$�eta _1$-paracompactness with respect to an ideal


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 135-145
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

β1-paracompactness with respect to an ideal

Abdo Qahis

Department of Mathematics, Faculty of Science and Arts, Najran University, Saudi Arabia

Abstract. The notion of β1-paracompactness in topological spaces is introduced and studied
in [1]. In this paper, we introduce and investigate the notion of β1-paracompact spaces with
respect to an ideal I which is a generalization of the notion of β1-paracompact spaces. We study
characterizations, subsets and subspaces of β1I-paracompact spaces. Also, we investigate the
invariants of β1I-paracompact spaces by functions.

2010 Mathematics Subject Classifications: 54A05, 54A08, 54D10

Key Words and Phrases: Ideal, β-open set, β1-paracompact, β1-paracompact modulo I, locally
finite collection, β-irresolute, β-continuous, strongly β-continuous.

1. Introduction and Preliminaries

In 2016, Heyam Al-Jarrah introduced and studied the concept of β1-paracompact
spaces. A space (X, τ) is said to be β1-paracompact space [1] if every β-open cover of
X has a locally finite open refinement. In this paper, we introduce a new class of spaces,
called β1I-paracompact spaces and investigate their properties and their relations with
other types of spaces.

The notion of ideals in topological spaces was first studied by Kuratowski [12] and
Vaidyanathaswamy [23]. An ideal I on a set X is a nonempty collection of subsets of X
which satisfies the following properties:

(i) A ∈ I and B ⊂ A implies B ∈ I;
(ii) A ∈ I and B ∈ I implies A ∪B ∈ I.

Given a topological space (X, τ) with an ideal I on X and if p(X) is the set of all subsets of
X, a set operator ()∗ : p(X)→ p(X), called a local function [10] of A with respect to τ and
I is defined as follows: for A ⊂ X, A∗(I, τ) = {x ∈ X : U ∩ A /∈ I for every U ∈ τ(x)}
where τ(x) = {U ∈ τ : x ∈ U}. A Kuratowski closure operator cl∗() for a topology
τ∗(I, τ) called ∗-topology finer than τ is defined by cl∗(A) = A ∪ A∗(I, τ) [10] and
β = {U \ I : U ∈ τ, I ∈ I} is a basis for τ∗ [10]. We simply write τ∗ for τ∗(I, τ). If
I is an ideal on X, then (X, τ, I) is called an ideal space. If β = τ∗, then we say I is
τ -simple [10]. A sufficient condition for I to be simple is the following: for A ⊂ X, if for

DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3552

Email address: cahis82@gmail.com (A. Qahis)

http://www.ejpam.com 135 c© 2019 EJPAM All rights reserved.



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 136

every a ∈ A there exists U ∈ τ(a) such that U ∩ A ∈ I, then A ∈ I. If (X, τ, I) satisfies
this condition, then τ is said to be compatible with respect to I [10] or I is said to be
τ -local, denoted by I ∼ τ . Given an ideal space (X, τ, I), we say I is τ -boundary [10]
or I-codense if I ∩ τ = ∅. An ideal I is said to be weakly τ -local [11] if A∗ = ∅ implies
A ∈ I. Some useful ideals in X are: (i) p(A), where A ⊆ X and (ii) If , the ideal of all
finite subsets of X.

By a space (X, τ), we always mean a topological space (X, τ) with no separation prop-
erties assumed. If A ⊆ X, we denote the closure of A and the interior of A by cl(A) and
int(A), respectively. A subset A of (X, τ) is siad to be semi-open[13] (resp., α-open[17],
regular open [21]) if A ⊂ cl(int(A)), (resp., A ⊂ int(cl(int(A))), A = int(cl(A))). The
family of α-sets of a space (X, τ) denoted by τα forms a topology on X finer than τ [17].
Abd El-Monsef et al.[8] introduced and studied the concept of β-open sets in topological
spaces. A subset A of X is called β-open if A ⊂ cl(int(cl(A))). The complement of a
β-open set is said to be β-closed [8]. The family of all β-open (resp., β-closed) subsets
of X is denoted by βO(X, τ) (resp., βC(X, τ)). The union of all β-open subsets of X
contained in A is called β-interior of A and is denoted by βint(A) and the intersection
of all β-closed subsets of X containing A is called the β-closure of A and is denoted by
βcl(A). A set A is called β-regular [19] if it is both β-open and β-closed. A space (X, τ) is
said to be β-regular [14] if for each β-open set U and each x ∈ U , there exists a β-open set
V such that x ∈ V ⊂ βcl(V ) ⊂ U . For any space, βO(X, τα) = βO(X, τ) [2]. A collection
W = {Wα : α ∈ ∆} of subsets of a space (X, τ) is said to be locally finite if for each x ∈ X,
there exists an open set U containing x and U intersects at most finitely many members
of W. A subset A of space X is said to be N -closed relative to X (briery, N -closed)[7] if
for every cover {Uα : α ∈ ∆} of A by open subsets of X, there exists a finite subfamily
∆0 of ∆ such that A ⊂ ∪{int(cl(int(Uα))) : α ∈ ∆0}.

