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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
A note on Co-compactness and Co-paracompactness via Grills
Karthika.A and I.Arockiarani
Nirmala College for Women
Coimbatore
________________________________________________________________
Abstract
In this paper we introduce and investigate a new type of compactness namely
Cocompactness in grill topological space or simply G-Cocompactness and GCoparacompactness of a grill topological space (X, τ, G). Here we establish a strong and
equivalent property for Cocompactness using grills. Also the relationships among the
various forms of compact spaces are obtained.
Keywords and phrases: G -Cocompact, G -paracompact, G -Cocover.
2000 AMS Subject Classification: 54C30, 54C99.
________________________________________________________________
1. Introduction
The history of Grill was initiated by Choquet [3] in 1947. The idea of grills was found to
be very useful device like nets and filters. Also for the investigations of many topological
notions like compactifications, proximity spaces, theory of grill topology was used.
In 1968, M.K. Singal and Asha Rani [15] have introduced one of the most interesting
generalized forms of compactness namely almost compactness. Mashhour A.S. and Atis R.H.
introduced
a
new
version
of
compactness
namely
cocompactness
and
coparacompactness[2].The notion of paracompactness in ideals was initiated by Ham- lett et
al[6] in the year 1997. B.Roy and M.N. Mukherjee[12] extended the concept of para
compactness in terms of grills. Following their work we formulate the new definition of
G -Cocompactness and G-Coparacompactness via grills. Also we try to achieve a equivalent
property for cocompact , paracompact spaces.
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
Preliminaries
Definition 1.1[10]
A colletion G of nonempty subsets of a set X is called a grill if
1. A ∈ G and A ⊆ B ⊆ X implies that B ⊆ G, and
2. A ∪ B ∈ G (A, B ⊆ X) implies that A ∈ G or B∈ G.
Definition 1.2[12]
Let G be a grill on a topological space (X, τ). A cover {Uα : α ∈ Λ} of X is said to be a
G-cover if there exists a finite subset Λ0 of Λ such that X \ ∪α∈Λ0 Uα ∉ G.
Definition 1.3[13]
ED-space is a regular space in which every open set is Co-open.
Definition 1.4[13]
A subset A of a topological space (X, τ ) is Co-subset if clA is open.
Definition 1.5[13]
A subset A of a topological space (X, τ ) is Ic-subset if intA is closed.
Definition 1.6[13]
A cover η of a topological space X is called Co-open cover if each member of η is Co-open
set.
Definition 1.7[12]
Let G be a grill on a topological space (X, τ). Then the space X is said to be paracompact
with respect to the grill G or simply G-paracompact if every open cover U of X has a
precise locally finite open refinement U ∗(not necessarily a cover of X) such that
X \ ∪ U ∗ ∉G where the statement “a cover U = {Uα : α ∈ Λ} has a precise refinement “”
means as usual, that there exists a collection V = {Vα : α ∈ Λ} of subsets of X such that
Vα ⊆ Uα, for all α ∈ Λ.(Here refinement need not be a cover.)
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
Definition 1.8[14]
A topological space X is said to be lightly compact or Feebly compact if each collection of
locally finite nonempty open sets is finite.
2. Co-compactness on G-space
We formulate the new definition of G-Cocompactness via grills. Also we try to achieve
a equivalent property for cocompact , paracompact spaces.
Definition 2.1
Let G be a grill on a topological space (X, τ). A cover { Uα : α ∈ Λ } of X, whose
elements are co-open sets is said to be a G-co compact if there exists a finite subset Λ0 of Λ
such that X \ ∪α∈Λ0 Uα ∉ G.
Definition 2.2
A grill topological space (X, τ, G) is almost G-Cocompact if each co- open cover of
X has a finite subfamily, such that X \ ∪α∈Λ0 clUα ∉ G.
Remark 2.3
Let X be a topological space with the grill G, and A ⊆ X . Then
(i) X is G -compact ⇒ X is G -Cocompact ⇒ X is almost G -Cocompact.
(ii) X is almost G -compact ⇒ X is almost G -Cocompact.
Theorem 2.4
A topological space with the grill G is G -Cocompact if and only if each family of
Ic-closed sets which has a G -f.i.p has the arbitrary intersection belongs to the Grill. That is
∩α∈ΛAα ∈ G.
Proof
If η = {Aα : α ∈ Λ} is a family of Ic-closed subsets of a topological space (X, τ, G)
with G.f.i.p and the arbitrary intersection is also belongs to grill.
