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. Then the intersection ∩ 퐶푛is nonempty. 푛∈푍+

Proof Obviously {퐶푛}푛∈푍+ has finite intersection property. Therefore by the previous theorem ∩ 퐶푛is nonempty. 푛∈푍+ 32 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani

Theorem 3.16 Let (X, τ) and (Y, σ) be two topological spaces and 푓 ∶ (X, τ) → (Y, σ) be a function. Then, 1. 푓 is onto, pg**- continuous and 푋 is pg**- compact ⟹ 푌 is compact. 2. 푓 is onto, continuous and 푋 is pg**- compact ⟹ 푌 is compact. 3. 푓 is strongly pg**- irresolute and 푋 is compact ⟹ 푌 is pg**- compact. 4. 푓 is bijection and pg**- resolute then 푌 is pg**- compact ⟹ 푋 is compact. 5. 푓 is a bijection and pg**- open then 푌 is pg**- compact ⟹ 푋 is compact. 6. 푓 is onto, pg**- irresolute and 푋 is pg**- compact ⟹ 푌 is pg**- compact. 7. 푓 is a bijection and pg**- resolute then 푌 is pg**- compact ⟹ 푋 is pg**- compact.

−1 Proof (1) Let {푈훼}훼∈∆ be a pg**-open cover for 푌. Then {푓 (푈훼)}훼∈∆ is a pg**- open cover for 푋. Since 푋 is pg**-compact, there exists 훼1 , 훼2 , 훼3 , … , 훼푛 such that 푛 푛 −1 푋 ⊆ ∪ 푓 (푈훼 ). Then 푌 = 푓(푋) ⊆ ∪ 푈훼 . Therefore 푌 is compact. 푖 = 1 푖 푖 = 1 푖 Proofs for (2) to (7) are similar to the above.

Remark 3.17 The property of being pg**- compact, is a pg**- topological property. This follows from (6) and (7) of the above theorem.

Definition 3.18 A sequence 〈푥푛〉 in topological space(푋, 휏) is pg**- congregates to 푥 푝푔∗∗ in 푋 (〈푥푛〉 → 푥) is for every pg**-neighbourhood 푈 of 푥 there exists a positive integer 푁 such that 푥푛 ∈ 푈, ∀ 푛 ≥ 푁. We say that 푥 is the pg**-limit of the sequence 〈푥푛〉.

Result 3.19 In a topological space (X, τ), if a sequence 〈푥푛〉 pg**- congregates to 푥0 then the set 퐴 = {푥0, 푥1, 푥2,…} is pg**- compact in X.

Proof Let {푈훼}훼∈∆ be a pg**-open cover 풜 for 퐴, then 푥0 ∈ 푈훼0for some 훼0 ∈ ∆. 푝푔∗∗ Since 〈푥푛〉 → 푥, there exists a positive integer 푁 such that 푥푛 ∈ 푈훼0, ∀ 푛 ≥ 푁. Therefore the finite set {푥0, 푥1, … , 푥푁−1} can be covered by atmost a finite number of pg**-open sets 푈1, 푈2, … , 푈푁−1. Therefore the pg**-open cover 풜 contains a finite subcover 풞 = {푈훼0, 푈1, 푈2, … , 푈푁−1} of 퐴. Hence 퐴 pg**- compact in X.

Theorem 3.20 Let (X, τ) be a pg**- compact pg **- multiplicative space then every infinite subset of X has a pg**-cluster point. pg**- compact spaces 33

Proof Let (X, τ) be a pg**- compact pg**- multiplicative space and 퐴 be a subset of X. We desire to prove that, if 퐴 is infinite then 퐴 has a pg**-cluster point. We prove it contrapositively, that is ‘if 퐴 has no pg**-cluster point, then 퐴 must be finite’. Suppose 퐴 has no pg**-cluster point, then 퐴 is pg**-closed. Furthermore, for each

푎 ∈ 퐴 we can choose a pg**-neighbourhood 푈푎 of 푎 such that 푈푎 ∩ 퐴 = {푎}. Then the space X can be covered by {(푋 − 퐴), {푈푎}푎∈퐴}, being pg**- compact, it can be covered by finitely many of these sets. Since (푋 − 퐴) does not intersect 퐴 and each

푈푎 contains only one point of 퐴, the set 퐴 must be finite.

