CUBO A Mathematical Journal

Vol.19, No¯ 01, (01–15). March 2017

Almost ω-continuous functions defined by ω-open sets due
to Arhangel’skĭı

E. Rosas 1 , C. Carpintero2, M. Salas2, J. Sanabria2 and L. Vásquez
1 Department of Mathematics, Universidad De Oriente,

Núcleo De Sucre, Cumaná, Venezuela.

Departamento de Ciencias Naturales y Exactas, Universidad de la Costa

Barranquilla, Colombia.

2 Department of Mathematics, Universidad De Oriente,

Núcleo De Sucre, Cumaná, Venezuela.

ennisrafael@gmail.com, carpintero.carlos@gmail.com, salasbrown@gmail.com,

jesanabri@gmail.com, eligiovm85@gmail.com

ABSTRACT

In this paper, we apply the notion of ω-open sets due to Arhangel’skĭı [1] to present

and study a new class of functions called almost ω-continuous functions. Relationships

between this new class and other classes of functions are established.

RESUMEN

En este art́ıculo, aplicamos la noción de ω-conjuntos abiertos dada por Arhangel’skĭı

[1] para presentar y estudiar una nueva clase de funciones llamadas funciones casi ω-

continuas. Establecemos relaciones entre esta nueva clase y otras clases de funciones.

Keywords and Phrases: ω-open sets, almost ω-continuous functions.

2010 AMS Mathematics Subject Classification: 54D10.



2 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

1 Introduction and Preliminaries

Generalized open sets play a very important role in General Topology and they are now the research

topics of many topologist worldwide. Indeed, a significant theme in General Topology and Real

Analysis concerns the variously modified forms of continuity and separation axioms, by utilizing

generalized closed sets. Recently, as generalization of closed sets, the notion of β-closed sets were

introduced and studied by Noiri et al. [12] and the notion of ω-closed sets were introduced and

studied by Hdeib [8]. Let (X,τ) be a topological space and let A be a subset of X. We denote

the closure and the interior of A by Cl(A) and Int(A), respectively. 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. A subset A is

said to be ω-closed [8] if it contains all its condensation points. It is well known that a subset W

of a space (X,τ) is ω-open[8] if and only if for each x ∈ W, there exists U ∈ τ such that x ∈ U and

U \ W is countable. Other notion of ω-closed sets were introduced and studied by Arhangel’skĭi

[1]. A subset A of X is called ω-closed [1], if Cl(B) ⊂ A whenever B ⊂ A and B is a countable set.

The complement of an ω-closed set is said to be an ω-open set [1]. In the sequel, we will use the

ω-closed and ω-open sets in the sense of [1]. In the case that, we use the ω-closed and ω-open sets

in the sense of [8], this will be explicitly stated. The family of all ω-open subsets of a topological

space (X,τ) forms a topology on X which is finer than τ. The set of all ω-open sets of (X,τ) is

denoted by ωO(X). The set of all ω-open sets of (X,τ) containing a point x ∈ X is denoted by

ωO(X, x). The intersection of all ω-closed sets containing A is called the ω-closure of A and is

denoted by ωCl(A). The ω-interior of A is defined by the union of all ω-open sets contained in

A and is denoted by ωInt(A). A point x ∈ X is called a θ-cluster point of A if Cl(V) ∩ A ̸= ∅ for

every open set V 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). If A = Clθ(A), then A is said to be θ-closed. The complement of a

θ-closed set is said to be a θ-open set. The union of all θ-open sets contained in A is called the

θ-interior of A and is denoted by intθ(A). It follows from [17] that the collection of all θ-open sets

in a topological space (X,τ) forms a topology on X which is coarser than τ and is denoted by τθ.

A subset A of X is said to be regular open [16] if A = Int(Cl(A)). A subset A of X is said to be

δ-open [17] if it is the union of regular open sets of X. The complement of a regular open (resp.

δ-open) set is called regular closed (resp. δ-closed). The intersection of all δ-closed sets of (X,τ)

containing A is called the δ-closure [17] of A and is denoted by Clδ(A). A subset A of a topological

space (X,τ) is said to be β-open [2] (resp. semiopen [10], preopen [11]) if A ⊂ Cl(int(Cl(A)))

(resp. A ⊂ Cl(int(A)), A ⊂ int(Cl(A))). The complement of a semiopen (resp. preopen, β-open)

set is called a semiclosed (resp. preclosed, β-closed) set. The set of all regular open (resp. regular

closed, δ-open, δ-closed, β-open, preopen, semiclosed, preclosed, β-closed) sets of (X,τ) is denoted

by RO(X) (resp. RC(X), δO(X), δC(X), βO(X), PO(X), SC(X), PC(X), βC(X)). The intersection

of all semiclosed sets of (X,τ) containing A is called the semiclosure [5] of A and is denoted by

sCl(A). In this article, using the notions of ω-open sets given in [1], we introduce and study a new

class of functions called almost ω-continuous functions. The connections between these functions

and other existing well-known related functions are investigated.



CUBO
19, 1 (2017)

Almost ω-continuous functions 3

The following two examples shows that the notions of ω-open set in sense of [1] and ω-open

set in sense of [8] are independent. That means, the topologies τω generated by the ω-open sets

in the sense of [1] and [8] are different.

Example 1.1. Let X = R with the usual topology. Then A = R \ Q is an ω-open set in the sense

of [8], but A is not an ω-open set in the sense of [1].

Example 1.2. Consider the topology of the countable complement on X = R. Then A = {1} is an

ω-open set in the sense of [1], but A is not an ω-open set in the sense of [8].