Definition 1. A space (X, τ) is said to be:
(i) extremally disconnected (briefly e.d.) [24] if the closure of every open set in (X, τ) is
open;
(ii) submaximal [5] if each dense subset of X is open in X.

Lemma 1. [3] The union of a finite family of locally finite collection of sets in a space is
a locally finite family of sets.

Theorem 1. [16] Let (X, τ) be a space, A ⊂ B ⊂ X and B is β-open in (X, τ). Then A
is β-open in (X, τ) if and only if A is β-open in the subspace (B, τB).

Theorem 2. [4] If {Uα : α ∈ ∆} is a locally finite family of subsets in a space X and if
Vα ⊂ Uα for each α ∈ ∆, then the family {Vα : α ∈ ∆} is a locally finite in X.

Lemma 2. [9] If f : (X, τ) → (Y, σ) is a continuous surjective function and U = {Uα :
α ∈ ∆} is locally finite in Y , then f−1(U) = {f−1(Uα) : α ∈ ∆} is locally finite in X.

Lemma 3. [18] Let f : (X, τ) → (Y, σ) be almost closed surjection with N -closed point
inverse. If {Uα : α ∈ ∆} is a locally finite open cover of X, then {f(Uα) : α ∈ ∆} is a
locally finite cover of Y .



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 137

Lemma 4. [22] I is weakly τ -local implies I is τ -locally finite.

Lemma 5. [2] If V is open and A is semi-preopen (or β-open) then V ∩A is semi-preopen
(or β-open).

2. β1I-paracompact spaces

Recall that an ideal space (X, τ, I) is said to be I-paracompact [22] (resp., S1I-
paracompact [20]) if every open (resp., semi-open) cover U of X has a locally finite open
refinement V (not necessarily a cover) such that X \ ∪{V : V ∈ V} ∈ I.

Definition 2. An ideal space (X, τ, I) is said to be β1I-paracompact, or β1-paracompact
modulo an ideal I if every β-open cover U of X has a locally finite open refinement V (not
necessarily a cover) such that X \ ∪{V : V ∈ V} ∈ I. A family V of subsets of X such
that X \ ∪{V : V ∈ V} ∈ I is called an I-cover of X.

It follows from the definitions that

β1-paracompact ⇒ β1I-paracompact ⇒ S1I-paracompact ⇒ I-paracompact

The following examples show that the converses of the above implications need not be
true in general.

Example 1. Let X = R with the topology τ = {∅, X, {0}} and I = If . Then (X, τ) is
paracompact which implies that (X, τ, I) is I-paracompact. On the other hand (X, τ, I)
is not β1I-paracompact. For the β-open cover U = {{0, x} : x ∈ X,x 6= 0} , we can find
a locally finite open refinement V = {0} of U . But V does not I-cover of X. Therefore,
(X, τ, I) is not β1I-paracompact.

Example 2. Let X = {1, 2, 3, 4} with the topology τ = {∅, X, {1, 2}, {3, 4}} and I = If .
Then (X, τ, I) is S1I-paracompact, since SO(X, τ) = τ , but it is not β1I-paracompact
since U = {{1}, {2}, {3}, {4}} is a β-open cover of X which admits no locally finite open
refinement.

Example 3. Consider the ideal space (X, τ, I) where X = {1, 2, 3}, τ = {∅, X, {1}}
and I = {A ⊂ X : 1 /∈ A}. Then βO(X, τ) = {∅, X, {1}, {1, 2}, {1, 3}}. Therefore
(X, τ, I) is β1I-paracompact space. On the other hand, (X, τ) is not β1- paracompact
since U = {{1, 2}, {1, 3}} is a β1-open cover of (X, τ) which admits no locally finite open
refinement.

Example 4. Consider the ideal space (X, τ, I) where X = R, the set of all real numbers,
τ = {∅, X, {0}} and I = {A ⊂ X : 0 /∈ A}. Then (X, τ, I) is β1I-paracompact space but
(X, τ) is not β1-paracompact, since the β-open cover U = {{0, x} : x ∈ X,x 6= 0} admits
no locally finite open refinement.

Corollary 1. Let (X, τ) be a space with an ideal I = {∅}. Then (X, τ) is β1-paracompact
if and only if (X, τ, I) is β1I-paracompact.



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 138

Corollary 2. For an e.d. submaximal ideal space (X, τ, I), the following conditions are
equivalent:
(i) (X, τ, I) is β1I-paracompact;
(ii) (X, τ, I) is S1I-paracompact;
(iii) (X, τ, I) is I-paracompact.

Proof. This follows directly from the fact that if an ideal space (X, τ, I) is an e.d.
submaximal space, then τ = SO(X, τ) = βO(X, τ).

Proposition 1. If (X, τ, I) is β1I-paracompact, then (X, τα, I) is β1I-paracompact.

