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Mathematica
Balkanica
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New Series Vol. 23, 2009, Fasc. 1-2
co-γ-Compact Generalized Topologies and
c -Generalized Continunous Functions
S. Sagiroglu and A. Kanibir
Presented by P. Kenderov
This paper introduces the concepts of quasi co-γ-compact generalized topologies and
a larger collection named co-γ-compact generalized topologies. We show that on a topological
space (X, τ ) the co-γ-compact generalized topology for γ = intτ coincide with the cocompact
topology introduced by Gauld [6]. Also a new class of generalized continuous functions related
to co-γ-compact generalized topologies are defined and investigated.
Key Words: generalized topology, generalized continuity, γ-compactness, co-γ-compact
generalized topologies, cocompact topologies.
AMS Subj.Classification: 54A05, 54C05, 54A10
1. Introduction
Á. Császár introduced the notions of generalized topologies and
generalized continuous functions in [3] and also introduced the notions of γcompactness in [2]. The purpose of this paper is to consider the cocompact
topology which was defined by Gauld [6, 7] and implicitly by Gentry and Hoyle
[8] and to obtain a generalization of this concept on generalized topological
spaces by using γ-compactness.
In Section 3, we define the quasi co-γ-compact generalized topologies
and investigate some of their basic properties. In Section 4, on a topological space (X, τ ) we compare the quasi co-γ-compact generalized topology for
γ = intτ with cocompact topology. In Section 5, we define a larger collection
named co-γ-compact generalized topology of gγ on X which coincides with the
cocompact topology of τ on X for γ = intτ . We investigate its properties by
the help of quasi co-γ-compact generalized topologies. Also several of Gauld’s
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results are generalized. In Section 6, a new class of generalized continuous functions are defined and the connections between this new class and generalized
continuous functions are investigated.
2. Preliminaries
First, we recall the following concepts and notations defined in [1]. Let
X be a set and γ : exp X → exp X be a mapping from the power set exp X
of the underlying set X into itself, possessing the property of monotony (i.e.
such that A ⊂ B implies γA ⊂ γB). The collection of all mappings having this
property is denoted by Γ (X) (briefly Γ). For elements of Γ, further conditions
are denoted by Γn (n is an integer or one of the symbols + and −). We shall
use the conditions for n = 0, 1, 3 that are:
γ ∈ Γ0 (X) ⇐⇒ γ∅ = ∅
γ ∈ Γ1 (X) ⇐⇒ γX = X
and let (X, τ ) be a topological space, then
γ ∈ Γ3 (X) iff for A ⊂ X and any open set G ⊂ X, G ∩ γA ⊂ γ (G ∩ A) .
γ∈
Note that, for the sake of simplicity, we can write γ ∈ Γ013 instead of
Γn , ∆ = {0, 1, 3}. For γ ∈ Γ (X) , a subset A ⊂ X is said to be γ-open
n∈∆
iff A ⊂ γA.
In [3], a subfamily g of exp X is said to be a generalized topology (briefly
GT) on X iff ∅ ∈ g and any union of its elements belongs to g. The elements of
the GT g are called g-open and the pair (X, g) is called a generalized topological
space (briefly GTS). If X ∈ g, then g is said to be a strongly generalized topology
[5] on X. Clearly, if γ ∈ Γ then ∅ is γ-open and any union of γ-open sets is
γ-open [1]. Therefore the collection gγ of all γ-open sets of X for γ ∈ Γ (X) is a
GT. Let X and Y be two sets, g1 and g2 be two GT on X and Y, respectively.
Then a function f : (X, g1 ) −→ (Y, g2 ) is said to be generalized continuous [3] iff
V ∈ g2 implies f −1 (V ) ∈ g1 and f is said to be generalized open [11] iff U ∈ g1
implies f (U ) ∈ g2 .
