EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 410-418
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Invited Paper

Functions and weakly µH-compact spaces

Abdo Qahis1,∗, Takashi Noiri2

1 Department of Mathematics, Faculty of Science and Arts, Nagran University,
Saudi Arabia.
2 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, Japan.

Abstract. A GTS (X,µ) is said to be weakly µH-compact if for every µ-open cover {Vα : α ∈ ∆}
of X there exists a finite subset ∆0 of ∆ such that X \ ∪{cµ(Vα) : α ∈ ∆0} ∈ H. In this
paper we study the effect of functions on weakly µH-compact spaces. The main result is that the
θ(µ, ν)-continuous image of a weakly µH-compact space is weakly νf(H)-compact.

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

Key Words and Phrases: Generalized topology, hereditary class, µH-compact, weakly µH-
compact, θ(µ, ν)-continuity.

1. Introduction

The ideas of generalized topology and hereditary classes were introduced and studied
by Császár in [3] and [5], respectively. The strategy of using generalized topologies and
hereditary classes to extend classical topological concepts have been used by many au-
thors such as [5], [8], [15], and [18]. Moreover, investigations of continuity on generalized
topological spaces have been recently of major interest among general topologists. They
are studied by many authors, including Min [10], Al-omari and Noiri [1], Császár [3], and
Jayanthi [6]. In fact, mathematicians introduced in several papers different and inter-
esting new types of functions as well as generalized continuous functions in generalized
topological spaces. The purpose of this paper is to study the effect of functions on weakly
µH-compact spaces. We also show that some functions preserve this property. The main
result is that the image of a weakly µH-compact space under a θ(µ, ν)-continuous function
is weakly µH-compact.

∗Corresponding author.

Email addresses: cahis82@gmail.com (A. Qahis), t.noiri@nifty.com ( T. Noiri)

http://www.ejpam.com 410 c© 2017 EJPAM All rights reserved.



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 411

2. Preliminaries

Let X be a nonempty set and p(X) the power set of X. A subfamily µ of p(X) is
called a generalized topology [3] if ∅ ∈ µ and the arbitrary union of members of µ is
again in µ. The pair (X,µ) is called a generalized topological space (briefly GTS). The
elements of µ are called µ-open sets and the complement of µ-open sets are called µ-closed
sets. For A ⊆ X, we denote by cµ(A) the intersection of all µ-closed sets containing
A, i.e., the smallest µ-closed set containing A and by iµ(A) the union of all µ-open sets
contained in A, i.e., the largest µ-open set contained in A (see [3], [4]). A nonempty
subcollection H of p(X) is called a hereditary class (briefly HC) [5] if A ⊂ B, B ∈ H
implies A ∈ H. An HC H is called an ideal if H satisfies the additional condition:
A,B ∈ H implies A ∪ B ∈ H [9]. Some useful hereditary classes in X are: p(A), where
A ⊆ X and Hf , the HC of all finite subsets of X. A subset A of a GTS (X,µ) is said
to be weakly µ-compact [17] if any cover of A by µ-open sets of X has a finite subfamily,
the union of the µ-closures of whose members covers A. If A = X, then (X,µ) is called
a weakly µ-compact space. Given a generalized topological space (X,µ) with an HC
H, for a subset A of X, the generalized local function of A with respect to H and µ
[5] is defined as follows: A∗(H, µ) = {x ∈ X : U ∩ A /∈ H for all U ∈ µx}, where
µx = {U : x ∈ U and U ∈ µ}. If there is no confusion, we simply write A∗ instead of
A∗(H, µ). H is said to be µ-codense if µ ∩ H = ∅ [5]. And for a subset A of X, c∗µ(A) is
defined by c∗µ(A) = A∪A∗. The family µ∗ = {A ⊂ X : X \A = c∗µ(X \A)} is a GT on X
which is finer than µ [5]. The elements of µ∗ are said to be µ∗-open and the complement
of a µ∗-open set is called a µ∗-closed set. It is clear that a subset A is µ∗-closed if and
only if A∗ ⊂ A. We call (X,µ,H) a hereditary generalized topological space and briefly
we denote it by HGTS.

