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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 2, 2011

pp. 135-141

Mappings on weakly Lindelöf and weakly

regular-Lindelöf spaces

Anwar Jabor Fawakhreh and Adem Kılıçman

Abstract

In this paper we study the effect of mappings and some decompositions
of continuity on weakly Lindelöf spaces and weakly regular-Lindelöf
spaces. We show that some mappings preserve these topological prop-
erties. We also show that the image of a weakly Lindelöf space (resp.
weakly regular-Lindelöf space) under an almost continuous mapping is
weakly Lindelöf (resp. weakly regular-Lindelöf). Moreover, the image
of a weakly regular-Lindelöf space under a precontinuous and contra-
continuous mapping is Lindelöf.

2010 MSC: Primary: 54A05, 54C08, 54C10; Secondary: 54C05, 54D20.

Keywords: Lindelöf, weakly Lindelöf and weakly regular-Lindelöf spaces.
Almost continuous and almost precontinuous functions.

1. Introduction

Among the various covering properties of topological spaces a lot of attention
has been made to those covers which involve open and regular open sets. In
1959 Frolik [9] introduced the notion of a weakly Lindelöf space that afterward
was studied by several authors. In 1982 Balasubramanian [2] introduced and
studied the notion of nearly Lindelöf spaces. In 1984 Willard and Dissanayake
[18] gave the notion of almost Lindelöf spaces and in 1996 Cammaroto and
Santoro [4] introduced the notion of weakly regular-Lindelöf spaces on using
regular covers. Some generalizations of Lindelöf spaces have been recently
studied by the authors (see [7]) and by Song and Zhang [17].

Moreover, decompositions of continuity have been recently of major interest
among general topologist. They have been studied by many authors, including
Singal and Singal [16], Mashhour et al. [11], Abd El-Monsef et al. [1], Nasef



136 A. J. Fawakhreh and A. Kılıçman

and Noiri [12], Noiri and Popa [13], Dontchev [5], Dontchev and Przemski [6],
Kohli and Singh [10] and many other topologists.

Throughout the present paper, spaces mean topological spaces on which no
separation axioms are assumed unless explicitly stated otherwise. The interior
and the closure of any subset A of a space X will be denoted by Int(A) and
Cl(A) respectively. By regular open cover of X we mean a cover of X by regular
open sets in (X, τ).

The purpose of this paper is to study effect of mappings and decompositions
of continuity on weakly Lindelöf and weakly regular-Lindelöf spaces. We also
show that some mappings preserve these topological properties. We conclude
that the image of a weakly Lindelöf space (resp. weakly regular-Lindelöf space)
under an almost continuous mapping is weakly Lindelöf (resp. weakly regular-
Lindelöf). Moreover, the image of a weakly regular-Lindelöf space under a
precontinuous and contra-continuous mapping is Lindelöf.

2. Preliminaries

Recall that a subset A ⊆ X is called regular open (regular closed) if A =
Int(Cl(A)) (A = Cl(Int(A))). A space (X, τ) is said to be semiregular if the
regular open sets form a base for the topology. It is called almost regular if for
any regular closed set C and any singleton {x} disjoint from C, there exist two
disjoint open sets U and V such that C ⊆ U and x ∈ V . Note that a space X
is regular if and only if it is semiregular and almost regular [14]. Moreover, a
space X is said to be submaximal if every dense subset of X is open in X and
it is called extremally disconnected if the closure of each open set of X is open
in X. A space X is said to be nearly paracompact [15] if every regular open
cover of X admits an open locally finite refinement.

Definition 2.1. Let (X, τ) and (Y, σ) be topological spaces. A function f :
X −→ Y is said to be

(1) almost continuous [16] if f−1(V ) is open in X for every regular open
set V in Y .

(2) precontinuous [11]
(

resp. β-continuous [1]
)

if f−1(V ) ⊆ Int(Cl(f−1(V )))
(

resp. f−1(V ) ⊆ Cl(Int(Cl(f−1(V ))))
)

for every open set V in Y .