Proof. Suppose (X, τ, I) is β1I-paracompact. Let U = {Uα : α ∈ ∆} be a β-open
cover of (X, τα, I). Then U is a β-open cover of (X, τ, I). By hypothesis, there exist a
locally finite open refinement V = {Vλ : λ ∈ Λ} of U such that X ∪ \{Vλ : λ ∈ Λ} ∈ I.
Since τ ⊂ τα, the family V is a τα-locally finite τα-open refinement of U and so (X, τα, I)
is β1I-paracompact.

By replacing I by {∅} in Proposition 1, we have the following corollary.

Corollary 3. [1, Theorem 2.8(2)] If (X, τ) is β1-paracompact, then (X, τα) is β1-paracompact.

Theorem 3. Let (X, τ, I) be an ideal space. If I is codense, (X, τ∗, I) is β1I-paracompact
and I is τ -simple, then every β-open cover of (X, τ, I) has τ -locally finite β-open I-cover
refinement.

Proof. Let U = {Uα : α ∈ ∆} be a β-open cover of X. Since, βO(X, τ) ⊆ βO(X, τ∗).
Then U is a β-open cover of (X, τ∗, I) . By hypothesis, there exist τ∗-locally finite τ∗-open
refinement V = {Vλ \ Iλ : λ ∈ Λ, Vλ ∈ τ, Iλ ∈ I} of U such that X \∪{Vλ \ Iλ : λ ∈ Λ} ∈ I.
For each x ∈ X, there exists a τ∗-open set W containing x such that W ∩ (Vλ \ Iλ) = ∅
for λ 6= λ1, λ2, ..., λn. Since I is τ -simple, W = U \ I for some U ∈ τ and I ∈ I. Thus,
(U \ I) ∩ (Vλ \ Iλ) = ∅ for λ 6= λ1, λ2, ..., λn which implies that (U ∩ Vλ) \ (I ∪ Iλ) = ∅
for λ 6= λ1, λ2, ..., λn. Since I is codense, then U ∩ Vλ = ∅ for λ 6= λ1, λ2, ..., λn. Then
U ∩ (Vλ ∩Uα) = ∅ for λ 6= λ1, λ2, ..., λn. By Lemma 5, W = {Vλ ∩Uα : λ ∈ Λ} is τ -locally
finite β-open refinement of U . Since V refines U for every Vλ \ Iλ ∈ V, there exists Uα ∈ U
such that Vλ \ Iλ ⊂ Uα. Thus, Vλ \ Iλ = Uα ∩ (Vλ \ Iλ) = (Vλ ∩ Uα) \ Iλ ⊂ Vλ ∩ Uα ⊂ Uα.
Then X \ ∪{Vλ ∩ Uα : λ ∈ Λ, α ∈ ∆} ⊂ X \ ∪{Vλ \ Iλ : λ ∈ Λ} ∈ I which implies that
X \ ∪{Vλ ∩ Uα : λ ∈ Λ, α ∈ ∆} ∈ I.

Theorem 4. Let (X, τ, I) be an ideal space. If I is weakly τ -local and (X, τ, I) is β1I-
paracompact, then (X, τ∗, I) is β1I-paracompact.

Proof. Let U = {Uα \ Iα : α ∈ ∆, Uα ∈ τ, Iα ∈ I} be a β-open cover of (X, τ∗, I).
Then W = {Uα : α ∈ ∆} is a β-open cover of X and so it has locally finite open
refinement V = {Vλ : λ ∈ Λ} such that X \ ∪{Vλ : λ ∈ Λ} ∈ I. Now the family
{Vλ ∩ Iα : λ ∈ Λ} ⊂ I is locally finite. Since, I is weakly τ -local, ∪λ∈Λ(Vλ ∩ Iα) ∈ I, by
Lemma 4. Then X \ ∪λ∈Λ(Vλ \ Iα) ⊂ (X \

⋃
λ∈Λ Vλ)∪(

⋃
λ∈Λ (Vλ ∩ Iα)) ∈ I which implies



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 139

X \ ∪λ∈Λ(Vλ \ Iα) ∈ I. Since V is locally finite, V1 = {Vλ \ Iα : λ ∈ Λ} is locally finite.
Since τ∗ is finer than τ , V1 is τ∗-locally finite τ∗-open which refines U . Hence (X, τ∗, I) is
β1I-paracompact.

Theorem 5. Let (X, τ) be a β-regular space. If (X, τ, I) is β1I-paracompact, then every
β-open cover of X has a locally finite β-closed I-cover refinement.