Next we recall some known definitions, corollaries and theorems which will be used in
the work.

Theorem 1. [5] Let (X,µ) be a GTS, H a hereditary class on X and A be a subset
of X. If A is µ∗-open, then for each x ∈ A there exist U ∈ µx and H ∈ H such that
x ∈ U \H ⊂ A.

Definition 1. [17] Let A be a subset of a space (X,µ). Then A is said to be:
(1) µ-regular closed if A = cµ(iµ(A));
(2) µ-regular open if X \A is µ-regular closed.

Corollary 1. [14] Let f : (X,µ,H)→ (Y, ν) be a (µ, ν)-continuous surjection. If (X,µ,H)
is weakly µH-compact, then (Y, ν) is weakly νf(H)-compact.

Definition 2. A subset A of X is said to be µH-compact [2] (resp. µ-compact [7, 16]) if
for every cover {Uα : α ∈ ∆} of A by µ-open sets of (X,µ) there exists a finite subset Λ0

of Λ such that A \ ∪{Uα : α ∈ Λ0} ∈ H (resp. A ⊆ ∪{Uα : α ∈ ∆0} ). If A = X, then
(X,µ) is called a µH-compact (resp. µ-compact) space.

Definition 3. Let (X,µ) and (Y, ν) be two GTSs, then a function f : (X,µ) → (Y, ν) is
said to be.



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 412

(1) (µ, ν)-continuous [3] if U ∈ ν implies f−1(U) ∈ µ.
(2) almost (µ, ν)-continuous [13] if for each x ∈ X and each ν-open set V containing f(x),
there exists a µ-open set U containing x such that f(U) ⊆ iν(cν(V )).
(3) (µ, ν)-precontinuous [11] if f−1(V ) ⊆ iν(cν(f−1(V ))) for every ν-open set V in Y .
(4) δ(µ, ν)-continuous [10] (resp. almost δ(µ, ν)-continuous) if for each x ∈ X and each
ν-open set V of Y containing f(x), there exists a µ-open set U of X containing x such
that f(iµ(cµ(U))) ⊆ iν(cν(V )) (resp. f(iµ(cµ(U))) ⊆ cν(V )).
(5) θ(µ, ν)-continuous [3] (resp. strongly θ(µ, ν)-continuous [12]) if for every x ∈ X and
every ν-open subset V of Y containing f(x), there exists a µ-open subset U in X containing
x such that f(cµ(U)) ⊆ cν(V ) (resp. f(cµ(U)) ⊆ V ).
(6) contra-(µ, ν)-continuous [1] if f−1(V ) is µ-closed in X for every ν-open set V in Y .

3. Weakly µH-Compact Spaces

Firstly, we show some basic properties for weakly µH-compact spaces.

Definition 4. [14] Let (X,µ) be a GTS with HC. An HGTS (X,µ,H) is said to be weakly
µH-compact if for every cover {Vα : α ∈ ∆} of X by µ-open sets in X, there exists a finite
subset ∆0 of ∆ such that X \ ∪{cµ(Vα) : α ∈ ∆0} ∈ H.

The following lemma is used in the proof of the corollary stated below.

Lemma 1. An HGTS (X,µ,Hf ) is weakly µ-compact if and only if (X,µ,Hf ) is weakly
µHf -compact.

Proof. The necessity is clear and we prove the sufficiency. Assume that (X,µ,Hf ) is
weakly µHf -compact. Let {Vα : α ∈ ∆} be a cover of X by µ-open subsets of X. Then
by hypothesis, there exists a fnite subset ∆0 of ∆ such that X \

⋃
α∈∆0

cµ(Vα) ∈ Hf . Let

X \
⋃

α∈∆0

cµ(Vα) = {x1, x2, ..., xn}. For each 1 ≤ j ≤ n, choose Vαj such that xj ∈ Vαj .

