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Some Types of µ-semi-compactness Via
hereditary classes
Abdo Qahis
Department of Mathematics, Faculty of Science and Arts
Nagran university, Saudi Arabia
E-mail: [email protected]
Abstract
The aim of this paper is to introduce and study the notions of µ-semicompactness and strong µ-semi-compactness in generalized topological spaces
with respect to a hereditary class, called µH-semi-compactness and S-µH-semicompactness, respectively. Also several of their properties are presented. Finally some effects of various kinds of functions on them are studied..
Keywords: Generalized topology, hereditary class, µ-semi-covering, µHsemi-compact, S-µH-semi-compact.
2000 Mathematics Subject Classification. 54A05, 54A08, 54D10.
1
Preliminaries and Introduction
This work is developed around the concept of µ-semi-compactness with respect a hereditary class which was introduced by Jamal M. Mustafa in [8]. In
this research, we use the notions of generalized topology and hereditary class
introduced by Császár in [1] and [2], respectively, in order to define and characterize the µH-semi-compactness and S-µH-compactness spaces. Also some
properties of these spaces are obtained and the behavior of these spaces under
certain kinds of functions also is investigated. The strategy of using generalized topologies and hereditary classes to extend classical topological concepts
have been used by many authors such as [2], [9], [11], [13].
Let X be a non-empty set and 2X denote the power set of X. We call a class
µ ⊆ 2X a generalized topology [1] (briefly, GT) if ϕ ∈ µ and arbitrary union
of elements of µ belongs to µ. A set X with a GT is called a generalized
topological space (briefly, GTS) and is denoted by (X, µ). For a GTS (X, µ),
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
54
Abdo Qahis .
by iµ (A) the union of all µ-open sets contained in A, i.e., the largest µ-open
set contained in A (see [1], [3]). A subset A of X is said to be µ-semi-compact
if for every µ-semi-open cover U of A there exists a finite subfamily U ′ that
also covers A. X is said to be µ-semi-compact if X is µ-semi-compact as a
subset[8].
A nonempty family H of subsets of X is called a hereditary class [2] if
A ∈ H and B ⊂ A imply that B ∈ H. Given a generalized topological
space (X, µ) with a hereditary class H, for a subset A of X, the generalized local function of A with respect to H and µ [2] is defined as follows:
A∗ = {x ∈ X : U ∩ A ∈
/ H for all U ∈ µx }, where µx = {U : x ∈
U and U ∈ µ}. And for a subset A of X, is defined: c∗µ (A) = A ∪ A∗ . The
family µ∗ = {A ⊂ X : X \ A = c∗µ (X \ A)} is a GT on X. The elements of µ∗
are called µ∗ -open and the complement of a µ∗ -open set is called µ∗ -closed set.
It is clear that a subset A is µ∗ -closed if and only if A∗ ⊂ A. If the hereditary
class H satisfies the additional condition: if A, B ∈ H implies A ∪ B ∈ H,
then H is called an ideal on X [10]. We call (X, µ, H) a hereditary generalized
topological space and briefly we denote it by HGTS.
Definition 1.1 [3] Let (X, µ) generalized topological spaces. Then, A ⊆ X
is said to be µ-semi-open set if and only if A ⊂ cµ iµ (A) i.e., if there exist a
µ-open set V ∈ µ, such that V ⊆ A ⊆ cµ (V ).
The complement of a µ-semi-open set is said to be µ-semi-closed. The collection of all µ-semi-open set ( resp. µ-semi-closed set ) sets in X is denoted by
SOµ (X)(resp : SCµ (X)). The µ-semi-closure of A in (X, µ) is the intersection
of all µ-semi-closed set containing A and is denoted by scµ (A).
Definition 1.2 [11] Let (X, µ, H) hereditary generalized topological spaces
and A ⊂ X, Aµ (H, µ) = {x ∈ X : U ∩ A ̸∈ H f or every U ∈ SOµ (X, x)}
is called the generalized semi-local function of A with respect to H and µ, where
SOµ (X, x) = {U ∈ SOµ (X) : x ∈ U }.
