

(5)[implies] (1) Let x∈X and V be any open neighborhood of f(x) . Since Y-V is closed in Y , we have ωC_lθ (^f-1 (Y-V))⊆^f-1 (Cl(Y-V))=^f-1 (Y-V) . Therefore, ^f-1 (Y-V) is ω -θ -closed in X and ^f-1 (V) is an ω -θ -open set containing x . There exists U∈ωO(X,x) such that ωCl(U)⊆^f-1 (V) and hence f(ωCl(U))⊆V . This shows that f is strongly θ -ω -continuous.

Theorem 3.7.

Let Y be a regular space. Then, for a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is almost weakly ω -continuous;

(2) f is ω -continuous;

(3) f strongly θ -ω -continuous.

Proof.

(1)[implies] (2) Let x∈X and V be an open set of Y containing f(x) . Since Y is regular, there exists an open set W such that f(x)∈W⊆Cl(W)⊆V . Since f is almost weakly ω -continuous, there exists U∈ωO(X,x) such that f(U)⊆Cl(W)⊆V . Therefore f is ω -continuous.

(2)[implies] (3) Let x∈X and V be an open set of Y containing f(x) . Since Y is regular, there exists an open set W such that f(x)∈W⊆Cl(W)⊆V . Since f is ω -continuous, ^f-1 (W) is ω -open and ^f-1 (Cl(W)) is ω -closed. Set U=^f-1 (W) , then since x∈^f-1 (W)⊆^f-1 (Cl(W)) , U∈ωO(X,x) and ωCl(U)⊆^f-1 (Cl(W)) . Consequently, we have f(ωCl(U))⊆Cl(W)⊆V .

(3)[implies] (1) The proof follows immediately from the definition.

Corollary 3.8.

Let Y be a regular space. Then, for a function f:X[arrow right]Y , the following properties are equivalent:

(1) f is strongly θ -ω -continuous;

(2) f is ω -continuous;

(3) f is almost ω -continuous;

(4) f is θ -ω -continuous;

(5) f is almost weakly ω -continuous.

Theorem 3.9.

A space X is ^ω* -regular if and only if, for any space Y , any continuous function f:X[arrow right]Y is strongly θ -ω -continuous.

Proof

Sufficiency.

Let f:X[arrow right]X be the identity function. Then f is continuous and strongly θ -ω -continuous by our hypothesis. For any open set U of X and any points x of U , we have f(x)=x∈U and there exists G∈ωO(X,x) such that f(ωCl(G))⊆U . Therefore, we have x∈G⊆ωCl(G)⊆U . It follows from Lemma 2.16, that is, X is ^ω* -regular.

Necessity.

Suppose that f:X[arrow right]Y is continuous and X is ^ω* -regular. For any x∈X and any open neighborhood V of f(x) , ^f-1 (V) is an open set of X containing x . Since X is ^ω* -regular, there exists U∈ωO(X) such that x∈U⊆ωCl(U)⊆^f-1 (V) by Lemma 2.16. Therefore, we have f(ωCl(U))⊆V . This shows that f is strongly θ -ω -continuous.

Theorem 3.10.

Let f:X[arrow right]Y be a function and g:X[arrow right]X×Y the graph function of f defined by g(x)=(x,f(x)) for each x∈X . If g is strongly θ -ω -continuous, then f is strongly θ -ω -continuous and X is ^ω* -regular.

Proof.

Suppose that g is strongly θ -ω -continuous. First, we show that f is strongly θ -ω -continuous. Let x∈X and V be an open set of Y containing f(x) . Then X×V is an open set of X×Y containing g(x) . Since g is strongly θ -ω -continuous, there exists U∈ωO(X,x) such that g(ωCl(U))⊆X×V . Therefore, we obtain f(ωCl(U))⊆V . Next, we show that X is ^ω* -regular. Let U be any open set of X and x∈U . Since g(x)∈U×Y and U×Y is open in X×Y , there exists G∈ωO(X,x) such that g(ωCl(G))⊆U×Y . Therefore, we obtain x∈G⊆ωCl(G)⊆U and hence X is ^ω* -regular.

Proposition 3.11.

Let X be an ω -regular space. Then f:X[arrow right]Y is strongly θ -ω -continuous if and only if f is ω -continuous.

