4_Noiri.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 1, 2009, (73-84)

ISSN 1307-5543 – www.ejpam.com

Weak forms of ω-open sets and decompositions of continuity

Takashi Noiri1, Ahmad Al-omari2∗ and Mohd. Salmi Md. Noorani3

1 2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142 Japan

2Department of Mathematics, Faculty of Science, Mu’tah University, P.O.Box 7, Karak-Jordan

3School of mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,

43600 UKM Bangi, Selangor, Malaysia

Abstract. In this paper, we introduce some generalizations of ω-open sets and investigate some prop-

erties of the sets. Moreover, we use them to obtain decompositions of continuity.

AMS subject classifications: 54C05, 54C08, 54C10

Key words: b-open, ω-open, pre-ω-open, α-ω-open, decomposition of continuity.

1. introduction

Throughout this paper, (X ,τ) and (Y,σ) stand for topological 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. A is said to be ω-closed [8] 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 sets of a space (X ,τ), denoted by τω or

ωO(X ), forms a topology on X finer than τ. Theω-closure andω-interior, that can be defined

∗Corresponding author. Email addresses: t.noiri�nifty.
om (T. Noiri), omarimutah1�yahoo.
om (A. Al-

omari), msn�ukm.my (M. Noorani)

http://www.ejpam.com 73 c© 2009 EJPAM All rights reserved.



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 74

in the same way as Cl(A) and Int(A), respectively, will be denoted by Clω(A) and Intω(A),

respectively. Several characterizations of ω-closed sets were provided in [2,3,8,9,13].

Definition 1.1. A subset A of a space X is said to be

1. α-open [12] if A⊆ Int(Cl(Int(A)));

2. semi-open [10] if A⊆ Cl(Int(A));

3. pre-open [11] if A⊆ Int(Cl(A));

4. β -open [1] if A⊆ Cl(Int(Cl(A)));

5. b-open [5] if A⊆ Cl(Int(A))∪ Int(Cl(A)).

In this paper we introduce and investigate the new notions called b-ω-open sets , pre-ω-

open sets and α-ω-open sets which are weaker than ω-open. Moreover, we use these notions

to obtain decompositions of continuity.

2. Weak forms of ω-open sets

In this section we introduce the following notions.

Definition 2.1. A subset A of a space X is said to be

1. α-ω-open if A⊆ Intω(Cl(Intω(A)));

2. pre-ω-open if A⊆ Intω(Cl(A));

3. β -ω-open if A⊆ Cl(Intω(Cl(A)));

4. b-ω-open if A⊆ Intω(Cl(A))∪ Cl(Intω(A)).

Lemma 2.2. Let (X ,τ) be a topological space, then the following properties hold:

1. every ω-open set is α-ω-open.

2. every α-ω-open set is pre-ω-open.



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 75

3. every pre-ω-open set is b-ω-open.

4. every b-ω-open set is β -ω-open.

Proof. (1) If A is an ω-open set, then A= Intω(A). Since A⊆ Cl(A), then A⊆ Cl(Intω(A))

and A⊆ Intω(Cl(Intω(A))). Therefore A is α-ω-open.

(2) If A is an α-ω-open set, then A ⊆ Intω(Cl(Intω(A))) ⊆ Intω(Cl(A)). Therefore A is pre-

ω-open.

(3) If A is pre-ω-open, then A ⊆ Intω(Cl(A)) ⊆ Intω(Cl(A)) ∪ Cl(Intω(A)). Therefore, A is

b-ω-open.

(4) If A is b-ω-open, then A⊆ Intω(Cl(A))∪Cl(Intω(A))⊆ Cl(Intω(Cl(A)))∪Cl(Intω(A))⊆

Cl(Intω(Cl(A))). Therefore A is β -ω-open.

Since every open set is ω-open, then we have the following diagram for properties of

subsets.

open //

��

α-open //

��

preopen //

��

b-open //

��

β -open

��

ω-open // α-ω-open // pre ω-open // b-ω-open // β -ω-open

The converses need not be true as shown by the following examples.

