Weak and Strong Forms of ω-Continuous Functions

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


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 Velicko 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 τ ω 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 / 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 Cl θ A .A subset A is said to be θ-closed 2 if A Cl θ 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-θclosure of A and is denoted by ωCl θ A .A subset A is said to be ω-θ-closed if A ωCl θ 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 ωInt θ A .

θ-ω-Continuous Functions
We begin by recalling the following definition.Next, we introduce a relatively new notion.
Next, several characterizations of θ-ω-continuous functions are obtained.Theorem 2.3.For a function f : X → Y , the following properties are equivalent: Theorem 2.5.For a function f : X → Y , the following properties are equivalent: 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:   Theorem 2.9.For a function f : X → Y , the following properties are equivalent: 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

2.2
Therefore, we obtain 4 ⇒ 1 This is obvious by Theorem 2.5 since every open set is preopen.
Lemma 2.11.For a function f : X → Y , the following assertions are equivalent: Proposition 2.12.For a function f : X → Y , the following properties hold: 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 : Question.Is the converse of Proposition 2. 12

Proof
Necessity.Suppose that

Strongly θ-ω-Continuous Functions
We introduce the following relatively new definition.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: 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-ωcontinuous.
Theorem 3.7.Let Y be a regular space.Then, for a function f : X → Y , the following properties are equivalent: 3 ⇒ 1 The proof follows immediately from the definition.

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Corollary 3.8.Let Y be a regular space.Then, for a function f : X → Y , the following properties are equivalent: 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 → Y is strongly θ-ω-continuous.

Proof
Sufficiency.Let f : X → X be the identity function.Then f is continuous and strongly θω-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 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-ωcontinuous.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 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
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 Definition 3.14 see 9 .A space X is said to be ω-T 2 -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 Proof. 1 Suppose that Y is T 0 -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-T 2 .
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 {V α : α ∈ Λ} of K by ω-open sets of X, there exists a finite subset Λ 0 of Λ such that K ⊆ ∪{ωCl V α : α ∈ Λ 0 }.Theorem 3.16.Let f : X → 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 → Y is a strongly θ-ω-continuous function and K is ω-closed relative to X. Let {V α : α ∈ Λ} be an open cover of f K .For each point x ∈ K, there exists α x ∈ Λ such that f x ∈ V α x .Since f is strongly θ-ω-continuous, there exists U x ∈ ωO X, x such that f ωCl U x ⊆ V α x .The family {U x : 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 U x .Therefore, we obtain f K ⊆ x∈K * V α x .This shows that f K is compact.

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Recall that a subset A of a space X is quasi H-closed relative to X if for every cover {V α : α ∈ Λ} of A by open sets of X, there exist a finite subset Λ 0 of Λ such that A ⊆ ∪{Cl V α : α ∈ Λ 0 }.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 {V α : α ∈ Λ} be an open cover of f K .For each point x ∈ K, there exists α x ∈ Λ such that f x ∈ V α x .Since f is θ-ω-continuous, there exists U x ∈ ωO X, x such that f ωCl U x ⊆ Cl V α x .The family {U x : x ∈ K} is a cover of K by ω-open sets of X and hence there exists a finite subset Theorem 3.22.Let f : X → Y and g : Y → Z be functions.Then, the following properties hold.
Proof.We notice that a subspace A of a space X is Lindel öf if and only if for every cover {V α : α ∈ Λ} of A by open set of X, there exists a countable subset Λ 0 of Λ such that {V α : α ∈ Λ 0 } covers A.

Definition 3 . 1 Remark 3 . 3 .
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: ω-continuous almost weakly ω-continuous almost ω-continuous strongly θ-ω-continuous θ-ω-continuous 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.

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 {U α : α ∈ Λ} be an open cover of X.Since f −1 y is a Lindel öf subspace for each y ∈ Y , there exists a countable subset Λ y of Λ such that f −1 y ⊆ ∪{U α : α ∈ Λ y }.Let U y ∪{U α : 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-opencover of the Lindel öf space Y .By Theorem 3.23, there exist countable points of Y , says, y 1 , y 2 , . . ., y n , . . .such that Y n∈N V y n .Therefore, we haveX f −1 n∈N V y n n∈N f −1 V y n ⊆ n∈N U y n n∈N ∪{U α : α ∈ Λ y n } ∪{U α : α ∈ Λ y n , n ∈ N}.This shows that X is Lindel öf.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.
for every regular closed set R of Y .
This is obvious by Theorem 2.6 5 since Cl G is regular closed for every βopen 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.
1 This is obvious from Theorem 3.6.2Thisfollowsimmediatelyfrom Theorem 3.6 and Lemma 3.21.Definition 3.24 see 17 .A space X is said to be nearly Lindel öf if every regular open cover of X has a countably subcover.Letf : X → Y be an almost ω-continuous surjection.If X is Lindelöf, then Y is nearly Lindelöf.Proof.Let {V α : α ∈ Λ} be a regular open cover of Y. Since f is almost ω-continuous, {f −1 V α : α ∈ Λ} is an ω-open cover of X.Since X isLindel öf, by Theorem 3.23 there exists a countable subcover {f −1 V α n : n ∈ N} of X. Hence {V α n : n ∈ N} is a countable subcover of Y .Definition 3.26 see 18 .A topological space X is said to be almost Lindel öf if for every open cover {U α : α ∈ Λ} of X there exists a countable subset {α n: n ∈ N} ⊆ Λ such that X n∈N Cl U α n .Let f : X → Y be an almost weakly ω-continuous surjection.If X is Lindelöf, then Y is almost Lindelöf.Proof.Let {V α : α ∈ Λ} be an open cover of Y .Let x ∈ X and V α x be an open set in Y such that f x ∈ V α x .Since f is almost weakly ω-continuous, there exists an ω-open set U α x of X containing x such that f U α x ⊆ Cl V α x .Now {U α x : x ∈ X} is an ω-opencover of the Lindel öf space X.So by Theorem 3.23, there exists a countable subset {U α x n : n ∈ N} such that X n∈N U α x n .Thus Y f n∈N U α x n ⊆ n∈N f U α x n ⊆ n∈N Cl V α x n .This shows that Y is almost Lindel öf.
1 X is Lindelöf; 2 every ω-open cover of X has a countable subcover.