International Journal of Mathematics Research. ISSN 0976-5840 Volume 9, Number 1 (2017), pp. 27-43 © International Research Publication House http://www.irphouse.com pg**- compact spaces Mrs. G. Priscilla Pacifica Assistant Professor, St.Mary’s College, Thoothukudi – 628001, Tamil Nadu, India. Dr. A. Punitha Tharani Associate Professor, St.Mary’s College, Thoothukudi –628001, Tamil Nadu, India. Abstract The concepts of pg**-compact, pg**-countably compact, sequentially pg**- compact, pg**-locally compact and pg**- paracompact are introduced, and several properties are investigated. Also the concept of pg**-compact modulo I and pg**-countably compact modulo I spaces are introduced and the relation between these concepts are discussed. Keywords: pg**-compact, pg**-countably compact, sequentially pg**- compact, pg**-locally compact, pg**- paracompact, pg**-compact modulo I, pg**-countably compact modulo I. 1. Introduction Levine [3] introduced the class of g-closed sets in 1970. Veerakumar [7] introduced g*- closed sets. P M Helen[5] introduced g**-closed sets. A.S.Mashhour, M.E Abd El. Monsef [4] introduced a new class of pre-open sets in 1982. Ideal topological spaces have been first introduced by K.Kuratowski [2] in 1930. In this paper we introduce 28 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani pg**-compact, pg**-countably compact, sequentially pg**-compact, pg**-locally compact, pg**-paracompact, pg**-compact modulo I and pg**-countably compact modulo I spaces and investigate their properties. 2. Preliminaries Throughout this paper(푋, 휏) and (푌, 휎) represent non-empty topological spaces of which no separation axioms are assumed unless otherwise stated. Definition 2.1 A subset 퐴 of a topological space(푋, 휏) is called a pre-open set [4] if 퐴 ⊆ 푖푛푡(푐푙(퐴) and a pre-closed set if 푐푙(푖푛푡(퐴)) ⊆ 퐴. Definition 2.2 A subset 퐴 of topological space (푋, 휏) is called 1. generalized closed set (g-closed) [3] if 푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is open in (푋, 휏). 2. g*-closed set [7] if 푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is g-open in (푋, 휏). 3. g**-closed set [5] if 푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is g*-open in(푋, 휏). 4. pg**- closed set[6] if 푝푐푙(퐴) ⊆ 푈 whenever 퐴 ⊆ 푈 and 푈 is g*-open in(푋, 휏). Definition 2.3 A function 푓: (푋,휏) → (푌,휎) is called 1. pg**-irresolute[6] if 푓−1(푉) is a pg**-closed set of (푋, 휏) for every pg**-closed set 푉 of (푌, 휎). 2. pg**-continuous[6] if 푓−1(푉) is a pg**-closed set of (푋, 휏)for every closed set 푉 of (푌, 휎). 3. pg**-resolute[6] if 푓(푈) is pg**- open in 푌 whenever 푈 is pg**- open in 푋. Definition 2.4 An ideal[2] I on a nonempty set 푋 is a collection of subsets of 푋 which satisfies the following properties. (푖) 퐴 ∈ 퐼 , B ∈ I⟹ 퐴 ∪ 퐵 ∈ 퐼 (푖푖) 퐴 ∈ 퐼, 퐵 ⊂ 퐴 ⟹ 퐵 ∈ 퐼 . A topological space (푋, 휏) with an ideal 퐼 on 푋 is called an ideal topological space and is denoted by (푋, 휏, I ). Definition 2.5 [1] A collection 풞 of subsets of 푋 is said to have finite intersection property if for every finite subcolection {퐶1, 퐶2, … , 퐶푛} of 풞, the intersection 퐶1 ∩ 퐶2 ∩ … ∩ 퐶푛 is nonempty. pg**- compact spaces 29 3. pg**- Compact Space We introduce the following definitions Definition 3.1 Let 푋 be a topological space. A collection{푈훼}훼∈∆ of pg**-open subsets of 푋 is said to be pg**-open cover of 푋 if each point in 푋 belongs to at least one 푈훼 that is, if ∪ 푈훼 = 푋. 훼∈∆ Definition 3.2 The topological space(푋, 휏) is said to be pg**- compact if every pg**- open covering 풜 of 푋 contains a finite subcollection that also covers 푋. A subset 퐴 of 푋 is said to be pg**-compact if every pg**-open covering of 퐴 contains a finite subcollection that also covers퐴. Remark 3.3 A pg**- compact space is compact since every open set is pg**- open but not conversely. Any topological space having only finitely many points is necessarily pg**- compact, since in this case every pg**-open covering of 푋 is finite. Example 3.4 Let 푋 be an infinite set with cofinite topology. Then, 푃퐺∗∗푂(푋) = {휑,푋,퐴 /퐴푐푖푠 푓푖푛푖푡푒}. Let {푈훼}훼∈∆ be an arbitrary pg**-open cover for 푋. Let 푈훼0 be a pg**- open set in the pg**- open cover {푈훼}훼∈∆. Then 푋 − 푈훼0 is finite say { 푥1, 푥2, 푥3, … , 푥푛 }. Choose 푈훼푖 such