Theorem 3.21 (Generalization of extreme value theorem): Let 푓 ∶ 푋 → 푌 be pg**- continuous, where 푌 is an ordered set in the order topology. If 푋 is pg**- compact, then there exist points 푐 and 푑 in 푋 such that 푓(푐) ≤ 푓(푥) ≤ 푓(푑) ∀ 푥 ∈ 푋.

Proof Since 푓 is pg**- continuous and 푋 is pg**- compact, The set 퐴 = 푓(푋) is compact. Suppose A has no largest element then the collection {(−∞, 푎)/푎 ∈ 퐴} forms an open covering of A. Since A is compact it has some finite subcover {(−∞,

푎1), (−∞, 푎2), … , (−∞, 푎푛)}. Let 푎 = max {푎1 , 푎2 , 푎3 , … , 푎푛 }, then 푎 belong to none of these sets, contrary to the fact that they cover 퐴. ∴ 퐴 has a largest element 푀. Similarly it can be proved that it has the smallest element m. Therefore there exists c and d in 푋 such that 푓(푐) = 푚, 푓(푑) = 푀 and 푓(푐) ≤ 푓(푥) ≤ 푓(푑) ∀ 푥 ∈ 푋. Definition 3.22 A family 풞 of subsets of a topological space 푋 is said to be pg**- short if 풞 is not a pg**-open cover of 푋. 풞 is said to be finitely pg**-short if no finite subcollection of pg**-open covering 풞 covers 푋.

Theorem A topological space (X, τ) is pg**- compact if and only if each finitely pg**-short family of pg**-open sets in 푋 is pg**-short.

Proof Suppose 푋 is pg**- compact. Let 풞 be any finitely pg**-short family of pg**- open sets in 푋, then no finite subcollection of 풞 covers 푋. We show that 풞 does not cover 푋. Suppose 풞 is a pg**-open cover of 푋. Then 풞 is a pg**-open cover of 푋 which has no finite subcover which is a contradiction to 푋 is pg**- compact. Therefore 풞 is pg**-short. Conversely assume each finitely pg**-short family of pg**-open sets in 푋 is pg**-short. Suppose 푋 is not pg**- compact, then there exist a pg**-open cover 풜 of 푋 which has no finite subcover. Therefore 풜 is a finitely pg**- short family of pg**-open sets in 푋, then by hypothesis, 풜 is pg**-short that is 풜 does not cover 푋, which is a contradiction. Hence 푋 is pg**- compact.

34 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani

Definition 3.23 A function 푓: 푋 → 푌 between topological spaces is said to be pg**- proper if 푓−1(퐶) is pg**- compact for each pg**- compact subset 퐶 of 푌.

Definition 3.24 A function 푓:(푋,휏) → (푌,휎) is called pg**- resolve if 푓(퐹) is pg**- closed in 푌 whenever 퐹 is pg**- closed in 푋.

Theorem 3.25 If 푓: 푋 → 푌 is a pg**- resolve map between pg**- multiplicative spaces 푋 and 푌. Also 푓−1(푦) is pg**- compact for each 푦 ∈ 푌, then 푓 is pg**-proper.

Proof Let 퐶 ⊂ 푌 be pg**- compact and let {푈훼/훼 ∈ 퐴} be a collection of pg**-open −1 sets whose union contains 푓 (퐶). For any 푦 ∈ 퐶 there is a finite subset 퐴푦 ⊂ 퐴 such −1 that 푓 (푦) ⊂ ⋃{푈훼/훼 ∈ 퐴푦} . Take 푊푦 = ⋃{푈훼/훼 ∈ 퐴푦} and 푉푦 = 푌 − 푓(푋 − −1 푊푦), which is pg**-open. Now 푓 (푉푦) ⊂ 푊푦 and 푦 ∈ 푉푦. Since 퐶 is pg**- compact and is covered by 푉푦, there are points 푦1, 푦2, … , 푦푛 such that 퐶 ⊂ 푉푦1 ∪ 푉푦2 ∪ … ∪ 푉푦푛. −1 −1 −1 −1 Thus 푓 (퐶) ⊂ 푓 (푉푦1) ∪ 푓 (푉푦2) ∪ … ∪ 푓 (푉푦푛 ) ⊂ 푊푦1 ∪ 푊푦2 ∪ … ∪ 푊푦푛 . This −1 −1 implies 푓 (퐶) ⊂ ⋃{푈훼/훼 ∈ 퐴푦푖;푖 = 1,2,…,푛} which is finite. Therefore 푓 (퐶) is pg**- compact and hence 푓 is pg**-proper.