Definition 1.3. A topological space (X,τ) is said to be:

(1) ω-T1 (resp. r-T1 [7]) if for each pair of distinct points x and y of X, there exist ω-open (resp.

regular open) sets U and V such that x ∈ U, y /∈ U and x /∈ V, y ∈ V.

(2) ω-T2 (resp. r-T2 [7]) if for each pair of distinct points x and y of X, there exist ω-open (resp.

regular open) sets U and V such that x ∈ U, y ∈ V and U ∩ V = ∅.

Lemma 1.4. Let (X,τ) be a space and let A be a subset of X. The following statements are true:

(1) A ∈ PO(X) if and only if sCl(A) = int(Cl(A)) [9].

(2) A ∈ βO(X) if and only if Cl(A) is regular closed [3].

Definition 1.5. A function f : (X,τ) → (Y,σ) is said to be:

(1) ω-continuous [6] if f−1(V) is ω-open in X for every open set V of Y.

(2) almost continuous [15] if f−1(V) is open in X for every regular open set V of Y.

(3) R-map [4] if f−1(V) is regular open in X for every regular open set V of Y.

(4) weakly ω-continuous [6] if for each point x ∈ X and each open subset V in Y containing f(x),

there exists U ∈ ωO(X, x) such that f(U) ⊂ Cl(V).

The proof of the following Lemma is a direct consequence of Definition 1.5(1).

Lemma 1.6. A function f : (X,τ) → (Y,σ) is ω-continuous if and only if f−1(V) ∈ ωC(X) for

every closed set V of Y.

2 Almost ω-continuous functions

Definition 2.1. A function f : (X,τ) → (Y,σ) is said to be almost ω-continuous if for each

point x ∈ X and each open subset V of Y containing f(x), there exists U ∈ ωO(X, x) such that

f(U) ⊂ int(Cl(V)).



4 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

Theorem 2.2. For a function f : (X,τ) → (Y,σ), the following statements are equivalent:

(1) f is almost ω-continuous,

(2) f−1(V) ∈ ωO(X) for every V ∈ RO(Y),

(3) f−1(F) ∈ ωC(X) for every F ∈ RC(Y),

(4) f(ωCl(A)) ⊂ Clδ(f(A)) for every subset A of X,

(5) ωCl(f−1(B)) ⊂ f−1(Clδ(B)) for every subset B of Y,

(6) f−1(F) ∈ ωC(X) for every F ∈ δC(Y),

(7) f−1(V) ∈ ωO(X) for every V ∈ δO(Y).

Proof.

(1) ⇒ (2) Suppose that V ∈ RO(Y) and let x ∈ f−1(V), then f(x) ∈ V. Since V is an open set and

f is an almost ω-continuous function, there exists U ∈ ωO(X, x) such that f(U) ⊂ int(Cl(V)) = V.

Thus x ∈ U ⊂ f−1(f(U)) ⊂ f−1(V) and hence, we obtain that f−1(V) ∈ ωO(X).

(2) ⇒ (3) Let F ∈ RC(Y), then Y \ F ∈ RO(Y). By hypothesis, f−1(Y \ F) ∈ ωO(X) and since

f−1(Y \ F) = X \ f−1(F), we have X \ f−1(F) ∈ ωO(X). Therefore f−1(F) ∈ ωC(X).

(3) ⇒ (4) Suppose that K is a δ-closed set in Y containing f(A). Observe that K = Clδ(K) =
!
{F : K ⊂ F and F ∈ RC(Y)} and so f−1(K) =

!
{f−1(F) : K ⊂ F and F ∈ RC(Y)}. Now, by

part (3), we have that f−1(K) ∈ ωC(X) and A ⊂ f−1(K). Hence ωCl(A) ⊂ f−1(K), and it

follows that f(ωCl(A)) ⊂ K. Since this is true for any δ-closed set K containing f(A), we have

f(ωCl(A)) ⊂ Clδ(f(A)).

(4) ⇒ (5) Let B be a subset of Y, then f−1(B) is a subset of X. By part (4), f(ωCl(f−1(B))) ⊂

Clδ(f(f
−1(B))) ⊂ Clδ(B) and so, ωCl(f

−1(B)) ⊂ f−1(f(ωCl(f−1(B)))) ⊂ f−1(Clδ(B)).

(5) ⇒ (6) Suppose that F ∈ δC(Y), then

ωCl(f−1(F)) ⊂ f−1(Clδ(F)) = f
−1(F).

In consequence, ωCl(f−1(F)) = f−1(F) and hence f−1(F) ∈ ωC(X).

(6) ⇒ (7) Let V ∈ δO(Y), then Y \ V ∈ δC(Y). By hypothesis, f−1(Y \ V) ∈ ωC(X) and since

f−1(Y \ V) = X \ f−1(V), we have X \ f−1(V) ∈ ωC(X). Therefore f−1(V) ∈ ωO(X).

(7) ⇒ (1) Let x ∈ X and let V any open set in Y such that f(x) ∈ V. Put W = int(Cl(V)) and

U = f−1(W). Since Cl(V) is a closed set in Y, we have W = int(Cl(V)) ∈ δO(Y) and by part (7),

U = f−1(W) ∈ ωO(X). Now, f(x) ∈ V = int(V) ⊂ int(Cl(V)) = W, it follows that x ∈ f−1(W) = U

and f(U) = f(f−1(W)) ⊂ W = int(Cl(V)).

We note that Nour [13], has also defined a type of function which he calls almost ω-continuous.

But this definition is given by using the ω-open sets in sense of [8]. The following example shows

that the notions of almost ω-continuous function in the sense of this paper and almost ω-continuous

function in the sense of [13], are independent.