A space X is said to be γ-compact [2] iff γ ∈ Γ013 and X =
Aı , Aı ⊂
ı∈I
γAı imply the existence of a finite set F ⊂ I such that X =
Aı . Also a
ı∈F
subset of a space is said to be γ-compact iff it has this property equipped with
the subspace topology. On the other hand, let X be a set and C ⊂ X, the
mapping γC : exp C −→ exp C, γC A = γA ∩ C for A ⊂ C was defined in [1]
to introduce γC -open sets in the subspace C. So a subset A ⊂ C is said to be
co-γ-Compact Generalized Topologies ...
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γC -open iff A ⊂ γC A. Also, in [4], it was shown that A ⊂ C is γC -open iff it is γopen. Clearly C ⊂ X is said to be γC -compact iff γC ∈ Γ013 (C) and C =
Aı ,
ı∈I
Aı ⊂ γC Aı imply the existence of a finite set F ⊂ I such that C =
Aı .
ı∈F
The notion of c-continuous functions between topological spaces
was introduced by Gentry and Hoyle [8]. Let (X, τ1 ) and (Y, τ2 ) be topological
spaces. A function f : (X, τ1 ) −→ (Y, τ2 ) is defined to be c-continuous if for each
point x ∈ X and each open set V in Y containing f (x) and having compact
complement, there exists an open set U in X containing x such that f (U ) ⊂ V.
For a topological space (X, τ ) , the cocompact topology of τ on X is denoted by c (τ ) and defined by
c (τ ) = {∅} ∪ {A ∈ τ : X − A is compact in (X, τ )}
considered by Gauld [6]. It was shown that c (τ ) is a compact topology on X
and also (X, τ ) is compact iff c (τ ) = τ.
3. Quasi co-γ-compact generalized topologies
Let τ be a topology on X and γ ∈ Γ013 . First, we consider the collection
ϕ = {∅} ∪ {A ∈ gγ : X − A is γX−A -compact}
which is a subfamily of gγ . Then we show that ϕ need not be a generalized
topology on X by the following example.
E x a m p l e 3.1. Let X = {a, b, c} , τ = {X, ∅, {a} , {b} , {a, b}} and
γ : exp X −→ exp X, γ = ci with the operations i = intτ and c = clτ . Then
γ ∈ Γ013 by Theorem 1.11 and Proposition 2.1 of [1] since i and c ∈ Γ013 . It is
clear that γ∅ = ∅, γX = X, γ {a} = γ {a, c} = {a, c} , γ {b} = γ {b, c} = {b, c} ,
γ {c} = ∅, γ {a, b} = X and gγ = {X, ∅, {a} , {b} , {a, b} , {a, c} , {b, c}}. Thus
ϕ = {X, ∅, {a} , {b} , {a, c} , {b, c}} . For example, take {a} ∈ gγ then X − {a} =
{b, c} ∈ gγ and hence γ{b,c} ∈ Γ013 ({b, c}). Thus {b, c} is γ{b,c} -compact since
{b, c} is finite. So {a} ∈ ϕ. Similarly it can be shown for other sets. On the
other hand, {a, b} ∈
/ ϕ since X − {a, b} = {c} and γ{c} ∈
/ Γ1 ({c}) . Thus ϕ is not
a generalized topology on X, even if X is γ-compact since X is finite, therefore
(X, τ ) is compact so c (τ ) = τ . Also X is semi-regular.
Definition 3.2. [4] Let X be a set, γ ∈ Γ (X) and C ⊂ X, then C is
said to be γ-conservative iff A ∩ C is γ-open whenever A is γ-open.
For γ ∈ Γ (X) , we will denote the collection of all γ-conservative subsets
of X by Cγ . It is clear that if γ ∈ Γ1 , then Cγ ⊂ gγ and if γ ∈ Γ3 , then τ ⊂ Cγ
by Theorem 2.16 of [1]. Therefore if γ ∈ Γ13 , then τ ⊂ Cγ ⊂ gγ .
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Now we introduce the following subfamily of gγ which will be needed in
the sequel.