4. pg**- compact modulo I Definition 4.1 An ideal topological space (푋, 휏, I ) is said to be pg**- compact modulo I if for every pg**-open covering {푈훼}훼∈∆ of X, there exists a finite subset

∆0 of ∆ such that 푋 − ∪ 푈훼 ∈ I. 훼∈∆0

Remark 4.2 pg**- compactness implies pg**- compact modulo I for any ideal I but not conversely.

Example 4.3 Let (푋, 휏, I ) be an infinite indiscrete topological space where 퐼 =

퓅(푋) . Let {푈훼}훼∈∆ be a pg**-open cover for 푋 . Let 훼0 ∈ ∆ then, 푋 − 푈훼0 ∈ I. Therefore (푋, 휏, I ) is pg**- compact modulo I but not pg**- compact.

Note “pg**- compact modulo I ” and “pg**- compact” happens together when 퐼 = {휑}.

Remark 4.4 pg**- compact modulo I implies compact modulo I for any ideal I but not conversely. Proof is obvious, since 휏 ⊆ 퐺∗∗푂(푋). pg**- compact spaces 35

Example 4.5 An indiscrete space (푋, 휏, {휑}) is compact modulo I but not pg**- compact modulo I.

Theorem 4.6 Let X be an ideal topological space. Then the following are equivalent. 1) X is pg**- compact modulo I.

2) For every family {퐹훼/훼 ∈ Ω} of pg**-closed sets such that ∩ 퐹훼 = 휑, then 훼∈Ω 푛 푛 there exists a finite sub family {퐹훼 }푖=1 such that ∩ 퐹훼 ∈ 퐼. 푖 푖 = 1 푖 3) For every family {퐹훼/훼 ∈ Ω} of pg**-closed sets with 퐼-FIP ∩ 퐹훼 ≠ 휑 훼∈Ω

Proof

(1) ⇒ (2) Let X be pg**- compact modulo I and {퐹훼/훼 ∈ Ω} be a family of 푐 푐 pg**-closed sets such that ∩ 퐹훼 = 휑. Therefore ∪ 퐹훼 = 푋 where 퐹훼 is pg**-open. 훼∈Ω 푐 Hence {퐹훼 /훼 ∈ Ω} is a pg**-open cover for X, also since X is pg**- compact modulo 푛 푛 푐 I there exists 훼1 , 훼2 , 훼3 , … , 훼푛 such that 푋 − ∪ 퐹훼 ∈ 퐼. This implies ∩ 퐹훼 ∈ 퐼. 푖 = 1 푖 푖 = 1 푖 (2) ⇒ (3) Let {퐹훼/훼 ∈ Ω} be a family of pg**-closed sets with 퐼 -FIP. 푛 Suppose ∩ 퐹훼 = 휑, then there exists 훼1 , 훼2 , 훼3 , … , 훼푛 such that ∩ 퐹훼 ∈ 퐼. Which 훼∈Ω 푖 = 1 푖 contradicts the hypothesis. Therefore ∩ 퐹훼 ≠ 휑. 훼∈Ω 푐 (3) ⇒ (1) Let {푈훼/훼 ∈ Ω} be a pg**-open cover for X. Then ∩ 푈훼 = 휑. 훼∈Ω 푐 Therefore the family of pg**-closed sets {푈훼/훼 ∈ Ω} does not satisfy 퐼 -FIP and 푛 푛 푐 therefore there exists 훼1 , 훼2 , 훼3 , … , 훼푛 such that ∩ 푈훼 ∈ 퐼 (ie) 푋 − ∪ 푈훼 ∈ 퐼. 푖 = 1 푖 푖 = 1 푖 Therefore X is pg**-compact modulo I.

Theorem 4.7 If 퐼 ⊆ 퐽, then (푋, 휏, I ) is pg**- compact modulo I implies (푋, 휏, 퐽) is pg**-compact modulo 퐽. Proof is obvious.