Hence X = (
⋃

α∈∆0

cµ(Vα))∪ (
⋃

1≤j≤n
cµ(Vαi)). This implies that (X,µ) is weakly µ-compact.

Corollary 2. Let f : (X,µ,H)→ (Y, ν) be a (µ, ν)-continuous surjection. If (X,µ,H) is
weakly µH-compact and Y is a finite space, then (Y, ν) is weakly ν-compact.

Proof. Let f be a (µ, ν)-continuous surjection. By Corollary 1, (Y, ν) is weakly νf(H)-
compact. Since Y is a finite space, then the HC f(H) of finite subsets and and apply the
Lemma 1.

A subset A of a GTS (X,µ) is said to be µ-nowhere dense if iµ(cµ(A)) = ∅, and we
denote the HC of µ-nowhere dense sets by N (µ).

Proposition 1. If (X,µ,H) is weakly µH-compact and H is µ-condense, then (X,µ) is
weakly µ-compact.



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 413

Proof. Suppose that (X,µ,H) is weakly µH-compact and let {Vα : α ∈ ∆} be a
cover of X by µ-open subsets of X. There exists a finite subset ∆0 of ∆ such that
X \

⋃
α∈∆0

cµ(Vα) ∈ H. Since H is µ-codense, then iµ(X \
⋃

α∈∆0

cµ(Vα)) = ∅ which implies

X − iµ(X \
⋃

α∈∆0

cµ(Vα)) = cµ(
⋃

α∈∆0

cµ(Vα) =
⋃

α∈∆0

cµ(Vα) = X. Hence (X,µ) is weakly

µ-compact .

Theorem 2. For a GTS (X,µ), the following properties hold.
(1) (X,µ) is weakly µ-compact if and only if (X,µ,N (µ)) is µN (µ)-compact.
(2) (X,µ) is weakly µ-compact if and only if (X,µ,H) is µH-compact with respect to some
µ-codense HC.

Proof. (1) Necessity. Assume (X,µ) is weakly µ-compact and let {Vα : α ∈ ∆} be a
µ-open cover of X. Then by assumption there exists a finite subset ∆0 of ∆ such that
X =

⋃
α∈∆0

cµ(Vα) = cµ(
⋃

α∈∆0

Vα). Since X \ cµ(
⋃

α∈∆0

Vα) = ∅ and X \
⋃

α∈∆0

Vα is a µ-closed,

then iµ(X \
⋃

α∈∆0

Vα) = ∅. This implies X \
⋃

α∈∆0

Vα ∈ N (µ). Thus (X,µ,H) is µN (µ)-

compact.
Sufficiency. Assume (X,µ,N (µ)) is µN (µ)-compact and let {Vα : α ∈ ∆} be a µ-open
cover of X. By assumption, there exists a finite subset ∆0 of ∆ such that X \

⋃
α∈∆

Vα ∈

N (µ). This implies that iµ(X \
⋃
α∈∆

Vα) = ∅ and hence

X = cµ(
⋃
α∈∆0

Vα) =
⋃
α∈∆0

cµ(Vα).

(2) Necessity. From (1) H is a µ-codense.
Sufficiency. The proof is obvious by Proposition 1.

Theorem 3. Let f : (X,µ)→ (Y, ν,G) be a surjection onto a νG-compact. If µ = f−1(ν)
is the weak generalized topology on X induced by f and ν, then (X,µ) is µf−1(G)-compact.

Proof. Let {f−1(Vα) : α ∈ ∆} be a µ-open cover of X. Then {Vα : α ∈ ∆} is a ν-open
cover of Y and hence there exists a finite subset ∆0 of ∆ such that Y \

⋃
α∈∆

Vα ∈ G. Now

we have

f−1

Y \ ⋃
α∈∆0

Vα

 = X \
⋃
α∈∆0

f−1 (Vα) ∈ f−1(G).

Hence (X,µ) is µf−1(G)-compact.

The following lemma is used to prove the corollary which is stated below.

Lemma 2. If f : (X,µ) → (Y, ν,G) is a surjection and G is ν-codense, then f−1(G) is
f−1(ν)-codense, where µ = f−1(ν).