that 푥푖 ∈ 푈훼푖 for 푖 = 1,2,3,…,푛. Then 푋 = 푈훼0 ∪ 푈훼1 ∪ 푈훼2 ∪ … ∪ 푈훼푛. Hence the space 푋 is pg**- compact and hence compact. Example 3.5 The real line ℝ with usual topology is not pg**- compact since the pg**- open cover 픊 = {(푛,푛+2)/푛 ∈ ℤ} has no finite subcover. Example 3.6 Let 푋 be an infinite indiscrete topological space. Obviously it is compact. But {{푥}푥∈푋} is a pg**- open cover which has no finite sub cover. Hence it is not pg**- compact. Definition 3.7 A topological space (X, τ) is said to be pg**-discrete if every subset of 푋 is pg**-open. Equivalently every subset is pg**-closed. Theorem 3.8 A pg**-discrete space is pg**- compact if and only if the space is finite. 30 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani Proof Let (푋, 휏)be pg**- discrete space. Suppose 푋 is a finite set, then obviously it is pg**- compact. Hence in particular every pg**- discrete finite space is pg**- compact. Conversely suppose 푋 is pg**- compact and assume that 푋 is an infinite set. Then the collection 풜 = {{푥}: 푥 ∈ 푋} is a pg**- open cover of 푋. But 풜 does not contain any finite subcover for 푋. Therefore 푋 is not pg**- compact, contrary to our supposition. Thus 푋 is a finite set. Theorem 3.9 Every pg**- closed subset of a pg**- compact space is pg**- compact but not conversely. Proof Let A be a pg**- closed subset of the pg**- compact space 푋. Given a pg**- open covering {푈훼}훼∈∆ of 퐴. Let us form an pg**- open covering of 푋 by adjoining to {푈훼}훼∈∆ the single pg**- open set 푋 − 퐴, that is { {푈훼}훼∈∆ ∪ (푋 − 퐴) }. Some finite subcollection of {{푈훼}훼∈∆ ∪ (푋 − 퐴) } covers 푋. If this subcollection contains the set 푋 − 퐴 , discard 푋 − 퐴, otherwise leave the subcollection alone. The resulting collection is a finite subcollection of {푈훼}훼∈∆ that covers A. Example 3.10 Let 푋 = {푎, 푏, 푐}, 휏 = {휑, {푎}, {푏}, {푎, 푏}, 푋} . Here PG∗∗푂(푋) = {휑, {푎}, {푏}, {푎, 푏}, 푋}. 푋 is pg**- compact. 푌 = {푎, 푏} is pg**- compact but not pg**- closed. Theorem 3.11 Let 푋 and 푌 be topological spaces and 푓 ∶ (푋, 휏) → (푌, 휎) be a function. Then, 1. 푓 is pg**- irresolute and 퐴 is a pg**- compact subset of 푋 ⟹ 푓(퐴) is a pg**- compact subset of 푌. 2. 푓 is one-one pg**-resolute map and 퐵 is a pg**- compact subset of Y ⟹ 푓−1(퐵) is a pg**-compact subset of 푋. Proof (1) & (2) obvious from the definitions. The following theorems give several equivalent forms of pg**-compactness which are often easier to apply. Theorem 3.12 If 퐴1 , 퐴2 , 퐴3 , … , 퐴푛 are pg**-compact subsets of a pg**- multiplicative space 푋 then 퐴1 ∪ 퐴2 ∪ 퐴3 ∪ … ∪ 퐴푛 is also pg**-compact. pg**- compact spaces 31 푛 Proof Let 퐴 = ∪ 퐴푖. Suppose 풜 = {푈훼}훼∈∆ is an pg**-open cover for 퐴. Then 풜 푖 = 1 is a pg**-open cover for 퐴1 , 퐴2 , 퐴3 , … , 퐴푛 separately also. Since each 퐴푖 is pg**- 푛 compact there exists finite subcovers 풜푖 ’s of each 퐴푖 , then ∪ 풜푖 forms a finite 푖 = 1 푛 subcover of ∪ 퐴푖 = 퐴. Therefore 퐴 is pg**-compact. 푖 = 1 Theorem 3.13 A topological space is pg**-compact if and only if every collection of pg**-closed sets with empty intersection has a finite sub collection with empty intersection. Proof Follows from the fact that a collection of pg**-open sets is a pg**-open cover if and only if the collection of all their complements has empty intersection. Theorem 3.14 A topological space 푋 is pg**-compact ⇔ for every collection 풞 of pg**-closed sets in X having the finite intersection property, the intersection ∩ 퐶 of 푐∈풞 all the elements of 풞 is nonempty. Proof Let (푋, 휏)be pg**-compact and 풞 be a collection of pg**-closed sets with finite intersection property. Suppose ∩ 퐶 = 휑 then {푋 − 퐶}퐶∈풞 is a pg**-open 푐∈풞 푛 cover for 푋. Therefore there exists 퐶1 , 퐶2 , 퐶3 , … , 퐶푛 ∈ 풞 such that ∪ (푋 − 퐶푖) = 푖 = 1 푛 푋. This implies ∩ 퐶푖 = 휑 which is a contradiction. ∴ ∩ 퐶 ≠ 휑 . Conversely, 푖 = 1 푐∈풞 assume the hypothesis given in the statement. To prove 푋 is pg**-compact. Let {푈훼}훼∈∆ be a pg**-open cover for 푋. Then ∪ 푈훼 = 푋 ⟹ ∩ (푋 − 푈훼) = 휑. By the 훼∈∆ 훼∈∆ 푛 푛 hypothesis 훼1 , 훼2 , 훼3 , … , 훼푛 such that ∩ (푋 − 푈훼 ) = 휑. Therefore ∪ 푈훼 = 푋 푖 = 1 푖 푖 = 1 푖 and hence 푋 is pg**-compact. Corollary 3.15 Let (푋, 휏)be pg**-compact space and let 퐶1 ⊃ 퐶2 ⊃ ⋯ ⊃ 퐶푛 ⊃ 퐶푛+1 ⊃ ⋯ be a nested sequence of non-empty pg**-closed sets in X.