Example 2.3. Let X = {a, b, c} and τ = {X ,φ, {a}, {b}, {a, b}}. Then {c} is an ω-open (since

X is a countable set) set but it is not β -open.

Example 2.4. Let X = R with the usual topology τ. Let A= Q∩ [0,1]. Then A is a β -open set

which is not b-ω-open.

Example 2.5. Let X = R with the usual topology τ. Let A= (0,1]. Then A is a b-open set which

is not pre-ω-open.

Example 2.6. Let X = R with the usual topology τ. Let A= Q be the set of all rational numbers.

Then A is a preopen set which is not α-ω-open.

Example 2.7. Let X be an uncountable set and let A, B, C and D be subsets of X such that each

of them is uncountable and the family {A, B, C , D} is a partition of X . We defined the topology



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 76

τ={φ, X , {A},

{B}, {A, B}, {A, B, C}}. Then {A, B, D} is an α-open set which is not ω-open.

Lemma 2.8. [7] If U is an open set, then Cl(U ∩ A) = Cl(U ∩ Cl(A)) and hence U ∩ Cl(A) ⊆

Cl(U ∩ A) for any subset A.

Theorem 2.9. If A is a pre-ω-open subset of a space (X ,τ) such that U ⊆ A⊆ Cl(U) for a subset

U of X , then U is a pre-ω-open set.

Proof. Since A ⊆ Intω(Cl(A)), U ⊆ Intω(Cl(A)). Also Cl(A) ⊆ Cl(U) implies that

Intω(Cl(A))⊆ Intω(Cl(U)). Thus U ⊆ Intω(Cl(A))

⊆ Intω(Cl(U)) and hence U ia a pre-ω-open set.

Theorem 2.10. A subset A of a space (X ,τ) is semi-open if and only if A is β -ω-open and

Intω(Cl(A))⊆ Cl(Int(A)).

Proof. Let A be semi-open. Then A ⊆ Cl(Int(A)) ⊆ Cl(Intω(Cl(A))) and hence A is β -ω-

open. In addition Cl(A)⊆ Cl(Int(A)) and hence Intω(Cl(A))⊆ Cl(Int(A)). Conversely let A be

β -ω-open and Intω(Cl(A))

⊆ Cl(Int(A)). Then A ⊆ Cl(Intω(Cl(A))) ⊆ Cl(Cl(Int(A))) = Cl(Int(A)). And hence A is

semi-open.

Proposition 2.11. The intersection of a pre-ω-open set and an open set is pre-ω-open.

Proof. Let A be a pre-ω-open set and U be an open set in X . Then A ⊆ Intω(Cl(A)) and

Intω(U) = U, by Lemma 2.8, we have U ∩ A⊆ Intω(U) ∩ Intω(Cl(A)) ⊆ Intω(U ∩ Cl(A)) ⊆

Intω(Cl(U ∩ A)). Therefore, A∩ U is pre-ω-open.

Proposition 2.12. The intersection of a β -ω-open set and an open set is β -ω-open.

Proof. Let U be an open set and A a β -ω-open set. Since every open set is ω-open, by



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 77

Lemma 2.8, we have

U ∩ A⊆ U ∩ Cl(Intω(Cl(A)))

⊆ Cl(U ∩ Intω(Cl(A)))

= Cl(Intω(U)∩ Intω(Cl(A)))

= Cl(Intω(U ∩ Cl(A)))

⊆ Cl(Intω(Cl(U ∩ A))).

This shows that U ∩ A is β -ω-open.

We note that the intersection of two pre-ω-open (resp. b-ω-open, β -ω-open) sets need

not be pre-ω-open (resp. b-ω-open, β -ω-open) as can be seen from the following example:

Example 2.13. Let X = R with the usual topology τ. Let A = Q and B = (R\Q) ∪ {1}, then

A and B are pre-ω-open, but A∩ B = {1} which is not β -ω-open since Cl(Intω(Cl({1}))) =

Cl(Intω({1}) = Cl({φ}) = φ.