We simply write Aµ instead of Aµ (H, µ) is this case there is no ambiguity. It is
given in [14], µ∗g (H) is a generalized topology on X, generated by the subbases
{U − H : U ∈ SOµ (X) and H ∈ H} or equivalently µ∗g (H) = {U ⊂ X :
∗g
c∗g
µ (X − U ) = X − U }. The closure operator cµ for a generalized topology
∗g
µ∗g (H) is defined as follows: for A ⊆ X, c∗g
µ (A) = A ∪ Aµ . iµ denotes the
∗g
∗
∗g
interior of the set µ (H). It is know that µ ⊆ µ (H) ⊆ µ (H).
Definition 1.3 [14] A subset of A of an hereditary generalized topological
spaces (X, µ, H) is said to be:
1. µ-semi-∗-perfect if A = Aµ ;
2. µ-semi dense in-itself if A ⊆ Aµ ;
55
Some Types of µ-semi-compactness Via hereditary Classes
3. µ-semi-∗-closed if Aµ ⊆ A.
Definition 1.4 Let (X, µ) be a GTS. Then a subset A of X is said to be
µ-dense [6] if cµ (A) = X. The space (X, µ) is said to be µ-submaximal [7] if
every µ-dense subset is µ-open in X.
Definition 1.5 [4] A GTS (X, µ) is said to be µ-extremally disconnected if
the µ-closure of every µ-open set is µ-open
Definition 1.6 [1] Let (X, µ) and (Y, ν) be two GTS’s, then a function
f : (X, µ) → (Y, ν) is said to be (µ, ν)-continuous if U ∈ ν implies f −1 (U ) ∈ µ.
Definition 1.7 [8] Let (X, µ) and (Y, ν) )be GTS’s. A function f : (X, µ) →
(Y, ν) is said to be µ-pre-semi-open if it maps µ-semi-open subsets of X onto
µ-semi-open subsets of Y .
2
µH-Semi-Compactness Spaces
Definition 2.1 A subset A of a HGTS (X, µ, H) is said to be µH-semicompact if for every µ-semi-open cover {Vα }α∈Λ of A there exists a finite subset
∪
Λ0 of Λ such that A \
Vα ∈ H. The HGTS (X, µ, H) is said to be µHα∈Λ0
semi-compact if X is µH-semi-compact as a subset.
Remark 2.2
1. It is clear that (X, µ) is µ-semi-compact if and only if
(X, µ, {ϕ}) is µ{ϕ}-semi-compact.
2. If (X, µ) is a µ-semi-compact then (X, µ, H) is µH-semi-compact. The
converse is not true as shown by the following example.
Example 2.3 Let X = [0, 1), µ = {ϕ} ∪ {[0, a) : 0 < a ≤ 1} and H = {A :
A ⊆ [0, 1)}, then:
1. (X, µ, H) is µH-semi-compact because for any µ-semi-open cover {Vα }α∈Λ
∪
we have X \
Vα ∈ H;
α∈Λ
2. (X, µ) is not µ-semi-compact. In fact if we take Vn = [0, 1 − n1 ) for all
∪
integer number n ≥ 2, then X =
[0, 1 − n1 ) but if k ∈ Z + where Z +
n≥2
is the set of positive integers, and n1 , n2 , ..., nk are integer numbers with
ni ≥ 2, for all i ∈ {1, 2, ..., k}, then X ̸=
k [
∪
i=1
0, 1 −
1
ni
)
.
It is clear that the family SOµ (X) of all µ-semi-open subsets of X forms a
generalized topology on X which is finer than µ.
56
Abdo Qahis .
Proposition 2.4 Let (X, µ, H) be a HGTS and B be a base for the generalized topology SOµ (X), then (X, µ, H) is µH-semi-compact if and only if for
∪
all family {Vα }α∈Λ of µ-semi-open sets in B, if X =
Vα then there exists
Λ0 ⊆ Λ, finite, with X \
∪
α∈Λ0
α∈Λ
Vα ∈ H.