Proof. Let U be a β-open cover of X. For each x ∈ X, let Ux ∈ U such that x ∈ Ux.
Since (X, τ) is β-regular, there exists Vx ∈ βO(X, τ) such that x ∈ Vx ⊂ βcl(Vx) ⊂ Ux.
Then the family V = {Vx : x ∈ X} is a β-open cover refinement of U . By hypothesis,
there exist a locally finite open refinement W = {Wα : α ∈ ∆} which refine V such
that X \ ∪{Wα : α ∈ ∆} ∈ I. The family βcl(W) = {βcl(Wα) : α ∈ ∆} is locally
finite for each α ∈ ∆. Now X \ ∪{βcl(Wα) : α ∈ ∆} ⊆ X \ ∪{Wα : α ∈ ∆} implies
X \ ∪{βcl(Wα) : α ∈ ∆} ∈ I. Hence βcl(W) is I-cover. Let βcl(Wα) ∈ βcl(W). Since
W refines V, there is some Vx ∈ V such that Wα ⊂ Vx and so βcl(Wα) ⊂ βcl(Vx) ⊂ Ux
implies that βcl(Wα) ⊂ Ux. Hence βcl(W) refines U . Thus, βcl(W) = {βcl(Wα) : α ∈ ∆}
is a locally finite β-closed I-cover refinement of U .

If I = {∅} in Theorem 5, then we have the following corollary.

Corollary 4. [1, Theorem 2.12] Let (X, τ) be a β-regular space.. If each β-open cover
of the space X has a locally finite refinement, then each β-open cover of X has a locally
finite β-closed refinement

Recall that a function f : (X, τ) → (Y, σ) is said to be β-continuous [8] (resp., β-
irresolute [15]) if f−1(V ) ∈ βO(X, τ) for each open (resp., β-open) set V in (Y, σ).

Theorem 6. Let f : (X, τ, I)→ (Y, σ) be an open, β-irresolute and almost closed surjec-
tive function with N -closed point inverse. If (X, τ, I) is β1I-paracompact, then (Y, σ, f(I))
is β1f(I)-paracompact.

Proof. Let U = {Uα : α ∈ ∆} be a β-open cover of Y . Since f is β-irresolute,
U1 = {f−1(Uα) : α ∈ ∆} is a β-open cover of X. By hypothesis, there exists a τ -locally
finite τ -open refinement V1 = {Vλ : λ ∈ Λ} of U1 such that X \ ∪{Vλ : λ ∈ Λ} ∈ I. Then
f(X\∪{Vλ : λ ∈ Λ}) ∈ f(I). Now, f(X)\∪{f(Vλ) : λ ∈ Λ} ⊆ f(X\∪{Vλ : λ ∈ Λ}) implies
that f(X) \ ∪{f(Vλ) : λ ∈ Λ} ∈ f(I) which implies that Y \ ∪{f(Vλ) : λ ∈ Λ} ∈ f(I).
Since f is open and V1 is τ -locally finite, V = {f(Vλ) : λ ∈ Λ} is σ-locally finite by Lemma
3. Let f(Vλ) ∈ V. Then Vλ ∈ V1. Since V1 refines U1, there exists f−1(Uα) ∈ U1 such
that Vλ ⊂ f−1(Uα). Thus f(Vλ) ⊂ f(f−1(Uα)) implies that f(Vλ) ⊂ Uα for some Uα ∈ U .
Hence V refines U . Therefore, (Y, σ, f(I)) is β1f(I)-paracompact.

Since every compact set is N -closed and every closed map is almost closed, we conclude
the following corollary.

Corollary 5. Let f : (X, τ, I)→ (Y, σ) be an open, β-irresolute, closed surjective function
with compact point inverse. If (X, τ, I) is β1I-paracompact, then (Y, σ, f(I)) is β1f(I)-
paracompact.



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 140

A function f : (X, τ)→ (Y, σ) is said to be strongly β-continuous [1] if f−1(V ) ∈ τ for
each V ∈ βO(Y, σ).

Theorem 7. Let f : (X, τ, I)→ (Y, σ) be an open, strongly β-continuous, almost closed,
surjective function with N -closed point inverse. If (X, τ, I) is I-paracompact, then (Y, σ, f(I))
is β1f(I)-paracompact.

Proof. Let U = {Uα : α ∈ ∆} be a β-open cover of Y . Since f is strongly β-continuous,
U1 = {f−1(Uα) : α ∈ ∆} is an open cover ofX. By hypothesis, there exists a τ -locally finite
τ -open refinement V1 = {Vλ : λ ∈ Λ} which refines U1 such that X \ ∪{Vλ : λ ∈ Λ} ∈ I.
Then f(X \∪{Vλ : λ ∈ Λ}) ∈ f(I). Now f(X) \∪{f(Vλ) : λ ∈ Λ} ⊆ f(X \∪{Vλ : λ ∈ Λ})
implies that f(X) \ ∪{f(Vλ) : λ ∈ Λ} ∈ f(I) which implies that Y \ ∪{f(Vλ) : λ ∈ Λ} ∈
f(I). Since f is open and V1 is τ -locally finite, V = {f(Vλ) : λ ∈ Λ} is σ-locally finite by
Lemma 3. Let f(Vλ) ∈ V. Then Vλ ∈ V1. Since V1 refines U1, there exists f−1(Uα) ∈ U1

such that Vλ ⊂ f−1(Uα). Thus f(Vλ) ⊂ f(f−1(Uα)) implies that f(Vλ) ⊂ Uα for some
Uα ∈ U . Hence V refines U . Therefore, (Y, σ, f(I)) is β1f(I)-paracompact.