Theorem 4.8 Let 퐼퐹 denote the ideal of all finite subsets of 푋. Then (푋, 휏) is pg**- compact if and only if (푋, 휏, 퐼퐹) is pg**- compact modulo 퐼퐹.

Proof

Necessity follows since {휑}∈ 퐼퐹. 36 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani

Sufficiency Let {푈훼}훼∈∆ be a pg**-open cover for 푋, then there exists a finite subset

∆0 of ∆ such that 푋 − ∪ 푈훼 ∈ 퐼퐹 . 푋 − ∪ 푈훼 = {푥1 , 푥2 , 푥3 , … , 푥푛 }. Choose 훼푖 훼∈∆0 훼∈∆0 푛 such that 푥푖 ∈ 푈훼푖 for 푖 = 1,2,3,… ,푛. Then 푋 = { ∪ 푈훼} ∪ { ∪ 푈훼푖 }. Therefore 푋 훼∈∆0 푖 = 1 is pg**- compact.

5. pg**- countably compact space Definition 5.1 A subset 퐴 of a topological space(X, τ) is said to be pg**- countably compact if every countable pg**-open covering of 퐴 has a finite sub cover.

Example 5.2 An infinite. cofinite topological space is pg**- countably compact.

Example 5.3 A countably infinite indiscrete topological space is not pg**- countably compact.

Remark 5.4 Every pg**- compact space is pg**- countably compact.

Theorem 5.5 Every pg**- closed subset of a pg**- countably compact space is pg**- countably compact. Proof is similar to theorem (3.9)

Theorem 5.6 A topological space 푋 is pg**- countably compact ⇔ for every collection 풞 of pg**-closed sets in 푋 having the finite intersection property, the intersection ∩ 퐶 of all the elements of 풞 is nonempty. 푐∈풞 Proof is similar to the proof of theorem (3.14)

Corollary 5.7 Let (푋, 휏) be pg**- countably compact space and let 퐶1 ⊃ 퐶2 ⊃ 퐶3 ⊃ ⋯ ⊃ 퐶푛 ⊃ 퐶푛+1 ⊃ ⋯ be a nested sequence of non-empty pg**-closed sets in 푋. Then the intersection ∩ 퐶푛 is nonempty. 푛∈푍+

Proof Obviously {퐶푛}푛∈푍+ has finite intersection property. Therefore by theorem (5.6) ∩ 퐶푛is nonempty. 푛∈푍+

Theorem 5.8 A topological space (푋, 휏) is pg**-countably compact if and only if every infinite subset has a pg**-cluster point. pg**- compact spaces 37

Proof Suppose if every infinite subset of (푋, 휏) has a pg**-cluster point and 픉 = 퐹푖 countable collection of pg**-closed sets with finite intersection property. The 푛 ∩ intersection 퐻푛 = 퐹푗 is nonempty for all 푛. Choose 푥푛 ∈ 퐻푛 for each 푛. Then the 푗 = 1 set 퐸 = {푥푛 / 푥푛 ∈ 퐻푛} has a pg**-cluster point 푥 . But 푥푛 ∈ 퐹푖 for all 푛 ≥ 푖, since 퐹푖 is pg**-closed 푥 ∈ 퐹푖 , ∀ 푖. Therefore 푥 ∈ ∩ 퐹푖, hence every countable collection 픉 of pg**-closed sets with finite intersection property has a nonempty intersection, then by theorem (5.6) 푋 is pg**- countably compact. Conversely let (푋, 휏)be pg**- countably compact and suppose that there exists an infinite subset 퐴 has no pg**-cluster point. Let 퐸 = {푥푛 /푛 ∈ 푁} be a countable subset of 퐴 . Since 퐸 has no pg**-cluster point of 퐸 , there exists a pg**- 푐 neighbourhood 푈푛 of 푥푛 such that 퐸 ∩ 푈푛 = {푥푛} . Therefore {퐸 , {푈푛}푛∈푁 } is a countable pg**-open cover for 푋. This pg**-open cover has no finite subcover, this is a contradiction for 푋 is pg**- countably compact. Therefore every infinite subset of 푋 has a pg**-cluster point.