(3) almost precontinuous
(

resp. almost β-continuous
)

[12] if for each x ∈

X and each regular open set V in Y containing f(x), there exists a set

U in X containing x with U ⊆ Int(Cl(U))
(

resp. U ⊆ Cl(Int(Cl(U)))
)

such that f(U) ⊆ V .
(4) contra-continuous [5] if f−1(V ) is closed in X for every open set V in

Y .

Note that almost continuity as well as precontinuity implies almost pre-
continuity and almost precontinuity as well as β-continuity implies almost β-
continuity but the converses, in general, are not true (see [6], [11] and [13]).



Mapping on weakly Lindelöf spaces 137

Definition 2.2. A topological space X is said to be nearly Lindelöf [2]
(

resp.

almost Lindelöf [18]
)

if, for every open cover {U
α
: α ∈ ∆} of X, there exists a

countable subset {α
n
: n ∈ N} ⊆ ∆ such that X =

⋃

n∈N
Int(Cl(U

αn
))

(

resp.

X =
⋃

n∈N
Cl(U

αn
)
)

.

3. Mappings on Weakly Lindelöf Spaces

Definition 3.1 ([9]). A topological space X is said to be weakly Lindelöf
if for every open cover {U

α
: α ∈ ∆} of X there exists a countable subset

{α
n
: n ∈ N} ⊆ ∆ such that X = Cl(

⋃

n∈N
U

αn
).

It is obvious that every nearly Lindelöf space is almost Lindelöf and every
almost Lindelöf space is weakly Lindelöf, but the converses are not true (see
[4]). Moreover, it is well known that the continuous image of a Lindelöf space is
Lindelöf and in [8] it was shown that a δ-continuous image of a nearly Lindelöf
space is nearly Lindelöf. For weakly Lindelöf spaces we give the following
theorem.

Theorem 3.2. Let (X, τ) and (Y, σ) be topological spaces. Let f : X −→ Y be
an almost continuous surjection from X into Y . If X is weakly Lindelöf then

Y is weakly Lindelöf.

Proof. Let {U
α
: α ∈ ∆} be an open cover of Y . Then {Int(Cl(U

α
)) : α ∈ ∆}

is a regular open cover of Y . Since f is almost continuous, f−1(Int(Cl(U
α
)))

is an open set in X. Thus {f−1(Int(Cl(U
α
))) : α ∈ ∆} is an open cover of the

weakly Lindelöf space X. So there exists a countable subset {α
n
: n ∈ N} ⊆ ∆

such that

X = Cl(
⋃

n∈N

f
−1(Int(Cl(U

αn
)))) ⊆ Cl(

⋃

n∈N

f
−1(Cl(U

αn
)))

= Cl(f−1(
⋃

n∈N

Cl(U
αn

))) ⊆ Cl(f−1(Cl(
⋃

n∈N

U
αn

))).

Since Cl(
⋃

n∈N
U
αn

) is regular closed in Y and f is almost continuous, we have

f−1(Cl(
⋃

n∈N
U
αn

)) is closed in X. So

X = Cl(f−1(Cl(
⋃

n∈N

U
αn

))) = f−1(Cl(
⋃

n∈N

U
αn

)).

Since f is surjective,

Y = f(X) = f(f−1(Cl(
⋃

n∈N

U
αn

))) = Cl(
⋃

n∈N

U
αn

).

Which implies that Y is weakly Lindelöf and completes the proof. �

Corollary 3.3. The almost continuous image of a weakly Lindelöf space is

weakly Lindelöf.



138 A. J. Fawakhreh and A. Kılıçman

Since every continuous function is almost continuous, (Proposition 3.5, [17])
becomes a direct corollary of Theorem 3.2 above.

Corollary 3.4. Weakly Lindelöf property is a topological property.

Note that a weakly Lindelöf, semiregular and nearly paracompact space X
is almost Lindelöf (see [4], Theorem 3.8). So depending on Theorem 3.2 above
we conclude the following two corollaries.

Corollary 3.5. Let f : X −→ Y be an almost continuous surjection from X
into Y . If X is weakly Lindelöf and Y is semiregular and nearly paracompact

then Y is almost Lindelöf.

Corollary 3.6. Let f : X −→ Y be an almost continuous surjection from X
into Y . If X is weakly Lindelöf and Y is regular and nearly paracompact then

Y is Lindelöf.