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 414

Proof. Assume f : (X, µ)→ (Y, ν,G) is a surjection and f−1(G) is not f−1(ν)-codense,
then there exists G ∈ G such that f−1(G) ∈ f−1(ν) \ {∅}, say f−1(G) = f−1(V ) where
V ∈ ν \ {∅}. Then G = V ∈ ν \ {∅} and G is not ν-codense. Then this contradicts to our
assumption.

Corollary 3. Let f : (X,µ) → (Y, ν,G) be a surjection and let µ denote the weak gener-
alized topology on X induced by f and ν. If G is ν-codense and (Y, ν,G) is νG-compact,
then (X,µ) is weakly µ-compact

Proof. If G is ν-codense and (Y, ν) is νG-compact, then by Theorem 3, (X,µ) is
µf−1(G)-compact. Since G is ν-codense. Then by Lemma 2, f−1(G) is µ = f−1(ν)-
codense. By Theorem 2(2), (X,µ) is weakly µ-compact.

Next we introduce the main result and prove that the θ(µ, ν)-continuous image of a
weakly µH-compact space is weakly νf(H)-compact. Note that if H is a hereditary class
on a set X and f : (X,µ) → (Y, ν) is a function, then f(H) = {f(H) : H ∈ H} is a HC
on Y [2].

Theorem 4. Let f : (X,µ,H)→ (Y, ν) be a θ(µ, ν)-continuous function. If A is a weakly
µH-compact subset of X, then f(A) is weakly νf(H)-compact.

Proof. Let V = {Vα : α ∈ ∆} be a cover of f(A) by ν-open subsets of Y . Let x ∈ A
and Vα(x) be a ν-open set in Y such that f(x) ∈ Vα(x). Since f is θ(µ, ν)-continuous,
there exists a µ-open set Uα(x) of X containing x such that f(cµ(Uα(x))) ⊆ cν(Vα(x)).
Now

{
Uα(x) : x ∈ A

}
is a cover of A by µ-open subsets of X. Since A is weakly µH-

compact, there exists a finite subset A0 of A such that A \
⋃

x∈A0

cµ(Uα(x)) ∈ H. Now

f(A \
⋃

x∈A0

cµ(Uα(x))) ∈ f(H). We know

f(A) \ f(
⋃
x∈A0

cµ(Uα(x))) ⊆ f(A \
⋃
x∈A0

cµ(Uα(x))).

This implies

f(A) \ f(
⋃
x∈A0

cµ(Uα(x))) = f(A) \
⋃
x∈A0

f(cµ(Uα(x))) ∈ f(H).

Since f(cµ(Uα(x))) ⊆ cν(Vα(x)) for each α(x), f(A)\
⋃

x∈A0

cν(Vα(x)) ⊆ f(A)\
⋃

x∈A0

f(cµ(Uα(x))).

Thus f(A) \
⋃

x∈A0

cν(Vα(x)) ∈ f(H). This implies that f(A) is weakly νf(H)-compact.

Corollary 4. Let f : (X,µ) → (Y, ν) be a θ(µ, ν)-continuous surjection. If (X,µ,H) is
weakly µH-compact, then (Y, ν, f(H)) is weakly νf(H)-compact.

The following lemma is used in the proofs of corollaries stated below.



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 415

Lemma 3. If f : (X,µ)→ (Y, ν) is almost (µ, ν)-continuous, then f is θ(µ, ν)-continuous.