Proof. (=⇒ ) Let {Vα }α∈Λ be a family of non-empty µ-semi-open sub∪
sets of X such that X =
Vα . Since B be a base for SOµ (X), then
α∈Λ
SOµ (X) = {∪B : B ⊆ B}. So, for each α ∈ Λ there exists a family
∪
∪
{Bαβ : β ∈ Λα } ⊆ B such that Vα =
Bαβ . Given that X =
Vα
and X =
such that
∪
∪
α∈Λ
β∈Λα
Bβα is µH-semi-compact there exist Bα1 β1 , Bα2 β2 , ..., Bαk βk
α∈Λ β∈Λα
k
∪
X \ Bαi βi
i=1
∈ H. But X \
k
∪
i=1
Vk ⊆ X \
k
∪
i=1
Bαi βi and so X \
k
∪
i=1
Vk ∈ H.
(⇐=) It is obvious.
Given a generalized topological space (X, µ), we denote by Hf the hereditary class of finite subsets of X. The following proposition is obvious and thus
the proof are omitted.
Proposition 2.5 For a HGTS (X, µ, H) the following are equivalent:
1. (X, µ) is µ-semi-compact;
2. (X, µ, Hf ) is µHf -semi-compact;
3. (X, µ, {ϕ}) is µ{ϕ}-semi-compact.
Theorem 2.6 A HGTS (X, µ, H) is µH-semi-compact if and only if for
∩
Gα = ϕ, there
any family {Gα }α∈Λ of µ-semi-closed subsets of X such that
exists a finite subset Λ0 of Λ with
∩
α∈Λ0
∩
α∈Λ
α∈Λ
Gα ∈ H.
Proof. Let {Gα }α∈Λ be a family of µ-semi-closed subsets of X such that
Gα = ϕ. Then {X \ Gα }α∈Λ is a µ-semi-open cover of X. By hypothesis,
there exists a finite subset Λ0 ⊂ Λ such that X \
implies that
∩
α∈Λ0
Gα ∈ H.
∪
α∈Λ0
(X\Gα ) ∈ H. This
Conversely, Let {Vα }α∈Λ be any µ-semi-pen cover of X. Then {X \ Vα }α∈Λ is a
∩
family of µ-semi-closed subsets of X such that
X \ Vα = ϕ. By hypothesis,
there exists a finite subset Λ0 ⊂ Λ such that
we obtain that, X \
∪
α∈Λ0
α∈Λ
∩
α∈Λ0
X\Vα ∈ H. Consequently,
Vα ∈ H and therefore (X, µ, H) is µH-semi-compact.
It is clear that the intersection of any two hereditary classes on a non-empty
set X is a heredity class.
57
Some Types of µ-semi-compactness Via hereditary Classes
Corollary 2.7 If (X, µ, H1 ∩H2 ) is µ(H1 ∩H2 )-semi-compact, then (X, µ, H1 )
is µH1 -semi-compact and (X, µ, H2 ) is µH2 -semi-compact.
Definition 2.8 [5] A HGTS (X, µ, H) is said to be µH-compact if for every
µ-open cover {Vα }α∈Λ of X there exists a finite subset Λ0 of Λ such that X \
∪
Vα ∈ H.
α∈Λ0
Proposition 2.9 Every µH-semi-compact space is µH-compact.
Proof. The proof is obvious since every µ-open set is µ-semi-open.
Proposition 2.10 If (X, µ) is µ-submaximal and extremally disconnected,
then µH-compact and µH-semi-compact are equivalent.
Proof. The proof comes immediately form the fact: In the extremally
disconnected µ-submaximal space µ = SOµ (X).
Next we study the behavior of some types of subspaces of a µH-semicompact space relative to X.
Theorem 2.11 If Ai , i = 1, 2 are µH-semi-compact subsets of a HGTS
(X, µ, H), and H is an ideal then A1 ∪ A2 is µH-semi-compact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that
∪
Vα . Then {Vα }α∈Λ is a µ-semi-open cover of Ai for i = 1, 2.
A1 ∪ A2 ⊆
α∈Λ
Since A1 and A2 are µH-semi-compact, then there exists finite subsets Λ0 and
∪
∪
Λ1 of Λ with A1 \
Vα ∈ H and A2 \
Vα ∈ H. This implies that
A1 \
∪
α∈Λ0 ∪Λ1
that
(
A1 \
α∈Λ0
Vα ∈ H and A2 \
∪
α∈Λ0 ∪Λ1
)
(
Vα ∪ A2 \
α∈Λ1
∪
α∈Λ0 ∪Λ1
Vα ∈ H and since H is an ideal we have
)
∪
α∈Λ0 ∪Λ1
Vα
∈ H. Thus A1 ∪ A2 \
So A1 ∪ A2 is µH-semi-compact.