CUBO
19, 1 (2017)

Almost ω-continuous functions 5

Example 2.3. Consider X = R with the countable complement topology τc and Y = R with

the discrete topology τd. Then, the function f : (X,τc) → (Y,τd) defined as f(x) = x, is almost

ω-continuous in the sense of this paper, but f is not almost ω-continuous in the sense [13].

Example 2.4. Let X = R with the topology τ = {∅, R, Q} and Y = {a, b, c} with the topology

σ = {∅, Y, {a}, {b}, {a, b}}. Consider the function f : (X,τ) → (Y,σ) defined as follows:

f(x) =

{
a, if x ∈ Q

b, if x /∈ Q.

Then, f is an almost ω-continuous function in the sense [13], but f is not almost ω-continuous in

the sense of this paper.

Proposition 2.5. Every almost ω-continuous function is weakly ω-continuous.

Proof.

Let x ∈ X and let V an open subset of Y such that f(x) ∈ V. Since f is an almost ω-continuous

function, there exists U ∈ ωO(X) such that x ∈ U and f(U) ⊂ int(Cl(V)) ⊂ Cl(V). Therefore, f is

a weakly ω-continuous function.

The following examples show that the converse of Proposition 2.5 is not true in general.

Example 2.6. Consider the function f in Example 2.4. It is easy to see that f is weakly ω-

continuous but is not almost ω-continuous.

Example 2.7. Let X = R with the topology τ = {∅, R, R \ Q} and Y = {a, b, c} with the topology

σ = {∅, Y, {a}, {b}, {a, b}}. Take A ⊂ Q and define the function F : (X,τ) → (Y,σ) as follows:

f(x) =

⎧
⎪⎪⎨

⎪⎪⎩

a, if x ∈ Q \ A.

c, if x ∈ R \ Q.

b, if x ∈ A.

Then, F is weakly ω-continuous, but F is not almost ω-continuous in in the sense [13].

Theorem 2.8. For a function f : (X,τ) → (Y,σ), the following statements 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) ⊂ sCl(V),

(3) 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,

(4) for each x ∈ X and each δ-open set V of Y containing f(x) there exists U ∈ ωO(X, x) such that

f(U) ⊂ V.



6 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

Proof. (1)⇒(2): Let x ∈ X and V be an open set of Y containing f(x). By part (1), there exists

U ∈ ωO(X, x) such that f(U) ⊂ int(Cl(V)). Since V is a preopen set, then by Lemma 1.4,

f(U) ⊂ sCl(V).

(2)⇒(3): Let x ∈ X and V be a regular open set of Y containing f(x). Then V is an open set of

Y containing f(x). By part (2), there exists U ∈ ωO(X, x) such that f(U) ⊂ sCl(V). Since V is a

preopen set, then by Lemma 1.4, f(U) ⊂ int(Cl(V)) = V.

(3)⇒(4). Let x ∈ X and V be a δ-open set of Y containing f(x). Then, there exists an open set

G containing f(x) such that G ⊂ int(Cl(G)) ⊂ V. Since int(Cl(G)) is a regular open set of Y

containing f(x), by part (3), there exists U ∈ ωO(X, x) such that f(U) ⊂ int(Cl(G)) ⊂ V.

(4)⇒(1). Let x ∈ X and V be an open set of Y containing f(x). Then int(Cl(V)) is a δ-open set of

Y containing f(x). By part (4), there exists U ∈ ωO(X, x) such that f(U) ⊂ int(Cl(V)). Therefore,

f is almost ω-continuous.

Theorem 2.9. For a function f : (X,τ) → (Y,σ), the following statements are equivalent:

(1) f is almost ω-continuous,

(2) f−1(int(Cl(V))) ∈ ωO(X) for every open set V of Y,

(3) f−1(Cl(int(F))) ∈ ωC(X) for every closed set F of Y.

Proof. (1)⇒(2): Let V be an open set in Y. We have to show that f−1(int(Cl(V))) is an ω-open

set in X. Let x ∈ f−1(int(Cl(V))). Then f(x) ∈ int(Cl(V)) and int(Cl(V)) is a regular open set in

Y. Since f is almost ω-continuous, there exists U ∈ ωO(X, x) such that f(U) ⊂ int(Cl(V)). These

implies that x ∈ U ⊂ f−1(int(Cl(V))), in consequence, f−1(int(Cl(V))) is ω-open set in X.

(2)⇒(3): Let F be a closed set of Y. Then Y \ F is an open set of Y. By part (2), we have

f−1(int(Cl(Y \ F)))) is ω-open set in X and as f−1(int(Cl(Y \ F))) = f−1(int(Y \ int(F))) = f−1(Y \

Cl(int(F))) = X \ f−1(int(Cl(F))) then f−1(int(Cl(F))) is an ω-closed set in X.

(3)⇒(1): Let F be a regular closed set of Y. Then F is a closed set of Y. By part (3), f−1(Cl(int(F)))

is an ω-closed set in X. Since F is a regular closed set, then f−1(Cl(int(F))) = f−1(F). Therefore,

f−1(F) is an ω-closed set in X. By Theorem 2.2, f is an almost ω-continuous function.

Theorem 2.10. A function f : (X,ωO(X)) → (Y,σ) is almost ω-continuous if and only if it is

almost continuous.

Proof. This is an immediate consequence of Theorem 2.2.