Proof. Let f be almost (µ, ν)-continuous. Let x ∈ X and V be a ν-open set in Y
such that f(x) ∈ V . Since f is almost (µ, ν)-continuous, there exists a µ-open set U of X
containing x such that f(U) ⊆ iν(cν(V )). This implies f(U) ⊆ iν(cν(V )) ⊆ cν(V ). We
have to show that f(cµ(U)) ⊆ cν(V ). For some x0 ∈ cµ(U) let f(x0) ∈ Y \ cν(V ). Then
by the almost (µ, ν)-continuity of f there exists a µ-open set W of X containing x0 such
that f(W ) ⊆ iν(cν(Y \ cν(V ))). But W ∩ U 6= ∅ and hence f(U) ∩ iν(cν(Y \ cν(V ))) 6= ∅.
Hence, we get a contradiction to the fact that f(U) ⊆ iν(cν(V )) ⊆ cν(iν(cν(V ))) ⊆ cν(V ).
Thus f(cµ(U)) ⊆ cν(V ). This implies that f is θ(µ, ν)-continuous.

Corollary 5. Let f : (X,µ) → (Y, ν) be an almost (µ, ν)-continuous surjection. If
(X,µ,H) is weakly µH-compact, then (Y, ν, f(H)) is weakly νf(H)-compact.

Since every (µ, ν)-continuous function is almost (µ, ν)-continuous and by Lemma 3, we
conclude the following corollary.

Corollary 6. Weakly µH-compact property is a GT property.

By taking H = {∅}, we get the following corollary

Corollary 7. Let f : (X,µ) → (Y, ν) be a θ(µ, ν)-continuous function. If A is a weakly
µ-compact subset of X, then f(A) is weakly ν-compact.

Proposition 2. Let f : (X,µ,H) → (Y, ν) be a strongly θ(µ, ν)-continuous function. If
A is a weakly µH-compact subset of X, then f(A) is νf(H)-compact.

Proof. Let {Vα : α ∈ ∆} be a cover of f(A) by ν-open subsets of Y . Let x ∈ A and
Vα(x) be a ν-open set in Y such that f(x) ∈ Vα(x). Since f is strongly θ(µ, ν)-continuous,
there exists a µ-open set Uα(x) of X containing x such that f(cµ(Uα(x))) ⊆ Vα(x). Now{
Uα(x) : x ∈ A

}
is a µ-open cover of the weakly µH-compact set A. So there exists a finite

subset A0 of A such that A \
⋃

x∈A0

cµ(Uα(x)) ∈ H.

Now f(A \
⋃

x∈A0

cµ(Uα(x))) ∈ f(H). We know

f(A) \ f(
⋃
x∈A0

cµ(Uα(x))) ⊆ f(A \
⋃
x∈A0

cµ(Uα(x))).

This implies

f(A) \ f(
⋃
x∈A0

cµ(Uα(x))) = f(A) \
⋃
x∈A0

f(cµ(Uα(x))) ∈ f(H).

Since f(cµ(Uα(x))) ⊆ Vα(x) for each α(x), f(A) \
⋃

x∈A0

Vα(x) ⊆ f(A) \
⋃

x∈A0

f(cµ(Uα(x))).

Thus f(A) \
⋃

x∈A0

Vα(x) ∈ f(H). Hence f(A) is νf(H)-compact.



A. Qahis, T. Noiri / Eur. J. Pure Appl. Math, 10 (3) (2017), 410-418 416

Corollary 8. Let f : (X,µ) → (X, ν) be a strongly θ(µ, ν)-continuous surjection. If
(X,µ,H) is weakly µH-compact, then (Y, ν, f(H)) is νf(H)-compact.

By taking H = {∅}, we get the following corollary.

Corollary 9. Let f : (X,µ)→ (X, ν) be a strongly θ(µ, ν)-continuous function. If A is a
weakly µ-compact subset of X, then f(A) is ν-compact.

Proposition 3. Let f : (X,µ,H) → (Y, ν) be a contra (µ, ν)-continuous and (µ, ν)-
precontinuous function. If A is weakly µH-compact, then f(A) is νf(H)-compact.