Let τ be a topology on X and γ ∈ Γ013 . Define the collection
qc (gγ ) = {∅} ∪ {A ∈ gγ : X − A ⊂ C such that C ∈ Cγ is γC -compact}
with qc (gγ ) ⊂ gγ .
Proposition 3.3.
Let (X, τ ) be a topological space and γ ∈ Γ013 .
Then qc (gγ ) is a strongly generalized topology on X.
P r o o f. It is clear that ∅ and X ∈ qc (gγ ) . Let A =
Aı such that
ı∈I
Aı ∈ qc (gγ ) for each ı ∈ I = ∅. Then A ∈ gγ and X −A =
(X − Aı ) ⊂ X −Aı
ı∈I
for each ı ∈ I. Since Aı ∈ qc (gγ ) , there exists a set Cı ∈ Cγ such that X−Aı ⊂ Cı
and Cı is γCı -compact . Hence A ∈ qc (gγ ) .
Definition 3.4. Let (X, τ ) be a topological space and γ ∈ Γ013 . The
collection qc (gγ ) is called the quasi co-γ-compact GT of gγ on X.
By the following example, we show that qc (gγ ) need not be a topology.
Also qc (gγ ) ϕ, even if it seems like the conditions given for construction of
qc (gγ ) is more complicated than for ϕ.
E x a m p l e 3.5.
Consider Example 3.1. It is easy to see that
qc (gγ ) = {X, ∅, {a} , {b} , {a, b} , {a, c} , {b, c}} . For example, take {a, b} ∈ gγ ,
then X − {a, b} = {c} ⊂ X ∈ Cγ . Thus {a, b} ∈ qc (gγ ) since X is γ-compact.
Similarly it can be shown for other sets. Hence qc (gγ ) need not be a topology
on X.
Let X be a set and g be a strongly GT on X. As an analogue of the
concept of compactness, let us say that (X, g) is a compact GTS iff
Aı , (Aı )ı∈I ⊂ g
X=
ı∈I
imply the existence of a finite set F ⊂ I such that
Aı .
X=
ı∈F
Then we can obtain that (X, qc (gγ )) is a compact GTS by the
following Theorem, as expected.
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Theorem 3.6. Let (X, τ ) be a topological space and γ ∈ Γ013 .
Then (X, qc (gγ )) is a compact GTS.
Aı , (Aı )ı∈I ⊂ qc (gγ ) . Then Aı ∈ gγ and there
P r o o f. Let X =
ı∈I
exists a γ-conservative subset Cı ⊂ X such that X − Aı ⊂ Cı and Cı is γCı compact for each ı ∈ I. Choose ı0 ∈ I, then X − Aı0 ⊂ Cı0 ⊂ X implies
that Cı0 =
(Aı ∩ Cı0 ) . Aı ∩ Cı0 ∈ gγ since Aı ∈ gγ and Cı0 ∈ Cγ . Thus
ı∈I
Aı ∩ Cı0 is γCı0 -open for each ı ∈ I. Then there is a finite set F ⊂ I such
(Aı ∩ Cı0 ) since Cı0 is γCı0 -compact. So X =
Aı . Hence
that Cı0 =
ı∈F
ı∈{ı0 }∪F
(X, qc (gγ )) is a compact GTS.
Proposition 3.7.
Let (X, τ ) be a topological space and γ ∈ Γ013 .
Then X is γ-compact iff (X, gγ ) is a compact GTS.
P r o o f. It is clear from the definitions of γ-compactness and compact
generalized topological spaces.
Theorem 3.8. Let (X, τ ) be a topological space and γ ∈ Γ013 . Then
(X, gγ ) is a compact GTS (X is γ-compact) iff qc (gγ ) = gγ .
P r o o f. It is clear that qc (gγ ) ⊂ gγ . Now let A ∈ gγ then X − A ⊂ X ∈
Cγ . Hence A ∈ qc (gγ ) since X is γ-compact. Conversely, if qc (gγ ) = gγ , then
(X, gγ ) is a compact GTS by Theorem 3.6.