Proof.

Suppose that f is ω -continuous. Let x∈X and V be any open set of Y containing f(x) . By the ω -continuity of f , we have ^f-1 (V)∈ωO(X,x) and hence there exists U∈ωO(X,x) such that ωCl(U)⊆^f-1 (V) . Therefore, we obtain f(ωCl(U))⊆V . This shows that f is strongly θ -ω -continuous.

Theorem 3.12.

Let f:X[arrow right]Y be a function and g:X[arrow right]X×Y the graph function of f defined by g(x)=(x,f(x)) for each x∈X . If f is strongly θ -ω -continuous and X is ω -regular, then g is strongly θ -ω -continuous.

Proof.

Let x∈X and W be any open set of X×Y containing g(x) . There exist open sets _U1 ⊆X and V⊆Y such that g(x)=(x,f(x))∈_U1 ×V⊆W . Since f is strongly θ -ω -continuous, there exists _U2 ∈ωO(X,x) such that f(ωCl(_U2 ))⊆V . Since X is ω -regular and _U1 ∩_U2 ∈ωO(X,x) , there exists U∈ωO(X,x) such that x∈U⊆ωCl(U)⊆_U1 ∩_U2 (by Lemma 2.16). Therefore, we obtain g(ωCl(U))⊆_U1 ×f(ωCl(_U2 ))⊆_U1 ×V⊆W . This shows that g is strongly θ -ω -continuous.

Theorem 3.13.

Suppose that the product of two ω -open sets of X is ω -open. If f:X[arrow right]Y is strongly θ -ω -continuous injection and Y is Hausdorff, then E={(x,y):f(x)=f(y)} is ω -θ -closed in X×X .

Proof.

Suppose that (x,y)∉E . Then f(x)≠f(y) . Since Y is Hausdorff, there exist open sets V and U containing f(x) and f(y) , respectively, such that U∩V=[varphi] . Since f is strongly θ -ω -continuous, there exist G∈ωO(X,x) and H∈ωO(X,y) such that f(ωCl(G))⊆V and f(ωCl(H))⊆U . Set D=G×H . It follows that (x,y)∈D∈ωO(X×Y) and ωCl(G×H)∩E⊆[ωCl(G)×ωCl(H)]∩E=[varphi] . By Proposition 2.4, E is ω -θ -closed in X×X .

Definition 3.14 (see [9]).

A space X is said to be ω -_T2 -space (resp., ω -Urysohn ) if for each pair of distinct points x and y in X , there exist U∈ωO(X,x) and V∈ωO(X,y) such that U∩V=[varphi] (resp., ωCl(U)∩ωCl(V)=[varphi] ).

Theorem 3.15.

If f:X[arrow right]Y is strongly θ -ω -continuous injection and Y is _T0 -space (resp., Hausdorff), then X is ω -_T2 -space (resp., ω -Urysohn).

Proof.

(1) Suppose that Y is _T0 -space. Let x and y be any distinct points of X . Since f is injective, f(x)≠f(y) and there exists either an open neighborhood V of f(x) not containing f(y) or an open neighborhood W of f(y) not containing f(x) . If the first case holds, then there exists U∈ωO(X,x) such that f(ωCl(U))⊆V . Therefore, we obtain f(y)∉f(ωCl(U)) and hence X-ωCl(U)∈ωO(X,y) . If the second case holds, then we obtain a similar result. Therefore, X is ω -_T2 .

(2) Suppose that Y is Hausdorff. Let x and y be any distinct points of X . Then f(x)≠f(y) . Since Y is Hausdorff, there exist open sets V and U containing f(x) and f(y) , respectively, such that U∩V=[varphi] . Since f is strongly θ -ω -continuous, there exist G∈ωO(X,x) and H∈ωO(X,y) such that f(ωCl(G))⊆V and f(ωCl(H))⊆U . It follows that f(ωCl(G))∩f(ωCl(H))=[varphi] , hence ωCl(G)∩ωCl(H)=[varphi] . This shows that X is ω -Urysohn .