∪
α∈Λ0 ∪Λ1
Vα ∈ H.
The following example shows that the previous theorem does not hold when
H is just a hereditary class, not an ideal.
Example 2.12 Let X = (0, 1), µ the usual topology, H = {A ⊂ X : A ⊂
(0, 21 ) or A ⊂ ( 12 , 1)} and if A = (0, 12 ) and B = ( 12 , 1), then:
1. It is clear that A = (0, 12 ) and B = ( 12 , 1) are µH-semi-compact subsets.
2. A ∪ B is not µH-semi-compact if {( n1 , 1 − n1 ) : n ∈ Z + } is a cover of
µ-semi-open subsets of X where Z + is the set of positive integers, but if
we choose a finite set n1 , ..., nk and we take N = max{n1 , ..., nk }, follows
that A ∪ B \
k (
∪
n=1
1
,
ni
1−
1
ni
)
=A∪B\
(
1
,
N
1−
1
N
)
∈
/ H.
58
Abdo Qahis .
A GTS (X, µ) is said to be µ-semi-Hausdroff [12] for each pair of distinct
points x and y in X, there exist µ-semi-open sets Ux and Vy containing x and
y, respectively, such that U ∩ V = ϕ.
/ Aµ if and only if (V \ H) ∩ A = ϕ, where V ∈ SOµ (X, x)
Lemma 2.13 x ∈
and H ∈ H.
Proof. Let x ∈
/ Aµ . Then there exist Ux ∈ SOµ (X, x) such that A ∩ V =
H ∈ H. It follows that (V \ H) ∩ A = ϕ.
Conversely, suppose that (V \ H) ∩ A = ϕ for some V ∈ SOµ (X, x) and
H ∈ H. Then V ∩ (X \ H) ∩ A = (A ∩ V ) ∩ (X \ H) = ϕ. This implies that
(A ∩ V ) ⊂ H ∈ H. Hence, x ∈
/ Aµ .
Theorem 2.14 Let (X, µ, H) be a HGTS such that (X, µ) is a µ-semiHausdorff. If A is µH-semi-compact subset of X, then A is µ-semi-∗-closed.
Proof. Suppose that A be a µH-semi-compact subset of a µ-semi-Hausdroff
HGTS (X, µ, H). Let x ∈
/ A then x ∈ X \ A. For each y ∈ A, there exist
two µ-semi-open sets Uy and Vy containing x and y, respectively, such that
Uy ∩ Vy = ϕ. Note that x ∈
/ scµ (Vy ). Then {Vy }y∈A is a µ-semi-open cover
of A relative to X. Since A is a µH- semi-compact, there exists a finite sub∪
set Λ0 of A such that A \ Vy ∈ H. Now x ∈
/ scµ (Vy ) for each y ∈ A
∪
implies x ∈
/
H = A \ scµ (
y∈Λ0
∪
y∈Λ0
Λ0
scµ (Vy ) = scµ (
Vy ) ⊂ A \
∪
y∈Λ0
∪
y∈Λ0
Vy ). Let U = X \ scµ (
∪
Vy ) and let
y∈Λ0
Vy = H1 , where H1 ∈ H. Since U ∈ SOµ (X, x)
and H ∈ H, U \ H ∈ µ∗g (H) containing x and (U \ H) ∩ A = ∅. By Lemma
2.13, x ∈
/ Aµ . Hence Aµ ⊂ A, so A is µ-semi-∗-closed.
Let (X, µ, H) be a HGTS and let A ⊆ X, A ̸= ϕ. We denote by HA the
collection {H ∩ A : H ∈ H} and by (A, µA ) the subspace of (X, µ) on A. It is
clear that the collection µA is a generalized topology on A and the collection
HA is a hereditary class of subsets in A. Then we have the following theorem.
Theorem 2.15 Let (X, µ, H) be a µH-semi-compact HGTS and A be a µsemi-closed subset of X then (A, µA , HA ) is semi-compact with respect to the
hereditary class HA .