Theorem 2.11. For a function f : (X,τ) → (Y,σ), the following statements are equivalent:

(1) f is almost ω-continuous,

(2) ωCl(f−1(V)) ⊂ f−1(Cl(V)) for every V ∈ βO(Y),

(3) f−1(int(F)) ⊂ ω int(f−1(F)) for every F ∈ βC(Y),



CUBO
19, 1 (2017)

Almost ω-continuous functions 7

(4) f−1(int(F)) ⊂ ω int(f−1(F)) for every F ∈ SC(Y),

(5) ωCl(f−1(V)) ⊂ f−1(Cl(V)) for every V ∈ SO(Y),

(6) f−1(V) ⊂ ω int(f−1(int(Cl(V)))) for every V ∈ PO(Y).

Proof. (1)⇒ (2): Let V be any β-open set of Y. Since Cl(V) ∈ RC(Y), by Theorem 2.2, f−1(Cl(V))

is ω-closed in X and f−1(V) ⊂ f−1(Cl(V)). Therefore, ωCl(f−1(V)) ⊂ f−1(Cl(V)).

(2)⇒(3): Let F be any β-closed set of Y. Then Y\F is β-open set of Y. By part (2), ωCl(f−1(Y\F)) ⊂

f−1(Cl(Y \F)) and ωCl(X\f−1(F)) ⊂ f−1(Y \int(F)) and hence, X\ωInt(f−1(F)) ⊂ X\f−1(int(F)).

Therefore, f−1(int(F)) ⊂ ωInt(f−1(F)).

(3)⇒(4): This is obvious since SC(Y) ⊂ βC(Y).

(4)⇒(5): Let V be any semiopen set of Y. Then Y \ V is a semiclosed set in Y. By part (4),

f−1(int(Y \ V)) ⊂ ω int(f−1(Y \ V)) and f−1(Y \ Cl(V)) ⊂ ω int(X \ f−1(V)) and hence, X \

f−1(Cl(V)) ⊂ X \ ωCl(f−1(V)). Therefore, ωCl(f−1(V)) ⊂ f−1(Cl(V)).

(5)⇒(1): Let K ∈ RC(Y). Then K ∈ SO(Y) and by part (5), ωCl(f−1(K)) ⊂ f−1(Cl(K)) = f−1(K).

Therefore, f−1(K) is ω-closed in X and hence f is almost ω-continuous by Theorem 2.2.

(1)⇒(6): Let V be any preopen set of Y. Since int(Cl(V)) ∈ RO(Y), by Theorem 2.2, we have

f−1(int(Cl(V))) ∈ ωO(X) and hence

f−1(V) ⊂ f−1(int(Cl(V))) = ω int(f−1(int(Cl(V)))).

(6)⇒(1): Let V be any regular open set of Y. Since V ∈ PO(Y), f−1(V) ⊂ ω int(f−1(int(Cl(V)))) =

ω int(f−1(V)) and hence f−1(V) ∈ ωO(X). It follows from Theorem 2.2, that f is almost ω-

continuous.

As a direct consequence of Theorem 2.11, we obtain the following two corollaries

Corollary 2.12. For a function f : (X,τ) → (Y,σ), the following statements are equivalent:

(1) f is almost ω-continuous,

(2) f−1(V) ⊂ ω int(f−1(sCl(V))) for each preopen set V of Y,

(3) ωCl(f−1(Cl(int(F)))) ⊂ f−1(F) for each preclosed set F of Y,

(4) ωCl(f−1(sInt(F))) ⊂ f−1(F) for each preclosed set F of Y.

Corollary 2.13. For a function f : (X,τ) → (Y,σ), the following statements are equivalent:

(1) f is almost ω-continuous,

(2) for each neighborhood V of f(x), x ∈ ω int(f−1(sCl(V))),

(3) for each neighborhood V of f(x), x ∈ ω int(f−1(int(Cl(V)))).



8 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

Theorem 2.14. Let f : (X,τ) → (Y,σ) be an almost ω-continuous function and let V be any open

subset of Y. If x ∈ ωCl(f−1(V)) \ f−1(V), then f(x) ∈ ωCl(V).

Proof. Let x ∈ X such that x ∈ ωCl(f−1(V)) \ f−1(V) and suppose f(x) /∈ ωCl(V). Then there

exists an ω-open set H containing f(x) such that H ∩ V = ∅. Then Cl(H) ∩ V = ∅ implies

int(Cl(H)) ∩ V = ∅ and int(Cl(H)) is a regular open set. Since f is almost ω-continuous, there

exists an ω-open set U in X containing x such that f(U) ⊂ int(Cl(H)). Therefore, f(U) ∩ V = ∅.

However, since x ∈ ωCl(f−1(V), U ∩ f−1(V) ̸= ∅ for every ω-open set U in X containing x, so that

f(U) ∩ V ̸= ∅. We have a contradiction. It follows that f(x) ∈ ωCl(V).

Recall that the family of all ω-open subsets of a topological space (X,τ) forms a topology on

X finer than τ. From this fact we obtain immediately the following result.

Lemma 2.15. Let A and B be subsets of a topological space (X,τ). If A ∈ ωO(X) and B ∈ τ,

then A ∩ B ∈ ωO(B).

Proof. Since τ is a topology, then the induced topology on B, denoted by τB is {K ∩ B : K ∈ τ}.

Let x ∈ A ∩ B, then x ∈ A and x ∈ B. Since A ∈ ωO(X), there exists U ∈ τ with x ∈ U

and U \ A is countable. Since U ∩ B is an open subset in τB with x ∈ U ∩ B, follows that

(U ∩ B) \ (A ∩ B) = (U \ A) ∩ B. Since U \ A is countable, (U ∩ B) \ (A ∩ B) is countable, in

consequence, A ∩ B ∈ ωO(B).