Proposition 2.14. The intersection of a b-ω-open set and an open set is b-ω-open.

Proof. Let A be b-ω-open and U be open, then A ⊆ Intω(Cl(A)) ∪ Cl(Intω(A)) and U =

Intω(U). Then we have

U ∩ A⊆ U ∩ [Intω(Cl(A))∪ Cl(Intω(A))]

= [U ∩ Intω(Cl(A))]∪ [U ∩ Cl(Intω(A))]

= [Intω(U)∩ Intω(Cl(A))]∪ [U ∩ Cl(Intω(A))]

⊆ [Intω(U ∩ Cl(A))]∪ [Cl(U ∩ Intω(A))]

⊆ [Intω(Cl(U ∩ A))]∪ [Cl(Intω(U ∩ A))].

This shows that U ∩ A is b-ω-open.

Proposition 2.15. The intersection of an α-ω-open set and an open set is α-ω-open.

Theorem 2.16. If {Aα : α ∈∆} is a collection of b-ω-open (resp. pre-ω-open, β -ω-open) sets of

a space (X ,τ), then ∪α∈∆Aα is b-ω-open (resp. pre-ω-open, β -ω-open).



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 78

Proof. We prove only the first case since the other cases are similarly shown. Since Aα ⊆

Intω(Cl(Aα))∪ Cl(Intω(Aα)) for every α ∈∆, we have

∪α∈∆Aα ⊆ ∪α∈∆[Intω(Cl(Aα))∪ Cl(Intω(Aα))]

⊆ [∪α∈∆ Intω(Cl(Aα))]∪ [∪α∈∆Cl(Intω(Aα))]

⊆ [Intω(∪α∈∆Cl(Aα))]∪ [Cl(∪α∈∆ Intω(Aα))]

⊆ [Intω(Cl(∪α∈∆Aα))]∪ [Cl(Intω(∪α∈∆Aα))].

Therefore, ∪α∈∆Aα is b-ω-open.

Proposition 2.17. Let A be a b-ω-open set such that Intω(A) = φ. Then A is pre-ω-open.

A space (X ,τ) is called a door space if every subset of X is open or closed.

Proposition 2.18. If (X ,τ) is a door space, then every pre-ω-open set is ω-open.

Proof. Let A be a pre-ω-open set. If A is open, then A is ω-open. Otherwise, A is closed and

hence A⊆ Intω(Cl(A)) = Intω(A)⊆ A. Therefore, A= Intω(A) and thus A is an ω-open set.

A topological space X is said to be anti-locally countable [4] if every non-empty open set

is uncountable.

Lemma 2.19. [4] If (X ,τ) is an anti-locally countable space, then Intω(A) = Int(A) for every

ω-closed set A of X and Clω(A) = Cl(A) for every ω-open set A of X .

Theorem 2.20. Let (X ,τ) be an anti-locally countable space and A a subset of X . Then, the

following properties hold:

1. if A is pre-ω-open, then it is pre-open.

2. if A is b-ω-open and ω-closed, then it is b-open.

3. if A is β -ω-open, then it is β -open.

Proof. (1) Let A be a pre-ω-open set. Then by Lemma 2.19 A⊆ Intω(Cl(A)) = Int(Cl(A))

since every closed set is ω-closed.



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 79

(2) Let A be a b-ω-open andω-closed set. By Lemma 2.19, we have Intω(Cl(A)) = Int(Cl(A)),

Cl(Intω(A)) = Cl(Int(A)) and hence A⊆ Intω(Cl(A))∪Cl(Intω(A)) = Int(Cl(A))∪Cl(Int(A)).

This shows that A is b-open.

(3) Let A be a β -ω-open set. Then, by Lemma 2.19, we have A ⊆ Cl(Intω(Cl(A))) =

Cl(Int(Cl(A))) and hence A is β -open.