Theorem 5.9 Let (X, τ) and (Y, σ) be two topological spaces and 푓 ∶ (X, τ) → (Y, σ) be a bijective function. Then, 1. 푓 is pg**- continuous and 푋 is pg**- countably compact ⟹ 푌 is countably compact. 2. 푓 is continuous and 푋 is pg**- countably compact ⟹ 푌 is countably compact. 3. 푓 is strongly pg**- irresolute and 푋 is countably compact ⟹ 푌 is pg**- countably compact. 4. 푓 is pg**- resolute and 푌 is pg**- countably compact ⟹ 푋 is countably compact. 5. 푓 is pg**-open and 푌 is pg**- countably compact ⟹ 푋 is countably compact. 6. 푓 is pg**- irresolute and 푋 is pg**- countably compact ⟹ 푌 is pg**- countably compact. 7. 푓 is pg**- resolute and 푌 is pg**- countably compact ⟹ 푋 is pg**- countably compact.

−1 Proof Let {푈훼}훼∈∆ be a pg**- countable open cover for 푌. Then {푓 (푈훼)}훼∈∆ is a pg**- countable open cover for 푋. Since 푋 is pg**- countably compact, there exists 푛 푛 −1 훼1 , 훼2 , 훼3 , … , 훼푛 such that 푋 ⊆ ∪ 푓 (푈훼 ). Then 푌 = 푓(푋) ⊆ ∪ 푈훼 . Therefore 푖 = 1 푖 푖 = 1 푖 푌 is countably compact. Proofs for (2) to (7) are similar to the above.

38 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani

Remark 5.10 The property of being pg**- countably compact, is a pg**- topological property. This follows from (6) and (7) of the above theorem.

6. pg**- countably compact modulo I Definition 6.1 An ideal topological space(푋, 휏 , I ) is said to be pg**-countably compact modulo I if for every countable pg**-open covering {푈훼}훼∈∆ of 푋, there exists a finite subset ∆0 of ∆ such that 푋 − ∪ 푈훼 ∈ I. 훼∈∆0 All the results from remark (4.2) to theorem are true in the case when (푋, 휏, I ) is pg**-countably compact modulo I.

7. Sequentially pg**- compact Definition 7.1 A subset 퐴 of a topological space (푋, 휏) is said to be sequentially pg**-compact if every sequence in 퐴 contains a subsequence which pg**- congregates to some point in 퐴.

Example 7.2 A finite topological space is sequentially pg**- compact.

Let 〈푥푛〉 be an arbitrary sequence in a finite topological space 푋. Since 푋 is finite 푥푛 = 푥0 except for finitely many 푛’s, therefore the constant sequence 푥0, 푥0,… is pg**- congregates to 푥0 in 푋. Hence 푋 is sequentially pg**- compact.

Example 7.3 An infinite indiscrete topological space is not sequentially pg**- compact.

Let (X, τ) be an infinite indiscrete topological space and let 푥0 ∈ 푋 be arbitrary. Since all the subsets are pg**-open in this space {푥0} is a pg**-neighbourhood of 푥0 , therefore no sequence other than the constant sequence 〈푥0〉 can pg**- congregates to 푥0 in 푋. Since 푥0 is arbitrary (X, τ) is not sequentially pg**- compact.

Remark 7.4 Sequentially pg**- compactness implies sequentially compactness, but the reverse implication is not true as seen in the following example.

Example 7.5 Every infinite indiscrete topological space is sequentially compact but not sequentially pg**- compact.

Theorem 7.6 A finite subset 퐴 of a topological space(X, τ) is sequentially pg**- compact. pg**- compact spaces 39

Proof Let 〈푥푛〉 be an arbitrary sequence in a finite subset 퐴. At least one element of the sequence say 푥0 must be repeated infinite number times, since 퐴 is finite. Hence the constant subsequence 푥0, 푥0,… is pg**- congregates to 푥0 in 푋.

Theorem 7.7 If a pg**- congregate sequence in a topological space has infinitely many distinct points, then its pg**-limit is a pg**-limit point of the set of points of the sequence.