Proof. Let {Vα }α∈Λ be a µA -semi-open cover of A where Vα = Uα ∩ A such
that Uα ∈ SOµ (X). Then {Uα }α∈Λ ∪ (X \ A) is a µ-semi-covering of X and
hence there exists a finite subset Λ0 of Λ such that X \ [∪{Uα : α ∈ Λ0 } ∪ (X \
A)] ∈ H. There exists H ∈ H with X \ [∪{Uλ : λ ∈ Λ0 } ∪ (X \ A)] = H. Since
A∩H = A∩(X \[∪{Uα : α ∈ Λ0 }∪(X \A)]) = A∩(X \∪{Uα : α ∈ Λ0 })∩A =
A ∩ (X \ ∪{Uα : α ∈ Λ0 }) = A \ ∪{Uα : α ∈ Λ0 } = A \ (A ∩ [∪{Uα : α ∈ Λ0 }]) =
59
Some Types of µ-semi-compactness Via hereditary Classes
A \ ∪{Uα ∩ A : α ∈ Λ0 }. Then, we have A \ ∪{Uα ∩ A : α ∈ Λ0 } = A ∩ H ∈ HA .
This shows that A is a µHA - semi-compact.
The well known result that a µ-semi-closed subspace of a µ-semi-compact
space is µ-semi-compact is a special case by taking H = {ϕ}.
Definition 2.16 Let (X, µ) be a GTS. Then a subset A of X is called a µsemi-generalized closed if scµ (A) ⊆ U whenever A ⊆ U where U is µ-semi-open
in X.
Theorem 2.17 If (X, µ, H) is µH-semi-compact and A ⊂ X is µ-semi
generalized-closed then A is µH-semi-compact.
Proof. Let {Vα }α∈Λ be any cover of A by µ-semi-open sets in X. Since A
is µ-semi generalized-closed, scµ (A) ⊂ ∪Vα . Then {Vα }α∈Λ ∪ (X \ scµ (A)) is
a µ-semi-open cover of X and hence there exists a finite subset Λ0 of Λ such
that [
]
X\
∪
α∈Λ0
Vα ∪ (X \ scµ (A) ∈ H. Now, we have X \ [∪{Vα : α ∈ Λ0 } ∪ (X \
scµ (A))] = [X \ ∪{Vα : α ∈ Λ0 }] ∩ scµ (A) ⊃ [X \ ∪{Vα : α ∈ Λ0 }] ∩ A =
A \ [∪{Vα : α ∈ Λ0 }]. This implies that A \ [∪{Vα : α ∈ Λ0 }] ∈ H, and so A is
µH-semi-compact relative to X.
Now we study the behavior of µH -semi-compactness under certain types
of functions.
Theorem 2.18 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) →
(Y, SOν (Y )) be a (SOµ (X), SOν (Y ))-continuous function. If (X, µ, H) is µHsemi-compact, and if G = {B ⊆ Y : f −1 (B) ∈ H} then:
1. G is a hereditary class on Y .
2. (Y, ν, G) is µH-semi-compact.
Proof. (1) Suppose that A ⊆ B ⊆ Y and B ∈ G . Since f −1 (A) ⊆
f (B) ∈ H, then f −1 (A) ∈ H, and so A ∈ G.
∪
∪ −1
(2) Let {Vα }α∈Λ ⊂ SOν (Y ) such that Y =
Vα . Since X =
f (Vα ) =
−1
(
f −1
∪
)
Vα
α∈Λ
Λ0 of Λ with f
α∈Λ
α∈Λ
and (X, µ, H) is µH-semi-compact, there exists a finite subset
(
−1
Y \
∪
α∈Λ0
)
Vα = X \
∪
α∈Λ0
f −1 (Vα ) ∈ H. Thus Y \
∪
α∈Λ0
Vα ∈ G.
The following lemma is very useful in studying the preservation of µHcompact by certain classes of functions
60
Abdo Qahis .
Lemma 2.19 [5] Let f : (X, µ) → (Y, ν) be a function. If H is a hereditary
class on X, then f (H) = {f (H) : H ∈ H} is a hereditary class on Y .
Theorem 2.20 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) →
(Y, SOν (Y )) be a surjection (SOµ (X), SOν (Y )-continuous function. If (X, µ, H)
is µH-semi-compact, then (Y, ν, f (H)) is νf (H)-semi-compact.