In [2] it was given that a γ-compact topological space (X, τ ) is also a
compact topological space. Then we have the following result by Theorem 3.8.
Corollary 3.9. Let (X, τ ) be a topological space and γ ∈ Γ013 . If
gγ = qc (gγ ) , then (X, τ ) is a compact topological space.
The next example shows that the reverse implication of Corollary 3.9 is
not true.
E x a m p l e 3.10. Let K be the set {1/n : n ∈ N} ∪ {0} equipped with
the subspace topology τ inherited from the usual (Euclidean) topology of R. It
is clear that (K, τ ) is a compact subspace of R. Define γ = ci by the operations
i = intτ and c = clτ . Then A = {1} ∈ gγ . Now we consider the γ-open cover
K = ({0} ∪ {1/2n : n ∈ N}) ∪∞
n=0 {1/2n + 1}
given in Example 3.1 of [2] for K ∈ Cγ . Similarly we obtain
K − A = ({0} ∪ {1/2n : n ∈ N}) ∪∞
n=1 {1/2n + 1}
a γK−A -open cover for the subset K − A = {1/n : n = 2, 3, ...} ∪ {0} ∈ Cγ . It
is clear that none of them admit any finite subcover. So A ∈
/ qc (gγ ). Hence
qc (gγ ) = gγ .
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4. Comparison of c (τ ) and qc (gγ ) for γ = i
Let (X, τ ) be a topological space, i = intτ and C ⊂ X. Recall that
iC : exp C −→ exp C, iC O = iO ∩ C for each O ⊂ C. Also consider Cγ for γ = i.
Since γ = i ∈ Γ13 , τ ⊂ Ci ⊂ gi . Therefore gi = τ implies gi = Ci = τ.
Lemma 4.1. Let (X, τ ) be a topological space, γ ∈ Γ013 , O ⊂ C ⊂ X
and consider the subspace topolgy τC on C. The following statements are valid:
(a) If C ∈ Cγ and O is open in (C, τC ), then O is γC -open.
(b) If C ∈ τ, then O is open in (C, τC ) iff O is iC -open.
P r o o f. (a) If O is open in (C, τC ), then there is a subset U ∈ τ such
that O = U ∩ C. Thus U ∈ gγ since γ ∈ Γ13 . Therefore C ∈ Cγ implies O ∈ gγ .
Then O ⊂ γO ∩ C = γC O. So O is γC -open.
(b) Necessity follows from (a) for γ = i since Ci = τ. Conversely, if O is iC -open,
then O ∈ τ . Hence O ∈ τC since O ⊂ C.
Corollary 4.2. Let (X, τ ) be a topological space, γ ∈ Γ013 and C ⊂ X.
The following statements are valid.
(a) If C ∈ Cγ is γC -compact, then C is compact in (X, τ ) .
(b) C ∈ τ is iC -compact iff C is compact in (X, τ ) .
P r o o f. (a) Let C =
Oı , (Oı )ı∈I ⊂ τC . Then Oı is γC -open for each
ı∈I
ı ∈ I by Lemma 4.1 (a) . Thus there is a finite set F ⊂ I such that C =
Oı
ı∈F
since C is γC -compact. Hence C is compact in (X, τ ) .
(b) (⇒) : Since Ci = τ, this follows from (a) for γ = i.
(⇐) : Let C =
Oı , Oı is iC -open for each ı ∈ I. Then Oı is open in (C, τC )
ı∈I
for each ı ∈ I by Lemma 4.1 (b) . Thus there is a finite set F ⊂ I such that
C=
Oı since C is compact in (X, τ ). Hence C is iC -compact.
ı∈F
Let τ be a topology on X and consider the collection qc (gγ ) for γ = i.
