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.
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 Velic̆ko [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 Xcontaining 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≠ϕ 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 Clθ(A). A subset A is said to be θ-closed [2] if A=Clθ(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≠ϕ 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 ωClθ(A). A subset A is said to be ω-θ-closed if A=ωClθ(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 ωIntθ(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→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→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→Y, the following properties are equivalent:
f is θ-ω-continuous;
ωClθ(f-1(B))⊆f-1(Clθ(B)) for every subset B of Y;
f(ωClθ(A))⊆Clθ(f(A)) for every subset A of X.
Proof.
(1)⇒(2) Let B be any subset of Y. Suppose that x∉f-1(Clθ(B)). Then f(x)∉Clθ(B) and there exists an open set V containing f(x) such that Cl(V)∩B=ϕ. 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=ϕ and ωCl(U)∩f-1(B)=ϕ. This shows that x∉ωClθ(f-1(B)). Thus, we obtain ωClθ(f-1(B))⊆f-1(Clθ(B)).
(2)⇒(1) Let x∈X and V be an open set of Y containing f(x). Then we have Cl(V)∩(Y-Cl(V))=ϕ and f(x)∉Clθ(Y-Cl(V)). Hence, x∉f-1(Clθ(Y-Cl(V))) and x∉ωClθ(f-1(Y-Cl(V))). There exists U∈ωO(X,x) such that ωCl(U)∩f-1(Y-Cl(V))=ϕ and hence f(ωCl(U))⊆Cl(V). Therefore, f is θ-ω-continuous.
(2)⇒(3) Let A be any subset of X. Then we have ωClθ(A)⊆ωClθ(f-1(f(A)))⊆f-1(Clθ(f(A))) and hence f(ωClθ(A))⊆Clθ(f(A)).
(3)⇒(2) Let B be a subset of Y. We have f(ωClθ(f-1(B)))⊆Clθ(f(f-1(B)))⊆Clθ(B) and hence ωClθ(f-1(B))⊆f-1(Clθ(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∉ωClθ(X-U)=X-U, and so there exists an ω-open set V with x∈V such that ωCl(V)∩(X-U)=ϕ. 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∈ωClθ(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)=ϕ. This is a contradiction since x∈ωClθ(X-U).
Theorem 2.5.
For a function f:X→Y, the following properties are equivalent:
f is θ-ω-continuous;
f-1(V)⊆ωIntθ(f-1(Cl(V))) for every open set V of Y;
ωClθ(f-1(V))⊆f-1(Cl(V)) for every open set V of Y.
Proof.
(1)⇒(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∈ωIntθ(f-1(Cl(V))). Therefore, we obtain f-1(V)⊆ωIntθ(f-1(Cl(V))).
(2)⇒(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=ϕ; hence Cl(W)∩V=ϕ. Therefore, we have f-1(Cl(W))∩f-1(V)=ϕ. Since x∈f-1(W), by (2) x∈ωIntθ(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)=ϕ and hence x∉ωClθ(f-1(V)). This shows that ωClθ(f-1(V))⊆f-1(Cl(V)).
(3)⇒(1) Suppose that x∈X and V are any open set of Y containing f(x). Then V∩(Y-Cl(V))=ϕ and f(x)∉Cl(Y-Cl(V)). Therefore x∉f-1(Cl(Y-Cl(V))) and by (3) x∉ωClθ(f-1(Y-Cl(V))). There exists U∈ωO(X,x) such that ωCl(U)∩f-1(Y-Cl(V))=ϕ. 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→Y, the following properties are equivalent:
f is θ-ω-continuous;
ωClθ[f-1(Int(Clθ(B)))]⊆f-1(Clθ(B)) for every subset B of Y;
ωClθ[f-1(Int(Cl(V)))]⊆f-1(Cl(V)) for every open set V of Y;
ωClθ[f-1(Int(K))]⊆f-1(K) for every closed set K of Y;
ωClθ[f-1(Int(R))]⊆f-1(R) for every regular closed set R of Y.
Proof.
(1)⇒(2) This follows immediately from Theorem 2.5(3) with V=Int(Clθ(B)).
(2)⇒(3) This is obvious since Clθ(V)=Cl(V) for every open set V of Y.