Theorem 2.16. Let f : (X,τ) → (Y,σ) be an almost ω-continuous function and A ⊂ X. If A ∈ τ,

then f|A : (A,τA) → (Y,σ) is almost ω-continuous.

Proof. It follows from Lemma 2.15.

Theorem 2.17. Let f : (X,τ) → (Y,σ) be a function and U = {Ui : i ∈ I} be an open cover of X.

If f|Ui is almost ω-continuous for each i ∈ I, then f is almost ω-continuous.

Proof. Suppose that V is a regular open set of Y. Since f|Ui is almost ω-continuous for each i ∈ I,

it follows that (f|Ui)
−1(V) is ω-open in Ui. Since f

−1(V) = X ∩ f−1(V) = (
"

i∈I
Ui) ∩ f

−1(V) =
"

i∈I
(Ui ∩ f

−1(V)) =
"

i∈I
(f|Ui)

−1(V), then f−1(V) ∈ ωO(X), which means that f is almost

ω-continuous.

Definition 2.18. Let (X,τ) be a topological space. A filter base Λ is said to be:

(1) ω-convergent to a point x in X if for every U ∈ ωO(X, x), there exists B ∈ Λ such that B ⊂ U.

(2) r-convergent to a point x in X if for every regular open set U of X containing x, there exists

B ∈ Λ such that B ⊂ U.

Theorem 2.19. If f : (X,τ) → (Y,σ) is an almost ω-continuous function, then for each point

x ∈ X and each filter base Λ in X ω-converging to x, the filter base f(Λ) is r-convergent to f(x).



CUBO
19, 1 (2017)

Almost ω-continuous functions 9

Proof. Let x ∈ X and Λ be any filter base in X, ω-converging to x. By Theorem 2.8, for any

regular open set V of (Y,σ) containing f(x), there exists U ∈ ωO(X, x) such that f(U) ⊂ V. Since

Λ is ω-converging to x, there exists B ∈ Λ such that B ⊂ U. This means that f(B) ⊂ V and hence

the filter base f(Λ) is r-convergent to f(x).

Definition 2.20. A net (x
λ
) is said to be ω-convergent to a point x if for every ω-open set V

containing x, there exists an index λ0 such that for λ ≥ λ0, xλ ∈ V.

Theorem 2.21. If f : (X,τ) → (Y,σ) is an almost ω-continuous function, then for each point

x ∈ X and each net (x
λ
) which is ω-convergent to x, the net f((x

λ
)) is r-convergent to f(x).

Proof. The proof is similar to that of Theorem 2.19.

Theorem 2.22. If f : (X,τ) → (Y,σ) is an almost ω-continuous injective function and (Y,σ) is

r-T1, then (X,τ) is ω-T1.

Proof. Suppose that (Y,σ) is r-T1. For any distinct points x and y in X, using the injectivity of

f, f(x) ̸= f(y) and then, there exist regular open sets V and W such that f(x) ∈ V, f(y) /∈ V,

f(x) /∈ W and f(y) ∈ W. Since f is almost ω-continuous, f−1(V) and f−1(W) are ω-open subsets

of (X,τ) such that x ∈ f−1(V), y /∈ f−1(V), x /∈ f−1(W) and y ∈ f−1(W). This shows that (X,τ)

is ω-T1.

Theorem 2.23. If f : (X,τ) → (Y,σ) is an almost ω-continuous injective function and (Y,σ) is

r-T2, then (X,τ) is ω-T2.

Proof. For any pair of distinct points x and y in X, using the injectivity of f, f(x) ̸= f(y) and then,

there exist disjoint regular open sets U and V in Y such that f(x) ∈ U and f(y) ∈ V. Since f is

almost ω-continuous, f−1(U) and f−1(V) are ω-open sets in X containing x and y, respectively.

Therefore, f−1(U) ∩ f−1(V) = ∅ because U ∩ V = ∅. This shows that (X,τ) is ω-T2.

Theorem 2.24. If f : (X,τ) → (Y,σ) is an almost continuous function and g : (X,τ) → (Y,σ) is an

almost ω-continuous function and Y is a r-T2-space, then the set E = {x ∈ X : f(x) = g(x)} is an

ω-closed set in (X,τ).

Proof. If x ∈ X \ E, then it follows that f(x) ̸= g(x). Since Y is r-T2, there exist disjoint regular

open sets V and W of Y such that f(x) ∈ V and g(x) ∈ W. Since f is almost continuous and g

is almost ω-continuous, then f−1(V) is open and g−1(W) is ω-open in X with x ∈ f−1(V) and

x ∈ g−1(W). Put A = f−1(V) ∩ g−1(W). Since ωO(X) is a topology on X finer than τ, we have

A is ω-open in X. Therefore, f(A) ∩ g(A) = ∅ and it follows that x /∈ ωCl(E). This shows that E

is ω-closed in X.

Definition 2.25. A function f : (X,τ) → (Y,σ) is said to be:

(1) ω-irresolute if f−1(V) is ω-open in X for every ω-open set V of Y.



10 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

(2) faintly ω-continuous if for each point x ∈ X and each θ-open set V of Y containing f(x), there

exists U ∈ ωO(X, x) such that f(U) ⊂ V.

Theorem 2.26. A function f : (X,τ) → (Y,σ) is faintly ω-continuous if and only if f−1(V) ∈

ωO(X) for every θ-open set V of Y.

Proof. Suppose that f is faintly ω-continuous. Let V be a θ-open set of Y and x ∈ f−1(V). Since

f(x) ∈ V and f is faintly ω-continuous, there exists U ∈ ωO(X, x) such that f(U) ⊂ V. It follows

that x ∈ U ⊂ f−1(V). Hence f−1(V) is ω-open in X.