Then we have
qc (gi ) = {∅} ∪ {A ∈ gi : X − A ⊂ C such that C ∈ Ci is iC -compact}
that is
qc (τ ) = {∅} ∪ {A ∈ τ : X − A ⊂ C such that C ∈ τ is compact in (X, τ )}
by gi = Ci = τ and Corollary 4.2 (b) .
Proposition 4.3.
topology on X.
Let (X, τ ) be a topological space. Then qc (τ ) is a
co-γ-Compact Generalized Topologies ...
91
P r o o f. Since qc (τ ) is a strongly generalized topology on X by Proposition 3.3, then ∅, X and any union of its elements belongs to qc (τ ) . Now let
n
Ak such that Ak ∈ qc (τ ) for each k = 1, 2, ..., n. Then Ak ∈ τ and there
A=
k=1
is an open set Ck ⊂ X such that X − Ak ⊂ Ck and Ck ∈ τ is compact in (X, τ )
n
for each k = 1, 2, ..., n. So X − A =
n
(X − Ak ) ⊂ C =
k=1
Ck ∈ τ and C is
k=1
compact in (X, τ ). Thus C is iC -compact by Corollary 4.2 (b) . Hence A ∈ qc (τ )
.
Proposition 4.4.
Let (X, τ ) be a topological space. Then qc (τ ) ⊂
c (τ ) and (X, qc (τ )) is a compact topological space.
P r o o f. Let A ∈ qc (τ ), then A ∈ τ and there is an open set C ⊂ X such
that X − A ⊂ C and C is compact in (X, τ ). Therefore X − A is compact in
(X, τ ) , because X − A is a closed subset of the compact subspace C of (X, τ ) .
Hence A ∈ c (τ ) . Also it is clear that (X, qc (τ )) is a compact topological space
since c (τ ) is a compact topology on X.
The following example shows that c (τ ) does not coincide with qc (τ ) .
E x a m p l e 4.5.
Let X = [0, 1] ∪ [2, 3[ equipped with the subspace
topology τ inherited from the usual (Euclidean) topology of R. Consider the
collections
c (τ ) = {∅} ∪ {A ∈ τ : X − A is compact in (X, τ )}
and
qc (τ ) = {∅} ∪ {A ∈ τ : X − A ⊂ C such that C ∈ τ is compact in (X, τ )} .
If we take A = ]5/2, 3[ ⊂ X, then it is clear that A ∈ c (τ ) but A ∈
/ qc (τ ).
We can obtain equality under certain conditions for the inclusion given
in Proposition 4.4.
Proposition 4.6.
Let (X, τ ) be a topological space. If (X, τ ) is
compact, then qc (τ ) = c (τ ) .
P r o o f. It is clear by Teorem 3.8 for γ = i and c (τ ) = τ .
Proposition 4.7. [9] Let (X, τ ) be a topological space. If (X, c (τ )) is
T2 , then (X, τ ) is compact.
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Proposition 4.8.
Let (X, τ ) be a topological space. If (X, qc (τ )) is
T2 , then qc (τ ) = c (τ ) .
P r o o f. It is clear by Proposition 4.4, 4.7 and 4.6, respectively.
We recall that for a topological space (X, τ ) , the α-open sets with respect
to τ constitute a topology τα finer than τ. It was shown that α-compactness with
respect to τ is the same as compactness with respect to τα , in [10].
Proposition 4.9.
Let (X, τ ) be a topological space. If (X, τ ) is
α-compact, then qc (τα ) = c (τ )α .
P r o o f. gγ = τα for γ = α. Then qc (τα ) = τα by Theorem 3.8. Hence it
is clear that qc (τα ) = c (τ )α since c (τ ) = τ .
5. co-γ-compact generalized topologies
Now we introduce a new collection c (gγ ) larger than qc (gγ ) , coincides
with c (τ ) for γ = i.
Let (X, τ ) be a topological space and γ ∈ Γ013 . Define the collection
c (gγ ) = c (τ ) ∪ {A ∈ gγ − τ : X − A ⊂ C such that C ∈ Cγ is γC -compact}
with c (gγ ) ⊂ gγ .