(3)⇒(4) For any closed set K of Y, Int(K)=Int(Cl(Int(K))) and by (3)
ωClθ(f-1(Int(K)))=ωClθ(f-1(Int(Cl(Int(K)))))⊂f-1(Cl(Int(K)))⊂f-1(K).
(4)⇒(5) This is obvious.
(5)⇒(1) Let V be any open set of Y. Since Cl(V) is regular closed, by (5) ωClθ(f-1(V)))⊂ωClθ(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→Y, the following properties are equivalent:
f is θ-ω-continuous;
ωClθ[f-1(Int(Cl(G)))]⊆f-1(Cl(G)) for every β-open set G of Y;
ωClθ[f-1(Int(Cl(G)))]⊆f-1(Cl(G)) for every semi-open set G of Y.
Proof.
(1)⇒(2) This is obvious by Theorem 2.6(5) since Cl(G) is regular closed for every β-open set set G.
(2)⇒(3) This is obvious since every semi-open set is β-open.
(3)⇒(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→Y, the following properties are equivalent:
f is θ-ω-continuous;
ωClθ[f-1(Int(Cl(G)))]⊆f-1(Cl(G)) for every preopen set G of Y;
ωClθ[f-1(G)]⊆f-1(Cl(G)) for every preopen set G of Y;
f-1(G)⊂ωIntθ(f-1(Cl(G))) for every preopen set G of Y.
Proof.
(1)⇒(2) The proof follows from Theorem 2.8 (2) since every preopen set is β-open.
(2)⇒(3) This is obvious by the definition of a preopen set.
(3)⇒(4) Let G be any preopen set of Y. Then, by (3) we have
X-ωIntθ(f-1(Cl(G)))=ωClθ(X-f-1(Cl(G)))=ωClθ(f-1(Y-Cl(G)))⊂f-1(Cl(Y-Cl(G)))=f-1(Y-Int(Cl(G)))⊂f-1(Y-G)=X-f-1(G).
Therefore, we obtain f-1(G)⊂ωIntθ(f-1(Cl(G))).
(4)⇒(1) This is obvious by Theorem 2.5 since every open set is preopen.
Definition 2.10.
A function f:X→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→Y, the following assertions are equivalent:
f is almost ω-continuous;
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));
f-1(F)∈ωC(X) for every F∈RC(Y);
f-1(V)∈ωO(X) for every V∈RO(Y).
Proposition 2.12.
For a function f:X→Y, the following properties hold:
if f is almost ω-continuous, then it is θ-ω-continuous;
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={ϕ,X,{A},{B},{A,B}}. Then, the function f:(X,τ)→(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≠ϕ.
Lemma 2.14.
For an ω-open set U in a space X, ωCl(U)=ωClθ(U).
Proof.
By definition, ωCl(U)⊆ωClθ(U). Let x∈ωClθ(U). Then for any ω-open set V containing x, ωCl(V)∩U≠ϕ. Let z∈ωCl(V)∩U. Then U∩V≠ϕ and x∈ωCl(U). Thus ωClθ(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→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→Y be a function and g:X→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→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→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={ϕ,X,{a,b}}, and σ={ϕ,X,{c}}. Define a function f:(X,τ)→(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={ϕ,X,{A},{B},{A,B}} and σ={ϕ,X,{A},{A,B}}. Then, the identity function f:(X,τ)→(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→Y, the following properties are equivalent:
f is strongly θ-ω-continuous;
f-1(V) is ω-θ-open in X for every open set V of Y;
f-1(F) is ω-θ-closed in X for every closed set F of Y;
f(ωClθ(A))⊆Cl(f(A)) for every subset A of X;
ωClθ(f-1(B))⊆f-1(Cl(B)) for every subset B of Y.
Proof.
(1)⇒(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)⇒(3) This is obvious.
(3)⇒(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 ωClθ(A)⊆ωClθ(f-1(f(A)))⊆ωClθ(f-1(Cl(f(A))))=f-1(Cl(f(A))). Therefore, we obtain f(ωClθ(A))⊆Cl(f(A)).
(4)⇒(5) Let B be any subset of Y. By (4), we obtain f(ωClθ(f-1(B)))⊆Cl(f(f-1(B)))⊆Cl(B) and hence ωClθ(f-1(B))⊆f-1(Cl(B)).
(5)⇒(1) Let x∈X and V be any open neighborhood of f(x). Since Y-V is closed in Y, we have ωClθ(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→Y, the following properties are equivalent:
f is almost weakly ω-continuous;
f is ω-continuous;
f strongly θ-ω-continuous.