3. Decompositions of continuity

Definition 3.1. A subset A of a space X is called

1. an ω-t-set if Int(A) = Intω(Cl(A));

2. an ω-B-set if A= U ∩ V , where U ∈ τ and V is an ω-t-set.

Proposition 3.2. Let A and B be subsets of a space (X ,τ). If A and B are ω-t-sets, then A∩ B is

an ω-t-set.

Proof. Let A and B be ω-t-sets. Then we have

Int(A∩ B) ⊆ Intω(Cl(A∩ B))

⊆ (Intω(Cl(A))∩ (Cl(B)))

= Intω(Cl(A)∩ Intω(Cl(B))

= Int(A)∩ Int(B)

= Int(A∩ B).

Then Int(A∩ B) = Intω(Cl(A∩ B)) and hence A∩ B is an ω-t-set.

From the following examples one can deduce that a pre-ω-open set and an ω-B-set are

independent.

Example 3.3. Let X = R with the usual topology τ. Then R\Q is pre-ω-open but it is not an

ω-B-set and (0,1] is an ω-B-set which is not pre-ω-open.

Proposition 3.4. For a subset A of a space (X ,τ), the following properties are equivalent:



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 80

1. A is open;

2. A is pre-ω-open and an ω-B-set.

Proof. (1) ⇒ (2): Let A be open. Then A = Int(A) ⊆ Intω(Cl(A)) and A is pre-ω-open.

Also A= A∩ X and hence A is an ω-B-set.

(2) ⇒ (1): Since A is an ω-B-set, we have A= U ∩ V , where U is an open set and Int(V ) =

Intω(Cl(V )). By the hypothesis, A is also pre-ω-open and we have

A⊆ Intω(Cl(A))

= Intω(Cl(U ∩ V ))

⊆ Intω(Cl(U)∩ Cl(V ))

= Intω(Cl(U))∩ Intω(Cl(V ))

= Intω(Cl(U))∩ Int(V ).

Hence

A= U ∩ V =(U ∩ V )∩ U

⊆ (Intω(Cl(U))∩ Int(V ))∩ U

= (Intω(Cl(U))∩ U)∩ Int(V )

= U ∩ Int(V ).

Therefore, A= (U ∩ V ) = (U ∩ Int(V )) and A is open.

Definition 3.5. A subset A of a space X is called

1. an ω-tα-set if Int(A) = Intω(Cl(Intω(A)));

2. an ω-Bα-set if A= U ∩ V , where U ∈ τ and V is an ω-tα-set.

Proposition 3.6. Let A and B be subsets of a space (X ,τ). If A and B are ω-tα-sets, then A∩ B

is an ω-tα-set.



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 81

Proof. Let A and B be ω-tα-sets. Then we have

Int(A∩ B) ⊆ Intω(Cl(Intω(A∩ B)))

⊆ (Intω(Cl(Intω(A)))∩ (Cl(Intω(B))))

= Intω(Cl(Intω(A))∩ Intω(Cl(Intω(B)))

= Int(A)∩ Int(B)

= Int(A∩ B).

Then Int(A∩ B) = Intω(Cl(Intω(A∩ B))) and hence A∩ B is an ω-tα-set.

From the following examples one can deduce that an α-ω-open set and an ω-Bα-set are

independent.

Example 3.7. Let X = R with the usual topology τ. Then R\Q is α-ω-open but it is not an

ω-Bα-set and (0,1] is an ω-Bα-set which is not α-ω-open.

Proposition 3.8. For a subset A of a space (X ,τ), the following properties are equivalent:

1. A is open;

2. A is α-ω-open and an ω-Bα-set.

Proof. (1) ⇒ (2): Let A be open. Then A = Intω(A) ⊆ Cl(Intω(A)) and A = Intω(A) ⊆

Intω(Cl(Intω(A)). Therefore A is α-ω-open. Also A= A∩ X and hence A is an ω-Bα-set.