Proposition 3.7. Let f : X −→ Y be an almost β-continuous surjection. If X
is submaximal, extremally disconnected and weakly Lindelöf then Y is weakly

Lindelöf.

Proof. This follows immediately from Theorem 3.2 above and ([12], Theorem
4.3). �

Proposition 3.8. Let f : X −→ Y be an almost precontinuous surjection. If
X is submaximal and weakly Lindelöf then Y is weakly Lindelöf.

Proof. This follows immediately from Theorem 3.2 above and ([12], Theorem
4.4). �

4. Mappings on Weakly Regular-Lindelöf Spaces

Definition 4.1 ([3]). An open cover {U
α
: α ∈ ∆} of a topological space X is

called regular cover if, for every α ∈ ∆, there exists a nonempty regular closed
subset C

α
of X such that C

α
⊆ U

α
and X =

⋃

α∈∆
Int(C

α
).

Definition 4.2 ([4]). A topological space X is said to be weakly regular-
Lindelöf if every regular cover {U

α
: α ∈ ∆} of X admits a countable subset

{α
n
: n ∈ N} ⊆ ∆ such that X = Cl(

⋃

n∈N
U

αn
).

Now we prove the following theorem.

Theorem 4.3. Let (X, τ) and (Y, σ) be topological spaces. Let f : X −→ Y be
an almost continuous surjection from X into Y . If X is weakly regular-Lindelöf

then Y is weakly regular-Lindelöf.

Proof. Let f : X −→ Y be an almost continuous function from the weakly
regular-Lindelöf space X onto Y . Let {U

α
: α ∈ ∆} be a regular cover of

Y . (i. e. for every α ∈ ∆ there exists a regular closed set C
α

⊆ U
α
with

Y =
⋃

α∈∆
Int(C

α
).) But

⋃

α∈∆

Int(C
α
) ⊆

⋃

α∈∆

C
α
⊆

⋃

α∈∆

U
α
⊆

⋃

α∈∆

Int(Cl(U
α
)).



Mapping on weakly Lindelöf spaces 139

So
⋃

α∈∆

f
−1(Int(C

α
)) ⊆

⋃

α∈∆

f
−1(C

α
) ⊆

⋃

α∈∆

f
−1(U

α
) ⊆

⋃

α∈∆

f
−1(Int(Cl(U

α
))).

Thus
X = f−1(Y ) = f−1(

⋃

α∈∆

Int(C
α
)) =

⋃

α∈∆

f−1(Int(C
α
)).

Since C
α
is regular closed and f is almost continuous, f−1(C

α
) is closed. Thus

Cl(Int(f−1(C
α
))) ⊆ f−1(C

α
) ⊆ f−1(Int(Cl(U

α
))).

Since Int(C
α
) is regular open in Y and f is almost continuous, f−1(Int(C

α
))

is open in X. Thus
⋃

α∈∆

Int(Cl(Int(f−1(C
α
)))) =

⋃

α∈∆

Int(f−1(C
α
))

⊇
⋃

α∈∆

Int(f−1(Int(C
α
))) =

⋃

α∈∆

f
−1(Int(C

α
)) = X.

It means that, for every α ∈ ∆, there exists a regular closed set Cl(Int(f−1(C
α
))) ⊆

f−1(Int(Cl(U
α
))) such that X =

⋃

α∈∆ Int(Cl(Int(f
−1(C

α
)))). Thus {f−1(Int(Cl(U

α
))) :

α ∈ ∆} is a regular cover of the weakly regular-Lindelöf space X. So there
exists a countable subset {α

n
: n ∈ N} ⊆ ∆ such that

X = Cl(
⋃

n∈N

f−1(Int(Cl(U
αn
)))) ⊆ Cl(

⋃

n∈N

f−1(Cl(U
αn

)))

= Cl(f−1(
⋃

n∈N

Cl(U
αn
))) ⊆ Cl(f−1(Cl(

⋃

n∈N

U
αn
))).