Conversely, let x ∈ X and V be a θ-open set of Y containing f(x), by hypothesis f−1(V) is an ω-open

set containing x. Take U = f−1(V), then f(U) ⊂ V. This shows that f is faintly ω-continuous.

As a direct consequence of the Theorem 2.26, we obtain the following corollary.

Corollary 2.27. A function f : (X,τ) → (Y,σ) is faintly ω-continuous if and only if f−1(V) ∈

ωC(X) for every θ-closed set V of Y.

Theorem 2.28. The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) hold for the following properties

of a function f : (X,τ) → (Y,σ):

(1) f is ω-continuous.

(2) f−1(Clδ(B)) is ω-closed in X for every subset B of Y.

(3) f is almost ω-continuous.

(4) f is weakly ω-continuous.

(5) f is faintly ω-continuous.

If, in addition, Y is regular, then the five properties are equivalent of one another.

Proof. (1) ⇒ (2): Since Clδ(B) is closed in Y for every subset B of Y, by Theorem 2.2, f
−1(Clδ(B))

is ω-closed in X.

(2) ⇒ (3): For any subset B of Y, f−1(Clδ(B)) is ω-closed in X and hence we have ωCl(f
−1(B))

⊂ ωCl(f−1(Clδ(B)) = f
−1(Clδ(B)). It follows from Theorem 2.2 that f is almost ω-continuous.

(3) ⇒ (4): This is obvious.

(4) ⇒ (5): Let F be any θ-closed set of Y. Since F is closed, it follows from Lemma 1.6 that, f−1(F)

is ω-closed in X and hence, by Theorem 2.26, f is faintly ω-continuous.

Suppose that Y is regular. We prove that (5) ⇒ (1). Let V be any open set of Y. Since Y is

regular, V is θ-open in Y. By the faintly ω-continuity of f, f−1(V) is ω-open in X. Therefore, f is

ω-continuous.

Definition 2.29. A function f : (X,τ) → (Y,σ) is said to be ω-preopen if f(U) ∈ PO(Y) for every

ω-open set U of X.



CUBO
19, 1 (2017)

Almost ω-continuous functions 11

Theorem 2.30. If a function f : (X,τ) → (Y,σ) is ω-preopen and weakly ω-continuous, then f is

almost ω-continuous.

Proof. Let x ∈ X and let V be an open set of Y containing f(x). Since f is weakly ω-continuous,

there exists U ∈ ωO(X, x) such that f(U) ⊂ Cl(V). Since f is ω-preopen, f(U) ⊂ int(Cl(f(U))) ⊂

int(Cl(V)) and hence f is almost ω-continuous.

Theorem 2.31. Let f : (X,τ) → (Y,σ) and g : (Y,σ) → (Z,η) be functions. Then the composition

g ◦ f : (X,τ) → (Z,η) is almost ω-continuous if f and g satisfy one of the following conditions:

(1) f is almost ω-continuous and g is R-map.

(2) f is ω-irresolute and g is almost ω-continuous.

(3) f is ω-continuous and g is almost continuous.

Proof. (1) Follows from Theorem 2.2 and Definition 1.5.

(2) Follows from Theorem 2.2 and Definition 2.25.

(3) Follows from Theorem 2.2 and Definition 1.5.

Theorem 2.32. If f : X →
∏

i∈I
Yi is almost ω-continuous function then pi ◦ f : X → Yi is almost

ω-continuous for each i ∈ I, where pi is the projection of
∏

i∈I
Yi onto Yi.

Proof. Let V be a regular open set of Yi. Since pi is continuous open, it is an R-map and hence

p−1
i

(V) is regular open in
∏

i∈I
Yi, it follows that f

−1(p−1
i

(V)) = (pi ◦ f)
−1(V) ∈ ωO(X). This

shows that pi ◦ f is almost ω-continuous for each i ∈ I.

Definition 2.33. A topological space (X,τ) is said to be almost regular [14] if for any regular

closed set F of X and any point x ∈ X \ F there exist disjoint open sets U and V such that x ∈ U

and F ⊂ V.

Theorem 2.34. If f : (X,τ) → (Y,σ) is a weakly ω-continuous function and Y is almost regular,

then f is almost ω-continuous.

Proof. Let x ∈ X and let V be an open set of Y containing f(x). By the almost regularity of Y,

there exists a regular open set G of Y such that f(x) ∈ G ⊂ Cl(G) ⊂ int(Cl(V)) [14, Theorem 2.2].

Since f is weakly ω-continuous, there exists U ∈ ωO(X, x) such that f(U) ⊂ Cl(G) ⊂ int(Cl(V)).

Therefore, f is almost ω-continuous.

Definition 2.35. An ω-frontier of a subset A of (X,τ), denoted by ωFr(A), is defined by ωFr(A) =

ωCl(A) ∩ ωCl(X \ A).



12 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

Theorem 2.36. The set of all points x ∈ X in which a function f : (X,τ) → (Y,σ) is not almost

ω-continuous is identical with the union of ω-frontier of the inverse images of regular open sets

containing f(x).

Proof. Suppose that f is not almost ω-continuous at x ∈ X. Then there exists a regular open

set V of Y containing f(x) such that U ∩ (X \ f−1(V)) ̸= ∅ for every U ∈ ωO(X, x). Therefore,

x ∈ ωCl(X \ f−1(V)) = X \ ω int(f−1(V)) and x ∈ f−1(V). Thus, x ∈ ωFr(f−1(U)). Conversely,

suppose that f is almost ω-continuous at x ∈ X and let V be a regular open set of Y containing f(x).