A subset K of a space X is said to be ω -closed relative to X if for every cover {_Vα :α∈Λ} of K by ω -open sets of X , there exists a finite subset _Λ0 of Λ such that K⊆∪{ωCl(_Vα ):α∈_Λ0 } .

Theorem 3.16.

Let f:X[arrow right]Y be strongly θ -ω -continuous and K ω -closed relative to X , then f(K) is a compact set of Y .

Proof.

Suppose that f:X[arrow right]Y is a strongly θ -ω -continuous function and K is ω -closed relative to X . Let {_Vα :α∈Λ} be an open cover of f(K) . For each point x∈K , there exists α(x)∈Λ such that f(x)∈_Vα(x) . Since f is strongly θ -ω -continuous, there exists _Ux ∈ωO(X,x) such that f(ωCl(_Ux ))⊆_Vα(x) . The family {_Ux :x∈K} is a cover of K by ω -open sets of X and hence there exists a finite subset _K* of K such that K⊆_...x∈K* ωCl(_Ux ) . Therefore, we obtain f(K)⊆_{...x∈K*Vα(x)} . This shows that f(K) is compact.

Recall that a subset A of a space X is quasi H -closed relative to X if for every cover {_Vα :α∈Λ} of A by open sets of X , there exist a finite subset _Λ0 of Λ such that A⊆∪{Cl(_Vα ):α∈_Λ0 } . A space X is said to be quasi H -closed (see [15]) if X is quasi H -closed relative to X .

Theorem 3.17.

Let f:X[arrow right]Y be θ -ω -continuous and Kω -closed relative to X , then f(K) is quasi H -closed relative to Y .

Proof.

Suppose that f:X[arrow right]Y is a θ -ω -continuous function and K is ω -closed relative to X . Let {_Vα :α∈Λ} be an open cover of f(K) . For each point x∈K , there exists α(x)∈Λ such that f(x)∈_Vα(x) . Since f is θ -ω -continuous, there exists _Ux ∈ωO(X,x) such that f(ωCl(_Ux ))⊆Cl(_Vα(x) ) . The family {_Ux :x∈K} is a cover of K by ω -open sets of X and hence there exists a finite subset _K* of K such that K⊆_...x∈K* ωCl(_Ux ) . Therefore, we obtain f(K)⊆_...x∈K* Cl(_Vα(x) ) . This shows that f(K) is quasi H -closed relative to Y .

Definition 3.18 (see [9]).

A function f:X[arrow right]Y is said to be pre-ω -open if f(U)∈ωO(Y) for every U∈ωO(X) .

Proposition 3.19.

Let f:X[arrow right]Y and g:Y[arrow right]Z be functions and let g[composite function]f:X[arrow right]Z be strongly θ -ω -continuous. If f:X[arrow right]Y is pre-ω -open and bijective, then g is strongly θ -ω -continuous.

Proof.

Let y∈Y and W be any open set of Z containing g(y) . Since f is bijective, y=f(x) for some x∈X . Since (g[composite function]f) is strongly θ -ω -continuous, there exists U∈ωO(X,x) such that (g[composite function]f)(ωCl(U))⊆W . Since f is pre-ω -open and bijective, the image f(A) of an ω -closed set A of X is ω -closed in Y . Therefore, we have ωCl(f(U))⊆f(ωCl(U)) and hence g(ωCl(f(U)))⊆(g[composite function]f)(ωCl(U))⊆W . Since f(U)∈ωO(Y,y) , g is strongly θ -ω -continuous.

Definition 3.20 (see [16]).

A function f:X[arrow right]Y is said to be ω -irresolute if ^f-1 (V)∈ωO(X) for each V∈ωO(Y) .

Lemma 3.21.

If f:X[arrow right]Y is ω -irresolute and V is ω -θ -open in Y , then ^f-1 (V) is ω -θ -open in X .

Proof.

Let V be an ω -θ -open set of Y and x∈^f-1 (V) . There exists W∈ωO(Y) such that f(x)∈W⊆ωCl(W)⊆V . Since f is ω -irresolute, we have ^f-1 (W)∈ωO(X) and ^f-1 (ωCl(W))∈ωC(X) . Therefore, we obtain x∈^f-1 (W)⊆ωCl(^f-1 (W))⊆^f-1 (ωCl(W))⊆^f-1 (V) . This shows that ^f-1 (V) is ω -θ -open in X .