Since f is almost continuous and Cl(
⋃

n∈N
U
αn

) is regular closed, f−1(Cl(
⋃

n∈N
U
αn

))

is closed in X. So X = Cl(f−1(Cl(
⋃

n∈N
U

αn
))) = f−1(Cl(

⋃

n∈N
U

αn
)). Thus

Y = f(X) = f(f−1(Cl(
⋃

n∈N

U
αn

))) ⊆ Cl(
⋃

n∈N

U
αn

).

This implies that Y is weakly regular-Lindelöf and completes the proof. �

Corollary 4.4. The almost continuous image of a weakly regular-Lindelöf

space is weakly regular-Lindelöf.

Corollary 4.5. Weakly regular-Lindelöf property is a topological property.

Note that every regular and weakly regular-Lindelöf space X is weakly Lin-
delöf and if X, moreover, is nearly paracompact then it is Lindelöf (see [7]).
So depending on Theorem 4.3 above we conclude the following two corollaries.

Corollary 4.6. Let f : X −→ Y be an almost continuous mapping from X
onto Y . If X is weakly regular-Lindelöf and Y is regular then Y is weakly

Lindelöf.

Corollary 4.7. Let f : X −→ Y be an almost continuous mapping from X
onto Y . If X is weakly regular-Lindelöf and Y is regular and nearly paracompact

then Y is Lindelöf.



140 A. J. Fawakhreh and A. Kılıçman

Next we prove the following proposition.

Proposition 4.8. The image of a weakly regular-Lindelöf space under a pre-

continuous and contra-continuous mapping is Lindelöf.

Proof. Let f : X −→ Y be a contra-continuous and precontinuous mapping
from the weakly regular-Lindelöf space X into Y . Let U = {U

α
: α ∈ ∆} be

an open cover of f(X). For each x ∈ X, let U
αx

∈ U such that f(x) ∈ U
αx

.
Since f is contra-continuous, f−1(U

αx
) is closed in X. Since f is precon-

tinuous, f−1(U
αx

) ⊆ Int(Cl(f−1(U
αx

))) = Int(f−1(U
αx

)). So f−1(U
αx

) =
Int(f−1(U

αx
)). It follows that f−1(U

αx
) is closed and open in X and hence

{f−1(U
αx

) : x ∈ X} is a regular cover of the weakly regular-Lindelöf space X.
Thus there exists a countable subfamily {x

n
: n ∈ N} such that

X = Cl(
⋃

n∈N

f−1(U
αxn

)) = Cl(f−1(
⋃

n∈N

U
αxn

)).

Since
⋃

n∈N
U
αxn

is open in Y and f is contra-continuous, f−1(
⋃

n∈N
U
αxn

) is
closed in X. Thus

Cl(f−1(
⋃

n∈N

U
αxn

)) = f−1(
⋃

n∈N

U
αxn

).

So X = f−1(
⋃

n∈N
U
αxn

). Since f is surjective,

f(X) = f(f−1(
⋃

n∈N

U
αxn

)) ⊆
⋃

n∈N

U
αxn

.

This implies that f(X) is Lindelöf and completes the proof. �

As in weakly Lindelöf spaces we give the following propositions.

Proposition 4.9. Let f : X −→ Y be an almost β-continuous surjection. If
X is submaximal, extremally disconnected and weakly regular-Lindelöf then Y

is weakly regular-Lindelöf.

Proof. The proof follows immediately from ([12], Theorem 4.3) and Theorem
4.3 above. �

Proposition 4.10. Let f : X −→ Y be an almost precontinuous surjection. If
X is submaximal and weakly regular-Lindelöf then Y is weakly regular-Lindelöf.

Proof. The proof follows immediately from ([12], Theorem 4.4) and Theorem
4.3 above. �

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Mapping on weakly Lindelöf spaces 141

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(Received December 2010 – Accepted March 2011)

A. J. Fawakhreh (abuanas7@hotmail.com)
Department of Mathematics, Collage of Science, Qassim University, P.O. Box
6644, Buraydah 51402, Saudi Arabia.

A. Kılıçman (akilicman@putra.upm.edu.my)
Department of Mathematics and Institute for Mathematical Research, Univer-
sity Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia.


	Mappings on weakly Lindelöf and weakly [3pt] regular-Lindelöf spaces. By A. J. Fawakhreh and A. Kılıçman