Then there exists U ∈ ωO(X, x) such that U ⊂ f−1(V). That is x ∈ ω int(f−1(V)). Therefore,

x ∈ X \ ωFr(f−1(V)).

Definition 2.37. A function f : (X,τ) → (Y,σ) is said to be complementary almost ω-continuous

if for each regular open set V of Y, f−1(Fr(V)) is ω-closed in X, where Fr(V) denotes the frontier

of V.

Theorem 2.38. If f : (X,τ) → (Y,σ) is weakly ω-continuous and complementary almost ω-

continuous, then f is almost ω-continuous.

Proof. Let x ∈ X and V be a regular open set of Y containing f(x). Then f(x) ∈ Y \ Fr(V) and

hence x ∈ X \ f−1(Fr(V)). Since f is weakly ω-continuous there exists G ∈ ωO(X, x) such that

f(G) ⊂ Cl(V). Put U = G ∩(X\f−1(Fr(V))). Then U ∈ ωO(X, x) and f(U) ⊂ f(G)∩(Y \Fr(V)) ⊂

Cl(V) ∩ (Y \ Fr(V)) = V. This shows that f is almost ω-continuous.

Theorem 2.39. If f : (X,τ) → (Y,σ) is almost ω-continuous, g : (X,τ) → (Y,σ) is weakly ω-

continuous and Y is Hausdorff, then the set {x ∈ X : f(x) = g(x)} is ω-closed in (X,τ).

Proof. Let A = {x ∈ X : f(x) = g(x)} and x ∈ X \ A. Then f(x) ̸= g(x). Since (Y,σ) is Hausdorff,

there exist open sets V and W of Y such that f(x) ∈ V, g(x) ∈ W and V ∩ W = ∅, hence

int(Cl(V)) ∩ Cl(W) = ∅. Since f is almost ω-continuous, there exists G ∈ ωO(X, x) such that

f(G) ⊂ int(Cl(V)). Since g is weakly ω-continuous, there exists H ∈ ωO(X) such that g(H) ⊂

Cl(W). Now put U = G ∩ H, then U ∈ ωO(X, x) and f(U) ∩ g(U) ⊂ int(Cl(V)) ∩ Cl(W) = ∅.

Therefore, we obtain U ∩ A = ∅ and hence A is ω-closed in X.

Theorem 2.40. Suppose that the product of two ω-open sets is ω-open. If f1 : (X1,τ) → (Y,σ)

is weakly ω-continuous, f2 : (X2,τ) → (Y,σ) is almost ω-continuous and (Y,σ) is Hausdorff, then

the set {(x1, x2) ∈ X1 × X2 : f1(x1) = f2(x2)} is ω-closed in X1 × X2.

Proof. Let A = {(x1, x2) ∈ X1 × X2 : f(x1) = f(x2)}. If (x1, x2) ∈ (X1 × X2) \ A, then we have

f(x1) ̸= f(x2). Since (Y,σ) is Hausdorff, there exist disjoint open sets V1 and V2 in Y such that

f(x1) ∈ V1 and f(x2) ∈ V2 and Cl(V1)∩int(Cl(V2)) = ∅. Since f1 (resp. f2) is weakly ω-continuous

(resp. almost ω-continuous), there exists U1 ∈ ωO(X1, x1) such that f(U1) ⊂ Cl(V1) (resp.

U2 ∈ ωO(X2, x2) such that f(ωCl(U1)) ⊂ int(Cl(V2))). Thus, we obtain (x1, x2) ∈ U1 × U2 ⊂

X1 × X2 \ A. Therefore, (X1 × X2) \ A is ω-open and so A is ω-closed in X1 × X2.



CUBO
19, 1 (2017)

Almost ω-continuous functions 13

Theorem 2.41. If g : (X,τ) → (Y,σ) is almost ω-continuous and S is a δ-closed subset of X × Y,

then pX(S ∩ G(g)) is ω-closed in X, where pX represents the projection of X × Y onto X and G(g)

denotes the graph of g.

Proof. Let S be a δ-closed set of X × Y and x ∈ ωCl(pX(S ∩ G(g))). Let U be an open set

of X containing x and V an open set of Y containing g(x). Since g is almost ω-continuous, we

have x ∈ g−1(V) ⊂ ω int(g−1(int(Cl(V)))) and U ∩ ω int(g−1(int(Cl(V)))) ∈ ωO(X, x). Since

x ∈ ωCl(pX(S ∩ G(g))), (U ∩ ω int(g
−1(int(Cl(V))))) ∩ pX(S ∩ G(g)) contains some point u of X.

This implies that (u, g(u)) ∈ S and g(u) ∈ int(Cl(V)). Thus, we have ∅ ̸= (U × int(Cl(V))) ∩ S ⊂

int(Cl(U × V)) ∩ S and hence (x, g(x)) ∈ Clδ(S). Since S is δ-closed, (x, g(x)) ∈ pX(S ∩ G(g)) and

x ∈ pX(S ∩ G(g)). Then pX(S ∩ G(g)) is ω-closed.

Corollary 2.42. If f : (X,τ) → (Y,σ) has a δ-closed graph and g : (X,τ) → (Y,σ) is almost

ω-continuous, then the set {x ∈ X : f(x) = g(x)} is ω-closed in X.

Proof. Since G(f) is δ-closed and pX(G(f) ∩ G(g)) = {x ∈ X : f(x) = g(x)} it follows from Theorem

2.41, that {x ∈ X : f(x) = g(x)} is ω-closed in X.