Theorem 3.22.

Let f:X[arrow right]Y and g:Y[arrow right]Z be functions. Then, the following properties hold.

(1) If f is strongly θ -ω -continuous and g is continuous, then the composition g[composite function]f is strongly θ -ω -continuous.

(2) If f is ω -irresolute and g is strongly θ -ω -continuous, then the composition g[composite function]f is strongly θ -ω -continuous.

Proof.

(1) This is obvious from Theorem 3.6.

(2) This follows immediately from Theorem 3.6 and Lemma 3.21.

Theorem 3.23 (see [3]).

For any space X , the following are equivalent:

(1) X is Lindelöf;

(2) every ω -open cover of X has a countable subcover.

Definition 3.24 (see [17]).

A space X is said to be nearly Lindelöf if every regular open cover of X has a countably subcover.

Proposition 3.25.

Let f:X[arrow right]Y be an almost ω -continuous surjection. If X is Lindelöf, then Y is nearly Lindelöf.

Proof.

Let {_Vα :α∈Λ} be a regular open cover of Y. Since f is almost ω -continuous, {^f-1 (_Vα ):α∈Λ} is an ω -open cover of X . Since X is Lindelöf, by Theorem 3.23 there exists a countable subcover {^f-1 (_Vαn ):n∈...} of X . Hence {_Vαn :n∈...} is a countable subcover of Y .

Definition 3.26 (see [18]).

A topological space X is said to be almost Lindelöf if for every open cover {_Uα :α∈Λ} of X there exists a countable subset {_αn :n∈...}⊆Λ such that X=_...n∈... Cl(_Uαn ) .

Theorem 3.27.

Let f:X[arrow right]Y be an almost weakly ω -continuous surjection. If X is Lindelöf, then Y is almost Lindelöf.

Proof.

Let {_Vα :α∈Λ} be an open cover of Y . Let x∈X and _Vα(x) be an open set in Y such that f(x)∈_Vα(x) . Since f is almost weakly ω -continuous, there exists an ω -open set _Uα(x) of X containing x such that f(_Uα(x) )⊆Cl(_Vα(x) ) . Now {_Uα(x) :x∈X} is an ω -open cover of the Lindelöf space X . So by Theorem 3.23, there exists a countable subset {_{Uα(xn )} :n∈...} such that X=_...n∈... (_{Uα(xn )} ) . Thus Y=f(_...n∈... (_{Uα(xn )} ))⊆_...n∈... f(_{Uα(xn )} )⊆_...n∈... Cl(_{Vα(xn )} ) . This shows that Y is almost Lindelöf.

We notice that a subspace A of a space X is Lindelöf if and only if for every cover {_Vα :α∈Λ} of A by open set of X , there exists a countable subset _Λ0 of Λ such that {_Vα :α∈_Λ0 } covers A .

Definition 3.28 (see [4]).

A function f:X[arrow right]Y is said to be ω -closed if the image of every closed subset of X is ω -closed in Y .

Theorem 3.29.

If f:X[arrow right]Y is an ω -closed surjection such that ^f-1 (y) is a Lindelöf subspace for each y∈Y and Y is Lindelöf, then X is Lindelöf.

Proof.

Let {_Uα :α∈Λ} be an open cover of X . Since ^f-1 (y) is a Lindelöf subspace for each y∈Y , there exists a countable subset Λ(y) of Λ such that ^f-1 (y)⊆∪{_Uα :α∈Λ(y)} . Let U(y)=∪{_Uα :α∈Λ(y)} and V(y)=Y-f(X-U(y)) . Since f is ω -closed, V(y) is an ω -open set containing y such that ^f-1 (V(y))⊆U(y) . Then {V(y):y∈Y} is an ω -open cover of the Lindelöf space Y . By Theorem 3.23, there exist countable points of Y , says, _y1 ,_y2 ,...,_yn ,... such that Y=_...n∈N V(_yn ) . Therefore, we have X=^f-1 (_...n∈N V(_yn ))=_...n∈N^f-1 (V(_yn ))⊆_...n∈N (U(_yn ))=_...n∈N (∪{_Uα :α∈Λ(_yn )})=∪{_Uα :α∈Λ(_yn ),n∈N} . This shows that X is Lindelöf.