Proposition 5.1.
Let (X, τ ) be a topological space and γ ∈ Γ013 .
Then c (gγ ) is a strongly GT on X.
Aı such that
P r o o f. It is clear that ∅ and X ∈ c (gγ ) . Let A =
ı∈I
Aı ∈ c (gγ ) for ı ∈ I. Then we have the following posibilities:
(a) If Aı ∈ c (τ ) for each ı ∈ I, then A ∈ c (τ ) ⊂ c (gγ ) since c (τ ) is a topology.
(b) If Aı ∈ c (gγ ) − c (τ ) for each ı ∈ I, then Aı ∈ gγ − τ for each ı ∈ I. Thus
A ∈ τ or A ∈ gγ − τ. If A ∈ τ, choose an ı ∈ I, then Aı ∈ gγ − τ and there
exists a subset Cı ⊂ X such that X − Aı ⊂ Cı and Cı ∈ Cγ is γCı -compact. So
Cı is compact in (X, τ ) by Corollary 4.2 (b) since Cı ∈ Cγ . This implies that
A ∈ c (τ ) since X − A is a closed subset of Cı . Hence A ∈ c (gγ ). If A ∈ gγ − τ,
then A ∈ c (gγ ) since X − A ⊂ X − Aı and there exists a subset Cı ⊂ X such
that X − Aı ⊂ Cı and Cı ∈ Cγ is γCı -compact for each ı ∈ I.
(c) If Aı ∈ c (τ ) for ı ∈ J I and Aı ∈ c (gγ ) − c (τ ) for ı ∈ I − J then Aı ∈ τ
for ı ∈ J I and Aı ∈ gγ − τ for ı ∈ I − J. So A ∈ c (gγ ) by a similar proof
in (b) . Hence c (gγ ) is a strongly GT.
It is clear that c (gγ ) need not be a topology on X by Example 3.1.
co-γ-Compact Generalized Topologies ...
93
Definition 5.2. Let (X, τ ) be a topological space and γ ∈ Γ013 . The
collection c (gγ ) is called the co-γ-compact GT of gγ on X.
the
(a)
(b)
(c)
Proposition 5.3. Let (X, τ ) be a topological space and i = intτ . Then
following statements are valid.
qc (gγ ) ⊂ c (gγ ) , γ ∈ Γ013 .
c (gγ ) = c (τ ) ∪ qc (gγ ) , γ ∈ Γ013 .
qc (τ ) ⊂ c (τ ) = c (τ ) , γ = i.
P r o o f. (a) Let A ∈ qc (gγ ) . Then A ∈ gγ and there exists a subset
C ⊂ X such that X − A ⊂ C and C ∈ Cγ is γC -compact. Clearly, A is an open
subset in (X, τ ) or A ∈ gγ − τ. If A ∈ τ, then X − A is compact in (X, τ ) since
γC -compactness of C implies compactness of C in (X, τ ) by Corollary 4.2 (a).
Thus A ∈ c (τ ) ⊂ c (gγ ) . If A ∈ gγ − τ, then it is clear that A ∈ c (gγ ) .
(b) c (gγ ) ⊂ c (τ ) ∪ qc (gγ ) follows from the definition of c (gγ ) . Also the reverse
inclusion is follows from (a) and c (τ ) ⊂ c (gγ ) .
(c) The inclusion follows from (a) for γ = i. The equality follows from (b) for
γ = i and Proposition 4.4.
It is clear that the reverse inclusions of Proposition 5.3 (a) and (c) need
not be true in general by Example 4.5.
Theorem 5.4. Let (X, τ ) be a topological space and γ ∈ Γ013 . Then
(X, c (gγ )) is a compact GTS.
Aı such that Aı ∈ c (gγ ) for each ı ∈ I. Then we
P r o o f. Let X =
ı∈I
have the following posibilities:
(a) If Aı ∈ c (τ ) for each ı ∈ I, then choose ı0 ∈ I with Aı0 = ∅. Since X − Aı0
is compact in (X, τ ) , there is a finite subset F ⊂ I such that X − Aı0 ⊂
Aı .