(2)⇒ (1): Since A is an ω-Bα-set, we have A= U ∩ V , where U is an open set and Int(V ) =

Intω(Cl(Intω(V )). By the hypothesis, A is also α-ω-open, and we have

A⊆ Intω(Cl(Intω(A)))

= Intω(Cl(Intω(U ∩ V ))

⊆ Intω(Cl(Intω(U)∩ Cl(Intω(V ))))

= Intω(Cl(U))∩ Intω(Cl(Intω(V )))

= Intω(Cl(U))∩ Int(V ).



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 82

Hence,

A= U ∩ V =(U ∩ V )∩ U

⊆ (Intω(Cl(U))∩ Int(V ))∩ U

= (Intω(Cl(U))∩ U)∩ Int(V )

= U ∩ Int(V ).

Therefore, A= (U ∩ V ) = (U ∩ Int(V )) and A is open.

Definition 3.9. A subset A of a space X is called an ω-set if A = U ∩ V , where U ∈ τ and

Int(V ) = Intω(V ).

From the following examples one can deduce that an ω-open set and an ω-set are inde-

pendent.

Example 3.10. Let X = R with the usual topology τ. Then R\Q is ω-open but it is not an ω-set

and A= (0,1)∩Q is an ω–set which is not ω-open.

Proposition 3.11. For a subset A of a space (X ,τ), the following properties are equivalent:

1. A is open;

2. A is ω-open and an ω-set.

Proof. (1)⇒ (2): This is obvious.

(2) ⇒ (1): Since A is an ω-set, we have A = U ∩ V , where U is an open set and Int(V ) =

Intω(V ). By the hypothesis, A is also ω-open and we have A = Intω(A) = Intω(U ∩ V ) =

Intω(U)∩ Intω(V ) = U ∩ Int(V ). Therefore, A is open.

Definition 3.12. A function f : X → Y is said to be ω-continuous [9] (resp. pre-ω-

continuous, ω-B-continuous, α-ω-continuous, ω-Bα-continuous, ω∗-continuous) if f −1(V )

is ω-open (resp. pre-ω-open, an ω-B-set, α-ω-open, an ω-Bα-set, an ω-set) for each open set

V in Y .



T. Noiri, A. Al-omari, and M. Noorani / Eur. J. Pure Appl. Math, 2 (2009), (73-84) 83

By Propositions 3.4, 3.8 and 3.11 we have an immediate result.

Theorem 3.13. For a function f : X → Y , the following properties are equivalent:

1. f is continuous;

2. f is pre-ω-continuous and ω-B-continuous;

3. f is α-ω-continuous and ω-Bα-continuous;

4. f is ω-continuous and ω∗-continuous.

Proposition 3.14. For a subset A of an anti-locally countable space (X ,τ), the following prop-

erties are equivalent:

1. A is regular open;

2. A= Intω(Cl(A));

3. A is pre-ω-open and an ω-t-set.

Proof. (1) ⇒ (2): Let A be regular open. Then by Lemma 2.19, we have Intω(Cl(A)) =

Int(Cl(A)) = A.

(2)⇒ (3): The proof is obvious.

(3)⇒ (1): Let A be pre-ω-open and an ω-t-set. Then A ⊆ Intω(Cl(A)) = Int(A) ⊆ A and

hence A= Intω(Cl(A)) = Int(Cl(A)).

Definition 3.15. A function f : X → Y is said to be completely continuous [6] (resp. ω-t-

continuous ) if f −1(V ) is regular open (resp. an ω-t-set) in X for each open set V of Y .

Theorem 3.16. Let (X ,τ) be an anti-locally countable space. A function f : X → Y is completely

continuous if and only if f is pre-ω-continuous and ω-t-continuous.

Proof. This is an immediate consequence of Proposition 3.14.

ACKNOWLEDGEMENTS. This work is financially supported by the Ministry of Higher Edu-

cation, Malaysia under FRGS grant no: UKM-ST-06-FRGS0008-2008.



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