Recall that a function f : (X, τ) → (Y, σ) is said to be pre β-open[15] if for every
β-open set V of (X, τ), f(V ) is β-open in (Y, σ).

Theorem 8. Let f : (X, τ)→ (Y, σ,J ) be a pre β-open, continuous, bijective function. If
(Y, σ,J ) is a β1J -paracompact, then (X, τ, f−1(J )) is β1f

−1(J )-paracompact.

Proof. Let U = {Uα : α ∈ ∆} be a β-open cover of X. Since f is a pre β-open,
f(U) = {f(Uα) : α ∈ ∆} is a β-open cover of Y and so it has a σ-locally finite σ-
open refinement W = {Wλ : λ ∈ Λ} of f(U) such that Y \ ∪{Wλ : λ ∈ Λ} ∈ J . Let
Y \ ∪{Wλ : λ ∈ Λ} = J ∈ J . This implies Y = (∪{Wλ : λ ∈ Λ}) ∪ J . Then f−1(Y ) =
(∪{f−1(Wλ) : λ ∈ Λ}) ∪ f−1(J) which implies X = (∪{f−1(Wλ) : λ ∈ Λ}) ∪ f−1(J).
It follows that X \ ∪{f−1(Wλ) : λ ∈ Λ} ∈ f−1(J ). Since f is continuous, by Lemma
2, V = {f−1(Wλ) : λ ∈ Λ} is is τ -open, τ -locally finite. Let f−1(Wλ) ∈ V. Then
Wλ ∈ W. Since W refines f(U), there exists f(Uα) ∈ f(U) such that Wλ ⊂ f(Uα). Thus
f−1(Wλ) ⊂ f−1(f(Uα)) implies that f−1(Wλ) ⊂ Uα for some Uα ∈ U . Hence V refines U .
Therefore, (X, τ, f−1(J )) is β1f

−1(J )-paracompact.

If I = {∅} in Theorem 8, then we have the following corollary.

Corollary 6. Let f : (X, τ) → (Y, σ) be a pre β-open, continuous, bijective function. If
(Y, σ) is β1-paracompact, then (X, τ) is β1-paracompact.

3. β1I-paracompact subsets

In this section, we define the subsets and subspaces of β1I-paracompact and study
some of their properties.

Definition 3. A subset A of an ideal space (X, τ, I) is said to be β1I-paracompact relative
to X (β1I-paracompact subset) if each cover U of A by β-open sets of X, there exists a



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 141

locally finite open refinement V of U such that A \ ∪{V : V ∈ V} ∈ I. A is said to be
β1AIA-paracompact (β1AIA-paracompact subspace) if (A, τA, IA) is β1AIA-paracompact
as a subspace, where τA is the usual subspace topology and IA = {A ∩ I : I ∈ I}.

A subset A of a space (X, τ) is said to be βg-closed [6] if βcl(A) ⊆ U whenever A ⊂ U
and U is any β-open set in (X, τ).

Theorem 9. Every βg-closed subset of a β1I-paracompact is β1I-paracompact.

Proof. Let A be a βg-closed subset of (X, τ, I) and U = {Uα : α ∈ ∆} be a cover of A
by β-open sets of X. Since A ⊆ ∪α∈∆Uα and A is a βg-closed, we have βcl(A) ⊆ ∪α∈∆Uα.
Then U1 = U ∪ {X \ βcl(A)} is a β-open cover of X. By hypothesis, there exist a
locally finite open family V = {Vλ : λ ∈ Λ} ∪ {V } which refines U1 ( Vλ ⊂ Uα for
some α ∈ ∆ and V ⊂ X \ βcl(A)) such that X \ ∪[{Vλ : λ ∈ Λ} ∪ {V }] ∈ I. Then
βcl(A) \ ∪{Vλ : λ ∈ Λ} = βcl(A) \ [V ∪ (∪{Vλ : λ ∈ Λ})] ⊂ X \ ∪[{Vλ : λ ∈ Λ} ∪ {V }] ∈ I.
Since A \ ∪{Vλ : λ ∈ Λ} ⊂ βcl(A) \ ∪{Vλ : λ ∈ Λ}, A \ ∪{Vλ : λ ∈ Λ} ∈ I, by heredity
property of I. Since V = {Vλ : λ ∈ Λ}∪{V } is a locally finite, the family V1 = {Vλ : λ ∈ Λ}
is locally finite. Thus, the family V1 is locally finite open and V1 refines U . Therefore, A
is βI-paracompact.

Theorem 10. Every regular open subset of a β1I-paracompact is β1AIA-paracompact.