ı∈F
Thus (X, c (gγ )) is a compact GTS since X = Aı0 ∪ (X − Aı0 ) .
(b) If Aı ∈ c (τ ) for ı ∈ J
I and Aı ∈ c (gγ ) − c (τ ) for ı ∈ I − J, then
X = Aı0 ∪ (X − Aı0 ) for an ı0 ∈ I − J. Thus (X, c (gγ )) is a compact GTS by a
similar proof in Theorem 3.6.
Theorem 2 of Gauld [6] follows as an immediate corollary of Proposition
5.3 (c) and Theorem 5.4 for γ = i.
Corollary 5.5. (X, c (τ )) is a compact topological space.
Theorem 5.6. Let (X, τ ) be a topological space and γ ∈ Γ013 . Then
(X, gγ ) is a compact GTS (X is γ-compact) iff c (gγ ) = gγ .
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P r o o f. Suppose that (X, gγ ) is a compact GTS. Then qc (gγ ) = gγ by
Theorem 3.8. Thus gγ ⊂ c (gγ ) by Proposition 5.3 (a). Also the reverse inclusion
is trival. Hence c (gγ ) = gγ . Conversely, if c (gγ ) = gγ , then (X, gγ ) is a compact
GTS by Theorem 5.4.
Corollary 3 of Gauld [6] follows as an immediate corollary of Theorem
5.6 for γ = i by Proposition 5.3 (c) .
Corollary 5.7.
compact iff c (τ ) = τ.
Let (X, τ ) be a topological space. Then (X, τ ) is
Proposition 5.8. Let (X, τ ) be a topological space and γ ∈ Γ013 . If
(X, gγ ) is a compact GTS, then c (gγ ) = qc (gγ ) .
P r o o f. It is clear by Theorem 3.8 and Theorem 5.6.
Corollary 5.9. If (X, τ ) is a compact topological space, then c (τ ) =
qc (τ ) = c (τ ).
P r o o f. It is clear by Proposition 5.8 for γ = i and Proposition 4.6,
respectively.
6. c -generalized continuous functions
In this section we relate c (gγ ) to classes of generalized continuous functions. Note that, (X, τ1 ) and (Y, τ2 ) will always denote topological spaces,
γ1 ∈ Γ (X) and γ2 ∈ Γ013 (Y ) .
Definition 6.1. Let f : (X, gγ1 ) −→ (Y, gγ2 ) be a function. Then f
is called c-generalized continuous if for each x ∈ X and V ∈ c (gγ2 ) containing
f (x) there is a gγ1 -open subset U containing x such that f (U ) ⊂ V.
Note that if γ1 = intτ1 and γ2 = intτ2 , then c-generalized continuous
functions coincide with c-continuous functions [8] between topological spaces.
Theorem 6.2. f : (X, gγ1 ) −→ (Y, gγ2 ) is c-generalized continuous iff
f : (X, gγ1 ) −→ (Y, c (gγ2 )) is generalized continuous.
P r o o f. Let V ∈ c (gγ2 ) . Then for each x ∈ f −1 (V ) , f (x) ∈ V ∈
c (gγ2 ) . So there is a gγ1 -open subset U containing x such that f (U ) ⊂ V
since f is c-generalized continuous. This implies that x ∈ U ⊂ f −1 (V ). Then
γ1 U ⊂ γ1 f −1 (V ) since γ1 ∈ Γ (X) . Therefore x ∈ γ1 f −1 (V ) since U is gγ1 -open.
Hence f −1 (V ) ∈ gγ1 . Conversely, it is clear by Definition 6.1.
co-γ-Compact Generalized Topologies ...