Proof. Let V = {Vα : α ∈ ∆} be a cover of f(A) by ν-open sets of (Y, ν). For each
x ∈ A, let Vα(x) ∈ V such that f(x) ∈ Vα(x). Since f is contra (µ, ν)-continuous and
(µ, ν)-precontinuous, f−1(Vα(x)) is µ-closed in X and f−1(Vα(x)) ⊆ iµ(cµ(f−1(Vα(x)))) =
iµ(f−1(Vα(x))). So f−1(Vα(x)) = iµ(f−1(Vα(x))). This implies f−1(Vα(x)) is µ-clopen and
hence {f−1(Vα(x)) : x ∈ A} is a µ-clopen cover of the weakly µH-compact subset A. There
exists a finite subset A0 of A such that

A \
⋃
x∈A0

cµ(f−1(Vα(x))) = A \
⋃
x∈A0

f−1(Vα(x)) ∈ H.

Now we have

f(A) \
⋃
x∈A0

Vα(x) ⊂ f(A) \ f(
⋃
x∈A0

f−1(Vα(x))) ⊂ f(A \
⋃
x∈A0

f−1(Vα(x))) ∈ f(H).

This implies f(A) \
⋃

x∈A0

Vα(x) ∈ f(H). Hence f(A) is νf(H)-compact.

Corollary 10. Let f : (X,µ)→ (Y, ν) be a contra (µ, ν)-continuous and (µ, ν)-precontinuous
surjection. If (X,µ,H) is weakly µH-compact, then (Y, ν, f(H)) is νf(H)-compact.

By taking H = {∅}, we get the following corollary.

Corollary 11. Let f : (X,µ)→ (Y, ν) be a contra (µ, ν)-continuous and (µ, ν)-precontinuous
function. If A is weakly µ-compact, then f(A) is ν-compact.

Theorem 5. Let f : (X,µ,H) → (Y, ν) be an almost δ(µ, ν)-continuous function. If for
every cover {Uα : α ∈ ∆} of A ⊆ X by µ-regular open sets in X there exists a finite subset
∆0 of ∆ such that A \

⋃
α∈∆0

Uα ∈ H, then f(A) is weakly νf(H)-compact.

Proof. Let V = {Vα : α ∈ ∆} be a cover of f(A) by ν-open sets of Y . Let x ∈ A
and Vα(x) ∈ V such that f(x) ∈ Vα(x). Then iµ(cµ(Vα(x))) is a µ-regularly open set in Y
containing f(x). Since f is almost δ(µ, ν)-continuous, then for every x ∈ A, there exists
a µ-open subset Uα(x) of X containing x such that f(iµ(cµ(Uα(x)))) ⊆ cν(Vα(x)). Then



REFERENCES 417

{iµ(cµ(Uα(x))) : x ∈ A} is a µ-regularly open cover of A. It follows that there exists a
finite subset A0 of A such that A \

⋃
x∈A0

iµ(cµ(Uα(x))) ∈ H. Now

f(A) \ f(
⋃
x∈A0

iµ(cµ(Uα(x)))) ⊆ f(A \
⋃
x∈A0

iµ(cµ(Uα(x)))) ∈ f(H).

This implies f(A) \
⋃

x∈A0

f(iµ(cµ(Uα(x)))) ∈ f(H). Therefore, we obtain

f(A) \
⋃
x∈A0

cν(Vα(x)) ⊆ f(A) \
⋃
x∈A0

f(iµ(cµ(Uα(x)))).

This implies f(A)\
⋃

x∈A0

cν(Vα(x)) ∈ f(H). This shows that f(A) is weakly νf(H)-compact.

Corollary 12. Let f : (X,µ,H) → (Y, ν) be an almost δ(µ, ν)-continuous surjection. If
for every cover {Uα : α ∈ ∆} of X by µ-regular open sets of X there exists a finite subset
∆0 of ∆ such that X \

⋃
α∈∆0

Uα ∈ H, then (Y, ν, f(H)) is weakly νf(H)-compact.

By taking H = {∅}, we get the following corollary.

Corollary 13. Let f : (X,µ) → (Y, ν) be an almost δ(µ, ν)-continuous function. If for
every cover {Uα : α ∈ ∆} of A ⊆ X by µ-regular open sets in X there exists a finite subset
∆0 of ∆ such that A ⊆

⋃
α∈∆0

Uα, then f(A) is weakly ν-compact.

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