Proof. Let {Vα }α∈Λ ⊂ SOν (Y ) such that Y =
∪
α∈Λ
Vα . Then {f −1 (Vα )}α∈Λ
is a µ-semi-open
( cover of X
) and hence there exists a finite subset Λ0 of Λ
∪
∪
∪
such that f −1 Y \
Vα = X \
f −1 (Vα ) ∈ H. Thus Y \
Vα =
f (f −1 (Y \
∪
α∈Λ0
α∈Λ0
Vα )) ∈ f (H).
α∈Λ0
α∈Λ0
Corollary 2.21 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) →
(Y, SOν (Y )) be a bijective µ-semi-open function. If (Y, ν, G) is νG-semi-compact,
then (X, µ, f −1 (G)) is µf −1 (G)-semi-compact
Proof. Since f : (X, SOµ (X)) → (Y, SOν (Y )) is a bijective µ-semi-open
function, f −1 : (Y, SOν (Y )) → (X, SOµ (X)) is a surjection (SOµ (X), SOν (Y ))continuous function. Since (Y, ν, G) is νG-semi-compact, by Theorem 2.20 we
obtain that (X, µ, f −1 (G)) is µf −1 (G)-semi-compact.
3
S-µH-Semi-Compactness Spaces
Definition 3.1 If (X, µ, H) is a HGTS and A ⊆ X, A is said to be strong
µH-semi-compact (briefly S-µH-semi-compact) relative to X if for every fam∪
ily {Vα }α∈Λ of µ-semi-open subsets of X with A \
Vα ∈ H there exists a
finite subset Λ0 of Λ, such that A ⊆
∪
α∈Λ
α∈Λ0
Vα . The HGTS (X, µ, H) is said to
be S-µH-semi-compact if X is S-µH-semi-compact.
Clearly, the following diagram follows immediately from the definitions and
facts.
↗
S − µH − semi − compact
↘
µH − semi − compact
↑
µ − semi − compact
Remark 3.2 We note that if (X, µ, H) is a HGTS and (X, µ∗ , H) is a SµH- semi-compact, then (X, µ, H) is S-µH-semi-compact, and that (X, µ, H) is
S-µH-semi-compact if and only if for any family {Gα }α∈Λ of µ-semi-closed sub∩
∩
sets of X, if
Gα ∈ H then there exists Λ0 ⊆ Λ, finite, such that
Gα = ϕ.
α∈Λ
α∈Λ0
61
Some Types of µ-semi-compactness Via hereditary Classes
Remark 3.3
1. It is clear that the GT (X, µ) is µ-semi-compact if and
only if (X, µ, {ϕ}) is S-µ{ϕ}-semi-compact.
2. If (X, µ, H) is a S-µH-semi-compact then (X, µ, H) is µH-semi-compact,
and (X, µ) is µ-semi-compact.
Definition 3.4 A subset A of a HGTS (X, µ, H) is said to be SOµ Hg -closed
if for every U ∈ SOµ with A \ U ∈ H then scµ (A) ⊆ U .
Theorem 3.5 Every SOµ Hg -closed subset of a S-µH-semi-compact space
is S-µH-semi-compact.
Proof. Let A be any SOµ Hg -closed subset of (X, µ, H) and {Vα }α∈Λ be
∪
a family of µ-semi-open subsets of X such that A \
Vα ∈ H. Since A
is SOµ Hg -closed, scµ (A) ⊆
∪
α∈Λ
α∈Λ
Vα . Then (X \ scµ (A)) ∪ (
[
semi-open cover of X and so X \ X \ cµ (A) ∪ (
]
∪
α∈Λ
∪
Vα ) is a µ-
α∈Λ
Vα ) = ϕ ∈ H. Given
that X is S-µH-semi-compact there exists s finite subset Λ0 of Λ such that
∪
∪
X = (X \ scµ (A) ∪
Vα ). Then A = A ∩ [(X \ scµ (A)) ∪
Vα ] =
A∩
∪
α∈Λ0
Vα ⊆
∪
α∈Λ0
α∈Λ0
Vα .