Proof Let (X, τ) be a topological space and let 〈푥푛〉 be a pg**- congregate sequence in 푋 with pg**-limit 푥. Assume that 푥 is not a pg**-limit point of the set 퐴 of points of the sequence, and show that the sequence has only finitely many distinct points. Since 푥 is not a pg**-limit point of 퐴 there exists a pg**-neighbourhood 푈 of 푥 contains no point of the sequence different from 푥. However, since 푥 is the pg**-limit of the sequence, all 푥푛’s must lie in 푈, hence must coincide with 푥. Therefore there are only finitely many distinct points in the sequence.

Theorem 7.8 A topological space (푋, 휏) is sequentially pg**- compact if and only if every infinite subset has a pg**-limit point.

Proof Assume that 푋 is sequentially pg**- compact and let 퐴 be an infinite subset of

X. Since 퐴 is infinite a sequence 〈푥푛〉 of distinct points can be extracted from 퐴. Since X is sequentially pg**- compact this sequence has a subsequence which pg**- congregates to a point 푥. Hence from theorem (7.7) 푥 is a pg**-limit point of the set of points of the subsequence, since this set is a subset of 퐴, 푥 is also a pg**-limit point of 퐴. Conversely assume that every infinite subset of X has a pg**-limit point.

Let 〈푥푛〉 be an arbitrary sequence in X. If 〈푥푛〉 has a point which is infinitely repeated, then it has a constant subsequence and this subsequence is pg**- congregate. If no point of 〈푥푛〉 is infinitely repeated, then the set 퐴 of points of this sequence is infinite, then we can extract the subsequence from 〈푥푛〉 which is pg**- congregate to 푥 since the set 퐴 has a pg**-limit point 푥.

Theorem 7.9 Let (X, τ) and (Y, σ) be two topological spaces, then X and Y are sequentially pg**- compact if and only if X × Y is sequentially pg**- compact.

Proof Suppose that X and Y are sequentially pg**- compact. If (푥푛, 푦푛) is a sequence 푝푔∗∗ in X × Y,then there exists a subsequence (푥푛푘) of (푥푛) such that (푥푛푘) → 푎 in X. 40 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani

푝푔∗∗ Similarly there is a subsequence (푦푛푘) of (푦푛) such that (푦푛푘) → 푏 in Y. Thus 푝푔∗∗ (푥푛푘, 푦푛푘) → (푎, 푏). Therefore X × Y is sequentially pg**- compact. To prove the converse consider the projection maps 휋푥: X × Y → X defined by 휋푥(푥, 푦) = 푥 and 휋푦: X × Y → Y defined by 휋푦(푥, 푦) = 푦 . Suppose X × Y is sequentially pg**- compact. If (푥푛) and (푦푛) are sequences of X and Y respectively then (푥푛, 푦푛 ) is a sequence in X × Y. Then there exists a subsequence (푥푛푘, 푦푛푘) of (푥푛, 푦푛) such that 푝푔∗∗ (푥푛푘, 푦푛푘) → (푎, 푏) since X × Y is sequentially pg**- compact. This implies 푝푔∗∗ 푝푔∗∗ (푥푛푘) → 푎 in X and (푦푛푘) → 푏 in Y. Therefore X and Y are sequentially pg**- compact.

Theorem 7.10 Let (X, τ) and (Y, σ) be two topological spaces and 푓 ∶ (X, τ) → (Y, σ) be a function. Then, 1. 푓 is a bijection and pg**- resolute then Y is sequentially pg**- compact ⟹ 푋 is sequentially pg**- compact. 2. 푓 is onto, pg**- irresolute and 푋 is sequentially pg**- compact ⟹ 푌 is sequentially pg**-compact. 3. 푓 is onto, strongly pg**- irresolute and 푋 is sequentially compact ⟹ Y is sequentially pg**- compact. 4. 푓 is onto, continuous and 푋 is sequentially pg**- compact ⟹ Y is sequentially compact.

Proof (1) If 〈푥푛〉 is a sequence in 푋, then 〈푓(푥푛)〉 is a sequence in 푌. Since Y is sequentially pg**- compact, 〈푓(푥푛)〉 has a pg**- congregate subsequence 〈푓(푥푛푘)〉 푝푔∗∗ such that 푓(푥푛푘) → 푦0 in Y. Then there exists 푥0 ∈ 푋 such that 푓(푥0) = 푦0. Let 푈 be a pg**-open set containing 푥0, then 푓(푈) is a pg**-open set containing푓(푥0).