Proof.
(1)⇒(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)⇒(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)⇒(1) The proof follows immediately from the definition.
Corollary 3.8.
Let Y be a regular space. Then, for a function f:X→Y, the following properties are equivalent:
f is strongly θ-ω-continuous;
f is ω-continuous;
f is almost ω-continuous;
f is θ-ω-continuous;
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→Y is strongly θ-ω-continuous.
Proof
Sufficiency.
Let f:X→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→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→Y be a function and g:X→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→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→Y be a function and g:X→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→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=ϕ. 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=ϕ. 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=ϕ (resp., ωCl(U)∩ωCl(V)=ϕ).
Theorem 3.15.
If f:X→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=ϕ. 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))=ϕ, hence ωCl(G)∩ωCl(H)=ϕ. 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→Y be strongly θ-ω-continuous and Kω-closed relative to X, then f(K) is a compact set of Y.
Proof.
Suppose that f:X→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→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→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→Y is said to be pre-ω-open if f(U)∈ωO(Y) for every U∈ωO(X).
Proposition 3.19.
Let f:X→Y and g:Y→Z be functions and let g∘f:X→Z be strongly θ-ω-continuous. If f:X→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∘f) is strongly θ-ω-continuous, there exists U∈ωO(X,x) such that (g∘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∘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→Y is said to be ω-irresolute if f-1(V)∈ωO(X) for each V∈ωO(Y).
Lemma 3.21.
If f:X→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→Y and g:Y→Z be functions. Then, the following properties hold.
If f is strongly θ-ω-continuous and g is continuous, then the composition g∘f is strongly θ-ω-continuous.
If f is ω-irresolute and g is strongly θ-ω-continuous, then the composition g∘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:
X is Lindelöf;
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→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∈ℕ} of X. Hence {Vαn:n∈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∈ℕ}⊆Λ such that X=⋃n∈ℕCl(Uαn).
Theorem 3.27.
Let f:X→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∈ℕ} such that X=⋃n∈ℕ(Uα(xn)). Thus Y=f(⋃n∈ℕ(Uα(xn)))⊆⋃n∈ℕf(Uα(xn))⊆⋃n∈ℕCl(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→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→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∈NV(yn). Therefore, we have X=f-1(⋃n∈NV(yn))=⋃n∈Nf-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→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.
FominS.Extensions of topological spaces194344471480MR0008686ZBL0061.39601N. V. VeličkoN.H-closed topological spaces1968782103118HdeibH. Z.ω-continuous functions1989162136142HdeibH. Z.ω-closed mappings1982161-26578MR677814ZBL0574.54008Al-OmariA.NooraniM. S. M.Regular generalized ω-closed sets20072007111629210.1155/2007/16292MR2306366NoiriT.Al-OmariA.NooraniM. S. M.On ωb-open sets and b-Lindelöf spaces20081339MR2446153NoiriT.Al-OmariA.NooraniM. S. M.Slightly ω-continuous functions20094197106MR2529036NoiriT.Al-OmariA.NooraniM. S. M.Weak forms of ω-open sets and decompositions
of continuity2009217384Al-OmariA.NooraniM. S. M.Contra-ω-continuous and almost contra-ω-continuous20072007134046910.1155/2007/40469MR2350525WillardS.1970Reading, Mass, USAAddison-Wesleyxii+369MR0264581LevineN.Semi-open sets and semi-continuity in topological spaces1963703641MR016675210.2307/2312781ZBL0113.16304MashhourA. S.Abd El-MonsefM. E.El-DeebS. N.On precontinuous and weak precontinuous functions1982534753Abd El-MonsefM. E.El-DeebS. N.MahmoudR. A.β-open sets and β-continuous mapping19831217790MR828081NoiriT.On δ-continuous functions1979/1980162161166MR577894PorterJ.ThomasJ.On H-closed and minimal Hausdorff spaces1969138159170MR023826810.2307/1994905Al-ZoubiK. Y.On generalized ω-closed sets20052005132011202110.1155/IJMMS.2005.2011MR2177691SingalM. K.MathurA.On nearly-compact spaces196924702710MR0257979ZBL0188.28005WillardS.DissanayakeU. N. B.The almost Lindelöf degree1984274452455MR763044ZBL0551.54003