95
It was shown that a function f : (X, τ1 ) −→ (Y, τ2 ) is c-continuous iff
f : (X, τ1 ) −→ (Y, c (τ2 )) is continuous, in [9]. So we can say that this follows
as an immediate corollary of Theorem 6.2 for γ1 = intτ1 and γ2 = intτ2 .
Clearly if f : (X, gγ1 ) −→ (Y, gγ2 ) is generalized continuous, then f :
(X, gγ1 ) −→ (Y, gγ2 ) is c-generalized continuous. But we show that the converse
implication is not true by the following example.
E x a m p l e 6.3. Consider Example 3.10, take the idempotent function
id : (K, τ ) −→ (K, gγ ) . It is clear that id is c-generalized continuous. But if
we take A = {0} ∪ {1/2n : n ∈ N} ∈ gγ , (id)−1 (A) = A ∈
/ τ , thus id is not
generalized continuous.
Corollary 6.4. Let (Y, gγ2 ) be a compact GTS. Then f : (X, gγ1 ) −→
(Y, gγ2 ) is c-generalized continuous iff f : (X, gγ1 ) −→ (Y, gγ2 ) is generalized
continuous.
P r o o f. It is clear by Theorem 5.6 and Theorem 6.2.
Theorem 6.5. Let (Z, σ) also be a topological space and γ3 ∈
Γ013 (Z) . If f : (X, gγ1 ) −→ (Y, gγ2 ) is generalized continuous and h :
(Y, gγ2 ) −→ (Z, gγ3 ) is c-generalized continuous, then h ◦ f : (X, gγ1 ) −→
(Z, gγ3 ) is c-generalized continuous.
P r o o f. Let V ∈ c (gγ3 ) . Then h−1 (V ) ∈ gγ2 since h : (Y, gγ2 ) −→
(Z, gγ3 ) is c-generalized continuous by Theorem 6.2. Thus (h ◦ f)−1 (V ) =
f −1 h−1 (V ) is gγ1 -open since f : (X, gγ1 ) −→ (Y, gγ2 ) is generalized continuous.
References
[1] Á . C s á s z á r , Generalized open sets. Acta Math. Hungar., 75 (1-2), 1997,
65-87.
[2] Á . C s á s z á r , γ-compact spaces. Acta Math. Hungar., 87 (1-2), 2000,
99-107.
[3] Á . C s á s z á r , Generalized topology, generalized continuity. Acta Math.
Hungar., 96(4), 2002, 351-357.
[4] Á . C s á s z á r , γ-connected sets. Acta Math. Hungar., 101 (4), 2003,
273-279.
[5] Á . C s á s z á r , Extremally disconnected generalized topologies. Annales
Univ. Sci. Budapest., 47 , 2004, 91-96.
[6] D . B . G a u l d, c-continuous functions and cocompact topologies. Kyungpook Math. J., 18 (2), 1978, 151-157.
96
S. Sagiroglu and A. Kanibir
[7] D . B . G a u l d , Topologies related to notions of near continuity. Kyungpook
Math. J., 21 (2), 1981, 195-204.
[8] K . R . G e n t r y a n d H . B . H o y l e I I I, c-continuous functions. Yokohama Math. Journal, 18, 1970, 71-76.
[9] P . E . L o n g a n d M . D . H e n d r i x, Properties of c-continuous
functions. Yokohama Math. Journal,, 22, 1974, 117-123.
[10] S . N . M a h e s h w a r i a n d S . S . T h a k u r, On α-compact spaces.
Bull. Inst. Math. Acad. Sinica, 13 (4), 1985, 341-347.
[11] W . K . M i n, Some results on generalized topological spaces and generalized systems. Acta Math. Hungar., 108 (1-2), 2005, 171-181.
S. Sagiroglu
Department of Mathematics
Ankara University
06570, Tandogan
Ankara, TURKEY
E-mail: [email protected]
A. Kanibir
Department of Mathematics
Hacettepe University
06532 Beytepe
Ankara, TURKEY
E-mail: [email protected]
Received 16.09.2008