Then there exists 푁 such that 푓(푥푛푘) ∈ 푓(푈), ∀ 푘 ≥ 푁. Therefore 푋 is sequentially pg**- compact. Proofs for (2) to (4) are similar to the above.

Remark 7.11 The property of being sequentially pg**-compact, is a pg**- topological property. This follows from (1) and (2) of the above theorem.

Theorem 7.12 Every sequentially pg**-compact is pg**- countably compact. pg**- compact spaces 41

Proof Let (X, τ) be sequentially pg**-compact. Suppose X is not pg**- countably compact, then there exists countable pg**-open covering {푈훼}훼∈∆such that ∪ 푈훼 = 훼∈∆ 푋 , which has no finite subcover. Choose 푥1 ∈ 푈1, 푥2 ∈ 푈2 − 푈1, 푥3 ∈ 푈3 − 푛−1 ∪ 푈푖 … 푥푛 ∈ 푈푛 − ∪ 푈푖 . Now {푥푛} is a sequence in 푋. Let 푥 ∈ 푋 be arbitrary, 푖 = 1,2 푖 = 1 then 푥 ∈ 푈푗 for some 푗. But by our choice of {푥푛}, 푥푖 ∉ 푈푗 ∀ 푖 > 푗. Therefore there exists no subsequence of {푥푛} which can pg**- congregates to 푥 , which is a contradiction to X is sequentially pg**-compact. Therefore X is pg**- countably compact.

8. pg**- locally compact Definition 8.1 A subset 퐴 of a topological space(X, τ) is said to be relatively pg**- compact if 푝푔∗∗푐푙(퐴) is pg**-compact in 푋.

Definition 8.2 A topological space(X, τ) is said to be pg**- locally compact if for every point 푥 of X there is some pg**-compact subset 퐶 of X that contains a pg**- neighbourhood of 푥 .

Example 8.3 Let X has discrete topology. Then for every 푥 ∈ 푋, {푥} is a pg**- neighbourhood of 푥 and pg**-compact. Therefore every discrete space is pg**- locally compact.

Remark 8.4 A pg**-compact space is pg**- locally compact, but the converse is not true as seen in the following example.

Example 8.5 Let X be an infinite discrete topological space. In this space for every 푥 ∈ 푋, {푥} is a pg**-neighbourhood of 푥. Also {푥} is pg**-compact. Therefore X is pg**- locally compact, but it is not pg**-compact.

Theorem 8.6 If 푓:(푋,휏) → (푌,휎) is pg**-irresolute, pg**-open map from a pg**- locally compact space 푋 onto a topological space 푌, then 푌 is pg**- locally compact.

Proof Let 푦 ∈ 푌 be arbitrary. Since 푓 is surjective we can find 푥 ∈ 푋 such that 푓(푥) = 푦. But 푋 is pg**- locally compact, therefore there exists a pg**-compact subset 퐶 of X such that 퐶 ⊇ 푈 where 푈 is a pg**-neighbourhood of 푥. Therefore 푥 ∈ 푈 ⊆ 퐶, this implies 푦 = 푓(푥) ∈ 푓(푈) ⊆ 푓(퐶) since 푓 is a pg**-open map, 푓(푈) is a pg**- 42 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani neighbourhood of 푓(푥) . Also 푓(퐶) is pg**-compact subset of 푌 since 푓 is pg**- irresolute. Thus 푌 is pg**- locally compact.

9. pg**- paracompact

Definition 9.1 A collection 풜 = {푈훼}훼∈∆ of subsets of a topological space 푋 is said to be pg**- locally finite if each point 푥 ∈ 푋 has a pg**- neighbourhood having nonempty intersection with atmost finitely many members of 풜.

Example 9.2 Consider ℝ with usual topology, then the collection of intervals 풜 = {(푛,푛+2)/ 푛 ∈ ℤ} is pg**- locally finite.

Theorem 9.3 Let (X, τ) be a topological space and let 풜 = {푈훼}훼∈∆ be a pg**- ∗∗ locally finite family of subsets of 푋. Then 풞 = {푝푔 푐푙(푈훼)}훼∈∆ is also pg**- locally finite.