Theorem 2.43. If for each pair of points x1, x2 distinct in a topological space (X,τ) there exists a

function f on (X,τ) into a Hausdorff space (Y,σ) such that f(x1) ̸= f(x2), f is weakly ω-continuous

at x1 and f is almost ω-continuous at x2, then X is ω-T2.

Proof. Since (Y,σ) is Hausdorff, for each pair of point x1, x2 distinct, there exist disjoint open sets

V1 and V2 of Y containing f(x1) and f(x2), respectively; hence Cl(V1) ∩ int(Cl(V2)) = ∅. Since f

is weakly ω-continuous at x1, there exists U1 ∈ ωO(X, x1) such that f(U1) ⊂ Cl(V1). Since f is

almost ω-continuous at x2, there exists U2 ∈ ωO(X, x2) such that f(U2) ⊂ int(Cl(V2)). Therefore,

we obtain U1 ∩ U2 = ∅. This shows that (X,τ) is ω-T2.

Definition 2.44. A function f : (X,τ) → (Y,σ) is said to have an ω-strongly closed graph if for

each (x, y) ∈ (X × Y) \ G(f), there exists an ω-open subset U of X and an open subset V of Y such

that (U × Cl(V)) ∩ G(f) = ∅.

As a consequence of Definition 2.44, we obtain easily the following Lemma.

Lemma 2.45. A function f : (X,τ) → (Y,σ) has ω-strongly closed graph G(f) if and only if for

each (x, y) ∈ (X × Y) \ G(f) there exists an ω-open set U and an open set V containing x and y,

respectively such that f(U) ∩ Cl(V) = ∅.

Theorem 2.46. If f : (X,τ) → (Y,σ) is an almost ω-continuous function and (Y,σ) is Hausdorff,

then f has an ω-strongly closed graph.

Proof. Let (x, y) ∈ X × Y such that y ̸= f(x). Since (Y,σ) is Hausdorff, there exist open sets V

and W of Y containing f(x) and y, respectively, such that V ∩ W = ∅. Then f(x) ∈ Y \ Cl(W) and



14 E. Rosas, C. Carpintero, M. Salas, J. Sanabria, L. Vásquez CUBO
19, 1 (2017)

Y \Cl(W) is regular open in Y. Since f is almost ω-continuous function, there exists U ∈ ωO(X, x)

such that f(U) ⊂ Y \ Cl(W) and hence f(U) ∩ Cl(W) = ∅. Therefore, by Lemma 2.45 f has an

ω-strongly closed graph.

The following corollary is an immediate consequence of Lemma 2.45, as we can see.

Corollary 2.47. If f : (X,τ) → (Y,σ) is an ω-continuous function and (Y,σ) is Hausdorff, then f

has an ω-strongly closed graph.

Proof. Let (x, y) ∈ X × Y such that y ̸= f(x). Since (Y,σ) is Hausdorff, there exist open sets V

and W of Y containing f(x) and y, respectively such that V ∩ W = ∅. Then f(x) ∈ Y \ Cl(W) and

Y \ Cl(W) is an open set in Y. Since f is weakly ω-continuous function, there exists U ∈ ωO(X, x)

such that U ⊂ f−1(Y \ Cl(W)) and then f(U) ⊂ Y \ Cl(W), hence f(U) ∩ Cl(W) = ∅. Therefore, by

Lemma 2.45 f has an ω-strongly closed graph.

References

[1] A. V. Arhangel’skĭı, Bicompacta that satisfy the Suslin condition hereditarily. Tightness and

free sequences, Dokl. Akad. Nauk SSSR, 199 (1971), 1227-1230.

[2] M. E. Abd El-Monsef, S. N. El-Deep and R. A. Mahmoud, β-open sets and β-continuous

functions, Bull. Fac. Sci. Assiut Univ. A, 12 (1983), 77-90.

[3] D. Andrijevic, Semi-preopen sets, Math. Vesnik, 38 (1986), 24-32.

[4] D. Carnahan, Some properties related to compactness in topological spaces, Ph. D. Thesis,

Univ. Arkansas (1973).

[5] S. G. Crossley and S. K. Hildebrand, Semi-closure, Texas J. Sci., 22 (1971), 99-112.

[6] E. Ekici and S. Jafari, On ω∗-closed sets and their topology, Acta Univ. Apulensis, 22 (2010),

175-184.

[7] E. Ekici, Generalization of perfectly continuous, Regular set-connected and clopen functions,

Acta. Math. Hungar., 107 (3) (2005), 193-206.

[8] H. Z. Hdeib, ω-closed mappings, Rev. Colomb. Mat., 16 (1-2) (1982), 65-78.

[9] D. S. Jankovic, A note on mappings of extremally disconnected spaces, Acta Math. Hungar.,

46 (1985), 83-92.

[10] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,

70 (1963), 36-41.



CUBO
19, 1 (2017)

Almost ω-continuous functions 15

[11] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deep, On precontinuous and weak pre-

continuous mappings, Proc. Math. Phys. Soc. Egypt, 53 (1982), 47-53.

[12] T. Noiri and V. Popa, On Almost β-continuous functions, Acta Math. Hungar., 79 (4) (1998),

329-339.

[13] T. M. Nour, Almost ω-continuous functions, European J. Sci. Res., 8 (1) (2005), 43-47.

[14] M. K. Singal and S. P. Arya, On almost regular spaces, Glasnik Mat., 4 (24) (1969), 89-99.

[15] M. K. Singal and A. R. Singal, Almost-continuous mappings, Yokohama Math. J., 16 (1968),

63-73.

[16] M. Stone, Applications of the theory of boolean rings to general topology, Trans. Amer. Math.

Soc., 41 (1937), 374-381.

[17] N. V. Veličko, H-closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), 103-118.