Theorem 3.30 (see [3]).

Let f be an ω -continuous function from a space X onto a space Y . If X is Lindelöf, then Y is Lindelöf.

Corollary 3.31.

Let f:X[arrow right]Y be an ω -closed and ω -continuous surjection such that ^f-1 (y) is a Lindelöf subspace for each y∈Y . Then X is Lindelöf if and only if Y is Lindelöf.

Proof.

Let X be Lindelöf. It follows from Theorem 3.30 that Y is Lindelöf. The converse is an immediate consequence of Theorem 3.29.

Acknowledgments

This work is financially supported by the Ministry of Higher Education, Malaysia, under FRGS grant no. UKM-ST-06-FRGS0008-2008. The authors would like to thank the referees for useful comments and suggestions. Theorems 2.6, 2.8, and 2.9 are established by suggestions of one of the referees.

References

[1] S. Fomin, "Extensions of topological spaces," Annals of Mathematics. Second Series , vol. 44, pp. 471-480, 1943.

[2] N. N. V. Velicko, " H -closed topological spaces," American Mathematical Society Translations , vol. 78, no. 2, pp. 103-118, 1968.

[3] H. Z. Hdeib, " ω -continuous functions," Dirasat , vol. 16, no. 2, pp. 136-142, 1989.

[4] H. Z. Hdeib, " ω -closed mappings," Revista Colombiana de Matemáticas , vol. 16, no. 1-2, pp. 65-78, 1982.

[5] A. Al-Omari, M. S. M. Noorani, "Regular generalized ω -closed sets," International Journal of Mathematics and Mathematical Sciences , vol. 2007, 2007.

[6] T. Noiri, A. Al-Omari, M. S. M. Noorani, "On ωb -open sets and b -Lindelöf spaces," European Journal of Pure and Applied Mathematics , vol. 1, no. 3, pp. 3-9, 2008.

[7] T. Noiri, A. Al-Omari, M. S. M. Noorani, "Slightly ω -continuous functions," Fasciculi Mathematici , no. 41, pp. 97-106, 2009.

[8] T. Noiri, A. Al-Omari, M. S. M. Noorani, "Weak forms of ω -open sets and decompositions of continuity," European Journal of Pure and Applied Mathematics , vol. 2, no. 1, pp. 73-84, 2009.

[9] A. Al-Omari, M. S. M. Noorani, "Contra-ω -continuous and almost contra-ω -continuous," International Journal of Mathematics and Mathematical Sciences , vol. 2007, 2007.

[10] S. Willard General Topology Spaces , pp. xii+369, Addison-Wesley, Reading, Mass, USA, 1970.

[11] N. Levine, "Semi-open sets and semi-continuity in topological spaces," The American Mathematical Monthly , vol. 70, pp. 36-41, 1963.

[12] A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, "On precontinuous and weak precontinuous functions," Proceedings of the Mathematical and Physical Society of Egypt , vol. 53, pp. 47-53, 1982.

[13] M. E. Abd El-Monsef, S. N. El-Deeb, R. A. Mahmoud, " β -open sets and β -continuous mapping," Bulletin of the Faculty of Science. Assiut University , vol. 12, no. 1, pp. 77-90, 1983.

[14] T. Noiri, "On δ -continuous functions," Journal of the Korean Mathematical Society , vol. 16, no. 2, pp. 161-166, 1979/1980.

[15] J. Porter, J. Thomas, "On H -closed and minimal Hausdorff spaces," Transactions of the American Mathematical Society , vol. 138, pp. 159-170, 1969.

[16] K. Y. Al-Zoubi, "On generalized ω -closed sets," International Journal of Mathematics and Mathematical Sciences , vol. 2005, no. 13, pp. 2011-2021, 2005.

[17] M. K. Singal, A. Mathur, "On nearly-compact spaces," Bollettino dell'Unione Matematica Italiana , vol. 2, no. 4, pp. 702-710, 1969.

[18] S. Willard, U. N. B. Dissanayake, "The almost Lindelöf degree," Canadian Mathematical Bulletin , vol. 27, no. 4, pp. 452-455, 1984.