Proof. Let A be a regular open in (X, τ) and W = {Wα : α ∈ ∆} be a β-open cover
of A in (A, τA, IA). Since A is open in (X, τ, I), Wα is a β-open set in (X, τ, I) for each
α ∈ ∆, by Theorem 1. Then U = {Wα : α ∈ ∆} ∪ {X \ A} is a β-open cover of the β1I-
paracompact (X, τ, I) and so it has a locally finite open refinement V = {Vλ : λ ∈ Λ} such
that X \ ∪{Vλ : λ ∈ Λ} = I ∈ I. Then A ⊂ A ∩ [(∪{Vλ : λ ∈ Λ}) ∪ I] = (∪{Vλ ∩ A : λ ∈
Λ}) ∪ (I ∩A) = (∪{Vλ ∩A : λ ∈ Λ}) ∪ IA which implies that A \ ∪{Vλ ∩A : λ ∈ Λ} ∈ IA.
Let x ∈ A. Since V is locally finite, there exists V ∈ τ(x) such that Vλ ∩ V = ∅ for
λ 6= λ1, λ2, ..., λn. Then (Vλ∩V )∩A = ∅ for λ 6= λ1, λ2, ..., λn and so (Vλ∩A)∩(V ∩A) = ∅.
Therefore, VA = {Vλ ∩ A : λ ∈ Λ} is τA-locally finite. Let Vλ ∩ A ∈ VA. Since V refines
U , there is some Wα ∈ U such that Vλ ⊂ Wα which implies Vλ ∩ A ⊂ Wα. Therefore, VA
refines W. Hence A is β1AIA-paracompact.

Corollary 7. Every clopen subset of a β1I-paracompact is β1AIA-paracompact.

If I = {∅} in Theorem 9 and Theorem 10, then we have the following corollary.

Corollary 8. [1, Theorem 3.5] Let (X, τ) be a β1-paracompact space. Then:
(i) If A is regular open subset of (X, τ), then (A, τA) is β1A-paracompact;
(ii) If A is a βg-closed subset of (X, τ), then A is a β1-paracompact.

Theorem 11. Let A and B be subsets of an ideal space (X, τ, I) such that A ⊂ B ⊂ X.
Then the following conditions hold.
(i) If A is βI-paracompact and B is β-open in (X, τ), then A is β1BIB-paracompact.
(ii) If A is β1BIB-paracompact and B is open in (X, τ), then A is β1I-paracompact.



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 142

Proof. (i) Let U = {Uα : α ∈ ∆} be a cover of A such that Uα ∈ βO(B, τB). Since B ∈
βO(X, τ), U is a β-open cover of A in (X, τ), by Theorem 1. By hypothesis, there exists a
locally finite open family V = {Vλ : λ ∈ Λ} refines U such that A \∪{Vλ : λ ∈ Λ} = I ∈ I.
Then A ⊆ (∪{Vλ : λ ∈ Λ})∪ I and A = A∩B ⊆ [∪{Vλ : λ ∈ Λ}∪ I]∩B = ∪{Vλ ∩B : λ ∈
Λ}∪ (I ∩B) implies A\∪{Vλ∩B : λ ∈ Λ} ∈ IB. Let x ∈ B. Since V is locally finite, there
exits U ∈ τ(x) such that U ∩ Vλ = ∅ for λ 6= λ1, λ2, ..., λn. This implies (U ∩ Vλ) ∩B = ∅
for λ 6= λ1, λ2, ..., λn which implies (U ∩B)∩ (Vλ∩B) = ∅ for λ 6= λ1, λ2, ..., λn. Therefore,
the family VB = {Vλ ∩B : λ ∈ Λ} is τB-locally finite τB-open . Let Vλ ∩B ∈ VB. Since V
refines U there exits Uα ∈ U such that Vλ ⊂ Uα and so Vλ ∩B ⊂ Uα. Hence VA refines U .
Therefore A is β1BIB-paracompact.
(ii) Let U = {Uα : α ∈ ∆} be a cover of A by β-open subsets of X. Then the family
U1 = {B ∩ Uα : α ∈ ∆} is a β-open cover of A in (B, τB, IB). By hypothesis, exists τB-
locally finite τB-open family V = {Vλ : λ ∈ Λ} refines U1 such that A\∪{Vλ : λ ∈ Λ} ∈ IB,
where IB = I ∩B. It follows that A \ ∪{Vλ : λ ∈ Λ} ∈ I. Since B is open in X. Then by
Theorem 1, V is a locally finite open refinement of U . Therefore, A is β1I-paracompact.

If I = {∅} in Theorem 11, then we have the following corollary.

Corollary 9. [1, Theorem 3.6] Let A and B be subsets of an ideal space (X, τ) such that
A ⊂ B ⊂ X. Then:
(i) If A is β-paracompact and B is β-open in (X, τ), then A is β1B-paracompact.
(ii) If A is β1B-paracompact and B is open in (X, τ), then A is β1-paracompact.

Theorem 12. Let A be a clopen subspace of an ideal space (X, τ, I). Then A a is β1AIA-
paracompact if and only if it is β1I-paracompact.