α∈Λ0
Theorem 3.6 If A and B are S-µH-semi-compact subsets of a HGTS
(X, µ, H), then A ∪ B is S-µH-semi-compact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that
∪
∪
∪
∪
Vα ⊆
Vα and B \
Vα ⊆ A ∪ B \
Vα ∈ H. Since, A \
A∪B\
A∪B \
α∈Λ
∪
α∈Λ
Vα then A \
∪
α∈Λ
subsets Λ0 and Λ1 of Λ such that A ⊆
that A ⊆
∪
α∈Λ0 ∪Λ1
∪
α∈Λ
Vα ∈ H and B \
Vα and B ⊆
A ∪ B is S-µH-semi-compact.
∪
α∈Λ0 ∪Λ1
∪
α∈Λ
α∈Λ0
α∈Λ
α∈Λ
Vα ∈ H and so there exist finite
Vα and B ⊆
∪
Vα . This implies
α∈Λ1
Vα and so A ∪ B ⊆
∪
α∈Λ0 ∪Λ1
Vα . Hence
Theorem 3.7 Let (X, µ, H) be a HGTS, A ⊆ B ⊆ X and B ⊆ scµ (A). If
A is µ-semi-generalized closed and S-µH-semi-compact, then B is S-µH-semicompact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that
∪
∪
B\
Vα ∈ H. Since, A \
Vα ∈ H and A is S-µH-semi-compact, there
α∈Λ
α∈Λ
exists a finite subset Λ0 of Λ such that A ⊆
generalized closed, scµ (A) ⊆
∪
α∈Λ0
∪
α∈Λ0
Vα . Since A is µ-semi-
Vα and this implies B ⊆
∪
Vα .
α∈Λ0
Now we study the behavior of S-µH-semi-compactness under certain types
of functions.
62
Abdo Qahis .
Theorem 3.8 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) →
(Y, SOν (Y )) be a (SOµ (X), SOν (Y ))-continuous surjective function. If (X, µ, H)
is S- µH-semi-compact and G = {B ⊆ Y : f −1 (B) ∈ H} then (Y, ν, G) is SµH-semi-compact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open
subsets
of Y such that
(
)
Y\
∪
α∈Λ
Vα ∈ G. Since X \
∪
α∈Λ
f −1 (Vα ) = f −1 Y \
∪
α∈Λ
Vα ∈ H and (X, µ, H)
is S-µH-semi-compact, there exists a finite subset Λ0 of Λ, such that X =
∪
∪
f −1 (Vα ). Given that f is surjective we have Y =
Vα .
α∈Λ0
α∈Λ0
Theorem 3.9 Let (X, µ) and (Y, ν) be GTSs and let f : (X, SOµ (X)) →
(Y, SOν (Y )) be a bijective (SOµ (X), SOν (Y ))-continuous function. If (X, µ, H)
is S-µH-semi-compact then (Y, ν, f (H)) is S-νf (H)-semi-compact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of Y such that
∪
∪
Vα = f (H). Then
Vα ∈ f (H). There exists H ∈ H with Y \
Y \
α∈Λ
H = f −1 (f (H)) = X \
∪
α∈Λ
α∈Λ
f −1 (Vα ) ∈ H. Given that (X, µ, H) is S-µH-semi-
compact, there exists a finite subset Λ0 of Λ such that X =
f is surjective, Y =
∪
∪
f −1 (Vα ). Since
α∈Λ0
Vα .
α∈Λ0
Corollary 3.10 If f : (X, µ) → (Y, ν) is a bijective and µ-pre-semi-open
function and (Y, ν, G) is S-νG-semi-compact, then (X, µ, f −1 (G)) is S-µf −1 (G)semi-compact.
Proof. Let {Vα }α∈Λ be a family of µ-semi-open subsets of X such that
∪
∪
Vα = f −1 (G). Then
Vα ∈ f −1 (G). There exists G ∈ G with X \
X\
Y \
α∈Λ
∪
α∈Λ
α∈Λ
f (Vα ) = f (f −1 (G)), and given that (Y, ν, G) is S-νG-semi- compact
then there exists a finite subset Λ0 of Λ with Y =
that X =
∪
∪
f (Vα ). This implies
α∈Λ0
Vα .
α∈Λ0
ACKNOWLEDGEMENTS. The author would like to thank the referees for useful comments and suggestions.
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