Then
∩ α ∈ ΛAc ∈ G ⇒ X \ ∪α ∈ Λ A ∉ G ⇒ If X is not G -Cocompact, η has a
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
finite sub collection such that X \ ∪i=1n Ai∈ G ⇒ ∩i=1n Ai ∉ G, where A (complement set
of Co-open set) Ic-closed sets , which is a contradiction. So, for every cover of co-open sets
there exists a subcollection such that X \ ∪ i=1 nA ∉ G. Hence X is G –Cocompact.
Conversely let space X be G -Cocompact if each family η of co-open sets, such that
no finite subfamily of η is a cover of X, fails to be a G -Cocover of X. This is true if and
only if each family of Ic-closed sets which satisfies ∩i=1n Ai ∈ G ⇒ ∩α∈ΛAi ∈ G. Hence the
proof.
Theorem 2.5
For a grill topological space (X, τ, G), the following statements are equivalent.
(i) (X, τ, G) is an almost G -Cocompact space.
(ii) Each cover of X whose members are both open and closed has a finite subcllection such
that X \ ∪ i=1 nA ∉ G.
(iii) For each family J of Co-open subsets of X having the G.f.i.p., ∩{F c : F ∈ J } ∈ G.
Proof
(i) ⇒(ii): Since an open and closed set is Co-open, we can consider the cover as a
Cocover. By (i) X \ ∪α∈Λ0 clUα ∉ G. But the sets are both open and closed ⇒ clUα =
Uα . Therefore X \ ∪α∈Λ0 Uα ∉ G. Hence the proof.
(ii) ⇒(iii): Let J be a family of Co-open subset of X having the G.f.i.p. with the intersection
belongs to the grill......(1) Suppose that ∩α∈ΛF ∉ G ⇒
X \ ∪α∈Λ F ∉ G ⇒ {Fα : α ∈ Λ} is a G -Cocover of X in which every member is both
open and closed. So, there exists a finite subfamily such that
X \ {∪ i=1n Fi} ∉ G ⇒ ∩ Fi ∉ G, which is a contradiction to the fact (1).Hence the result.
(iii) ⇒(i): If X is not an almost G -Cocompact space, then there is a G -Cocover η of X
which has no finite subfamily, such that X \ ∪ i=1n Ui ∉ G,⇒ we have ∩i=1 n Uic ∉ G. ⇒
It is a contradiction to the fact that the collection {Uα : α ∈ λ} has G.f.i.p. Hence X is a
almost G -Cocompact space.
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
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ISSN: 2250-1797
Theorem 2.6
An Ic-closed subset of an G -Cocompact space is G -Cocompact.
Proof
Let A be an Ic-closed subset of the G -Cocompact space (X, τ) and η be any τACocover of A. Every member U ∈ η is of the form U = A ∩ V where V is τ -Co- open. Then
let W = {Uα : α ∈ Λ} ∪ (X \ A) where X \A is also Co-open and hence it is a G -Cocover of
X. Since X is G -Cocompact, W has a finite subfamily such that X \ ∪i=1n Ui ∪ (X \ A) ∉ G.
W is a G -Cocover. It also should cover A ⊆ X .
But A ⊈ X \ A ⇒ A ⊆ ∪i=1n Ui ⇒ A\ ∪i=1n Ui ∉ G. ⇒ A is G -Cocompact.
Corollary 2.7
An Ic-closed subset of an almost G -Cocompact space is also almost G -Cocompact.
Theorem 2.8
Let f : (X, τ1, G 1) → (Y, τ 2, G 2) be a homeomorphism from X to Y, where X is
almost G -Cocompact. Then the image f(X), with the properties G 2 ⊆f(G1), f(G1) is a σ-grill,
is almost G 2-Cocompact.
Proof
Let f : (X, τ1, G 1) → (Y, τ 2, G 2) be a Co-continuous map from the almost
G 1-
Cocompact space X on to a grill topological space Y and let η = {Uα : α ∈ Λ} be a cover of
Y whose members are both open and closed(by theorem 3.9). {f −1(Uα ) : U ∈ η} is a cover of
X, whose members are both open and closed. Then there is a finite subfamily {f −1(Ui) : i ∈
I } such that X \ ∪i=1n f −1(Ui) ∉ G
given that G
2
1
⇒ Applying f we get f (X )\ ∪i=1n Ui ∉ f(G1).It is
⊆ f(G 1). That is f (X )\ ∪i=1n Ui ∉ G2 . Hence by theorem 3.9, f(X) is
almost G 2-Cocompact.
Theorem 2.9
An almost G -Cocompact space which is semi-regular is G -Cocompact.
Proof
Let (X, τ, G) be a semi-regular grill topological space. Then every open set is regular
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
Available online on http://www.rspublication.com/ijca/ijca_index.htm
ISSN: 2250-1797
open. Suppose there is a collection {Uα : α ∈ Λ} of Co-open sets which covers X. A Coopen set which is regular open is both open and closed. The space is almost-G Cocompact.