Proof. To prove necessity, let U = {Uα : α ∈ ∆} be a cover of A by β-open subsets
of the ideal subspace (A, τA, IA). Since A is open, U is a cover of A by β-open subsets
of X and so it has a locally finite open refinement, say V = {Vλ : λ ∈ Λ} such that
A \ ∪{Vλ : λ ∈ Λ} = I ∈ I. Then A ⊆ (∪{Vλ : λ ∈ Λ}) ∪ I. Now A ⊆ A ∩ [(∪{Vλ : λ ∈
Λ}) ∪ I] = ∪{Vλ ∩ A : λ ∈ Λ} ∪ (A ∩ I). It follows that A \ ∪{Vλ ∩ A : λ ∈ Λ} ∈ IA.
Let x ∈ A. Since V = {Vλ : λ ∈ Λ} is locally finite, there exists W ∈ τ(x) such that
Vλ ∩W = ∅ for λ 6= λ1, λ2, ..., λn. Then (Vλ ∩W ) ∩ A = ∅ for λ 6= λ1, λ2, ..., λn which
implies (Vλ∩A)∩(W ∩A) = ∅ for λ 6= λ1, λ2, ..., λn. Thus the family VA = {Vλ∩A : λ ∈ Λ}
is τA-locally finite τA-open refitment of U . Hence A is β1AIA-paracompact.
To prove sufficiency, let U = {Uα : α ∈ ∆} be a cover of A by β-open subsets of an
ideal space (X, τ, I). Then U1 = {A ∩ Uα : α ∈ ∆} is a β-open cover of the β1AIA-
paracompact ideal subspace (A, τA, IA) and so it has a τA-locally finite τA-open refinement
V = {Vλ : λ ∈ Λ} such that A \ ∪{Vλ : λ ∈ Λ} ∈ IA. Then A \ ∪{Vλ : λ ∈ Λ} ∈ I. But A
is an open set in X, so Vλ is an open set for every λ ∈ Λ. Now τA ⊆ τ and X \ A is an
open set in X which intersects no member of V. Therefore V is locally finite and refines
U . Thus A is a β1I-paracompact.

If I = {∅} in Theorem 12, then we have the following corollary.

Corollary 10. [1, Theorem 3.8] Let A be a clopen subspace of a space (X, τ). Then A is
a β1A-paracompact if and only if it is β1-paracompact.



A. Qahis / Eur. J. Pure Appl. Math, 12 (1) (2019), 135-145 143

Theorem 13. If (X, τ, I) is T2 space and A is β1I-paracompact relative to X, then A is
closed in (X, τ∗).

Proof. Let x ∈ X \ A. For each y ∈ A, there exists U ∈ τ such that y ∈ Uy and
x /∈ cl(Uy). Therefore, the family U = {Uy : y ∈ A} is an open cover of A which is β1I-
paracompact relative to X. Since U is a β-open cover of A and so it has a locally finite
open refinement V = {Vλ : λ ∈ Λ} of U such that A \ ∪{Vλ : λ ∈ Λ} ∈ I. Now x /∈ cl(Vλ)
for each λ implies that x /∈ ∪{cl(Vλ) : λ ∈ Λ}. Since the locally finite family V is closure-
preserving, x /∈ ∪{cl(Vλ) : λ ∈ Λ} = cl(∪{Vλ : λ ∈ Λ}). Let U = X \ cl(∪{Vλ : λ ∈ Λ})
and J = A\cl(∪{Vλ : λ ∈ Λ}) ⊂ A\∪{Vλ : λ ∈ Λ} = I1, where I1 ∈ I. Then U \J ∈ τ∗(x)
and (U \ J)∩A = ∅ which implies x /∈ A∗. Hence A∗ ⊂ A. This shows that A is closed in
(X, τ∗).

If I = {∅} in Theorem 13, then we conclude the following corollary.

Corollary 11. Let A be a β1-paracompact relative subset of a T2 space (X, τ). Then A is
closed in (X, τ).

Theorem 14. In an ideal space (X, τ, I), if A and B are β1I-paracompact, then A ∪ B
is β1I-paracompact.

Proof. Let U = {Uα : α ∈ ∆} be a cover of A∪B by β-open sets in X. Then U is a β-
open cover of A and B. By hypothesis, there exist locally finite families V = {Vλ : λ ∈ Λ}
of A and W = {Wγ : γ ∈ Λ0} of B which refines U such that A \ ∪{Vλ : λ ∈ Λ} ∈ I and
B \ ∪{Wγ : γ ∈ Λ0} ∈ I. Then A ∪ B ⊂ (∪{Vλ : λ ∈ Λ} ∪ I1) ∪ (∪{Wγ : γ ∈ Λ0} ∪ I2),
where I1, I2 ∈ I which implies that A ∪ B ⊂ (∪{Vλ ∪Wγ : λ ∈ Λ, γ ∈ Λ0}) ∪ (I1 ∪ I2).
It follows that (A ∪ B) \ ∪{Vλ ∪Wγ : λ ∈ Λ, γ ∈ Λ0} ∈ I. Since the families V and W
are locally finite the family V ′ = {Vλ ∪Wγ : λ ∈ Λ, γ ∈ Λ0} is locally finite, by Lemma 1
which refines U . Therefore, A ∪B is β1I-paracompact.

Theorem 15. In an ideal space (X, τ, I), if A is β1I-paracompact and B is a β-closed
subset of X, then A ∩B is β1I-paracompact.