Proof Choose a point 푥 ∈ 푋 and a pg**- neighbourhood 퐺 of 푥 such that 푈훼 ∩ 퐺 = 휑 ∗∗ for all except for finitely many 훼 ∈ ∆, then 푈훼 ⊂ 푋 − 퐺 this implies 푝푔 푐푙(푈훼) ⊂ ∗∗ 푋 − 퐺 . Consequently 푝푔 푐푙(푈훼) ∩ 퐺 = 휑 for all except for finitely many 훼 ∈ ∆. Therefore 풞 is pg**- locally finite.

Theorem 9.4 In a topological space, if the collection 풜 = {푈훼}훼∈∆ is pg**- locally ∗∗ ∗∗ finite then ∪ {푝푔 푐푙(푈훼)}훼∈∆ = 푝푔 푐푙(∪ {푈훼}훼∈∆).

∗∗ ∗∗ Proof For every 훼 ∈ ∆ , we have 푈훼 ⊂ ∪ {푈훼}훼∈∆ , then 푝푔 푐푙(푈훼) ⊂ 푝푔 푐푙(∪ ∗∗ ∗∗ {푈훼}훼∈∆), this implies ∪ {푝푔 푐푙(푈훼)}훼∈∆ ⊂ 푝푔 푐푙(∪ {푈훼}훼∈∆). On the other hand, ∗∗ let 푥 ∈ 푝푔 푐푙(∪ {푈훼}훼∈∆) . Since 풜 is pg**- locally finite, there exists a pg**- neighbourhood 퐺푥 of 푥 such that 퐺푥 has nonempty intersection with atmost finitely many members of 풜. Let these finite members of 풜 be 푈훼1 , 푈훼2 , 푈훼3 , … , 푈훼푛 . Therefore 퐺푥 ∩ 푈훼 ≠ 휑 for 훼1 , 훼2 , 훼3 , … , 훼푛 . Let 퐻푥 be any pg**- neighbourhood of 푥 and let 퐼푥 = 퐺푥 ∩ 퐻푥 . Now 퐼푥 ∩ (푈훼1 ∪ 푈훼2 ∪ … ∪ 푈훼푛 ) ≠ 휑 , since 푥 ∈ ∗∗ 푝푔 푐푙(∪ {푈훼}훼∈∆). But 퐼푥 ⊂ 퐻푥, and 퐻푥 is an arbitrary pg**- neighbourhood of 푥 and ∗∗ ∗∗ ∗∗ 푥 ∈ 푝푔 푐푙(푈훼1 ∪ 푈훼2 ∪ … ∪ 푈훼푛 ) = 푝푔 푐푙(푈훼1 ) ∪ 푝푔 푐푙 (푈훼2 ) ∪ ∗∗ ∗∗ … ∪ 푝푔 푐푙(푈훼푛 ). Then 푥 ∈ 푝푔 푐푙(푈훼푖 ) for some 푖 where 1 ≤ 푖 ≤ 푛. Therefore 푥 ∈ ∗∗ ∗∗ ∗∗ ∪ {푝푔 푐푙(푈훼)}훼∈∆ . Therefore ∪ {푝푔 푐푙(푈훼)}훼∈∆ = 푝푔 푐푙(∪ {푈훼}훼∈∆). Hence the theorem. pg**- compact spaces 43

Definition 9.5 Let 풜 be a pg**-open covering of a topological space 푋. A collection 풫 is called a pg**-refinement of 풜 if 풫 is also pg**-open covering of 푋 and each member of 풫 is contained in some member of 풜.

Definition 9.6 A topological space 푋 is said to be pg**- paracompact if every pg**- open covering of 푋 has a pg**- locally finite pg**-refinement which is also a pg**- open covering of 푋.

Example 9.7 Every pg**- compact space is pg**- paracompact. Since every finite subcover 풫 of the pg**-open cover 풜 is a pg**- locally finite pg**-refinement of 풜.

Theorem 9.8 Every pg**- closed subset of a pg**- paracompact space is pg**- paracompact.

Proof Let 퐴 be a pg**- closed subset of the pg**- paracompact 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) }. Take a pg**- locally finite pg**-refinement of this pg**- open covering of 푋 and intersect it with 퐴. This gives a pg**- locally finite pg**-refinement of pg**- open covering {푈훼}훼∈∆ of 퐴.

References

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44 Mrs. G. Priscilla Pacifica and Dr. A. Punitha Tharani