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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. R S. Publication, [email protected] Page 12 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.) R S. Publication, [email protected] Page 13 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 R S. Publication, [email protected] Page 14 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. R S. Publication, [email protected] Page 15 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 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 R S. Publication, [email protected] Page 16 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. R S. Publication, [email protected] Page 17 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 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. R S. Publication, [email protected] Page 18 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 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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