Proof. Let U = {Uα : α ∈ ∆} be a cover of A ∩ B by β-open subsets of X. Then
UA = U ∪{X \B} is a cover of A by β-open sets in X. By hypothesis, there exists a locally
finite open refinement VA = {Vλ : λ ∈ Λ} ∪ {V } of UA, where Vλ ⊂ Uα and V ⊂ X \ B
such that A \ [(∪{Vλ : λ ∈ Λ}) ∪ {V }] ∈ I. Let A \ [(∪{Vλ : λ ∈ Λ}) ∪ {V }] = I. Then
I ∩ B = A \ [(∪({Vλ : λ ∈ Λ}) ∪ {V }] ∩ B = A ∩ (X \ [(∪{Vλ : λ ∈ Λ}) ∪ {V }]) ∩ B
implies that I ∩ B = A ∩ [X \ (∪{Vλ : λ ∈ Λ}) ∩ {X \ V } ∩ B]. It follows that I ∩ B =
(A ∩ B) \ ∪{Vλ : λ ∈ Λ} ∈ I. Since Vλ ⊂ Vλ ∪ V , V = {Vλ : λ ∈ Λ} is locally finite open
by Theorem 2 which refines U . Hence A ∩B is β1I-paracompact.

Corollary 12. Let f : (X, τ)→ (Y, σ,J ) be a pre β-open, continuous, bijective function.
If A is β1J -paracompact relative to Y , then f−1(A) is β1f

−1(J )-paracompact relative to
X.



REFERENCES 144

Acknowledgements

The author would like to thank the referees for their helpful suggestions.

References

[1] H. H. AlJarrah. β1-paracompact spaces. J. Nonlinear Sci. Appl, 9:1728–1734, 2016.

[2] D. Andrijević. Semi-preopen sets. Mat. Vesn., 38:24–32, 1986.

[3] A. V. Arkhangelski and V. I. Ponomarev. Fundamentals of General Topology Prob-
lems and Exercises. Int. Hindustan, India, 1984.

[4] N. Bourbaki. General Topology. Hermann Addison Wesley, Massachusets, 1966.

[5] N. Bourbaki. General Topology Part I. Addison-Wesley, Reading, Mass, 1966.

[6] M. Caldas and S. Jafari. Weak and strong forms of β-irresolutess. Arab. J. Sci. Eng,
31:31–39, 2006.

[7] D. Carnahan. Locally nearly-compact spaces. Boll. Un. Mat. Ital, 6:146–153, 1972.

[8] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud. β-open sets and β-
continuous mapping. Bull. Fac. Sci.Assint Univ, 12:77–90, 1983.

[9] T. R. Hamlett and D. Janković. On almost paracompact and para-H-closed spaces.
Quest. Answers Gen. Topology, 11:139–143, 1993.

[10] D. Janković and T. R. Hamlett. New Topologies from Old via Ideals. Amer. Math.
Monthly, 97(4):295–310, 1990.

[11] D. Janković and T. R. Hamlett. Compatible extensions of Ideals. Boll. U. M. I,
7(6-B):453–465, 1992.

[12] K. Kuratowski. Topology I, Warszawa. Warszawa, 1933.

[13] N. Levine. Semi-open sets and semi-continuity in topological spaces. Amer. Math.
Monthly, 70:36–41, 1963.

[14] A. N. Geaisa M. E. Abd El-Monsef and R. A. Mahmoud. β-regular spaces. Proc.
Math. Phys. Soc. Egypt, 60:705–717, 1985.

[15] R. A. Mahmoud and M. E. Abd El-Monsef. β-irresolute and β-topological invariant.
Proc. Pakistan Acad. Sci, 27:285–296, 1990.

[16] G. B. Navalagi. Semi-precontinuous functions and properties of generalized semi-
preclosed sets in topological spaces. Int. J. Math. Math. Sci, 29:85–98, 2002.

[17] O. Njastad. On some classes of nearly open sets. Pac. J. Math, 15:961–970, 1965.



REFERENCES 145

[18] T. Noiri. Completely continuous image of nearly paracompact space. Mat. Vesn,
29:59–64, 1977.

[19] T. Noiri. Weak and strong forms of β-irressolute functions. Acta Math. Hungar,
99:315–328, 2003.

[20] N. Sathiyasundari and V. Renukadevi. S1-Paracompactness with Respect to An Ideal.
Acta Universitatis Apulensis, 35:215–227, 2013.

[21] M. H. Stone. Applications of the theory of Boolean rings to general topology. Trans.
Amer. Math. Soc, 41:375–481, 1937.

[22] D. Rose T. R. Hamlett and D. Janković. Paracompactness with respect to an ideal.
Internat. J. Math. Math. Sci, 20(3), pages = 433-442,), 1997.

[23] R. Vaidyanathaswamy. Set Topology. Chelsea Publishing Company, New York, 1946.

[24] S. Willard. General Topology. Addison-Wesley Publishing Company, 1970.