So, There exists a subcollection such that
X \ ∪i=1n cl(Ui) ∉ G ⇒ Since the Co-open sets are both open and closed cl(Uα) = Uα
for every α ⇒ X \ ∪i=1n Ui ∉ G. Hence the space is G -Cocompact.
Theorem 2.10
A G -Cocompact, ED-space is G -Compact.
Proof
Let (X, τ, G) be a G -Cocompact space, and let η be any open cover of X. Since X is
an ED-space, η is also a Cocover of X. Therefore X is G -Compact.
Definition 2.11
A topological space (X, τ, G) is G -Co-paracompact if each Cocover U = {Uα : α ∈
Λ} of X has an open locally finite refinement U * such that ,
X \ ∪α∈Λ U * ∉ G where U * = {Vα : α ∈ Λ} and for every Uα ∈ U we can find a Vα
∈ V such that Vα ⊆ Uα. Here refinement need not be a cover.
Definition 2.12
A topological space (X, τ, G) is almost G -Co-paracompact if for each Cocover U of
X, there exists a locally finite family U* of Co-open sets refines U such that, X \ ∪α∈ U∗
cl(Uα) ∉ G.
Remarks 2.13
Let (X, τ, G) be a topological space. Then,
(i) X is almost G -Cocompact ⇒ X is almost G -Co-paracompact.
(ii)X is G -Cocompact ⇒ X is G -Co-paracompact ⇒ X is almost G -Co-paracompact.
(iii)X is G -paracompact ⇒ X is G -Co-paracompact.
(iv)X is almost G -paracompact ⇒ X is almost G -Co-paracompact.
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International Journal of Computer Application
Issue 3, Volume 5 (September - October - 2013)
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ISSN: 2250-1797
Theorem 2.15
An Ic-closed subset of a G -Co-paracompact space is G -Co-paracompact.
Proof
Let A be an Ic-closed subset of the G -Co-paracompact space (X, τ),. Let µ be the
family defined by µ = {U=A ∩ V : where V is τ -Co-open } and denote by W, the family of
all members of µ and (X \ A). Since A is Ic-closed, (X \ A) is Co-open set. Then W is a τ Co-open cover of X. The family W has a τ -Co-open locally finite refinement K. Denoting
by γ, the members of K which refines µ. In this case C = {A ∩ G : G ∈ γ } refines U and C
is τA-locally finite. But this collection C should be a Cocover A. That is A\ ∪ {A ∩ G : G ∈
γ } ∉ G.⇒ A\ ∪ {C : C ∈ C } ∉ G. Thus A is G -Co-paracompact.
Theorem 2.16
A G -Co-paracompact space, which is lightly compact is G -Cocompact.
Proof
Let X be G -Co-paracompact space, and let γ be any Co-open cover of X ⇒ γ has an
Co-open locally finite refinement γ∗={Aα : α ∈ Λ} such that X \ ∪α∈Λ Aα ∉ G. Since X is
lightly compact every locally finite collection is finite. That is X \ ∪i=1n Ai ∉ G. Hence γ is a
G -Cocover. Finally X is G -Cocompact.
Theorem 2.17
An almost G -Co-paracompact space, which is lightly compact is almost G Cocompact.
Proof
Let X be almost G -Co-paracompact, and let η be any Cocover of X. Then there
exists a locally finite family J of Co-open sets which refines η such that
X \ ∪α∈J cl(Aα) ∉ G . Since X is lightly compact, J is finite. So, There exists a family
such that X \ ∪i=1n cl(Ui) ∉ G. Hence η is a G - Cocover and so X is almost G -Coparacompact.
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Theorem 2.18
An almost G -Co-paracompact space which is semi regular is G -Co- paracompact.
Proof
Let X is a semi regular space. Then every open set is regular open. Since a Co-open
set which is regular open is both open and closed, every Cocover consists of sets which are
both open and closed. If X is an almost G -Co-paracompact and V is a Cocover of X, then
there exists a locally finite refinement V ∗ such that X \ ∪A∈V ∗ cl(A) ∉ G. But clA = A, ⇒
X is G -Co-paracompact.
Theorem 2.19
A G -Co-paracompact, E.D space is G -paracompact.
Proof
Let X be a G -Co-paracompact , E.D space. Let η be any open cover of X. Then η is
a Cocover of X. So, η has an open locally finite refinement such that
X \ ∪α∈Λ (Aα) ∉ G. But X is E.D space. Hence X is simply G -paracompact.
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