About this Journal Submit a Manuscript Table of Contents 
ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 682829, 9 pages
doi:10.5402/2012/682829
Research Article

πœƒ - ℐ 𝑔 - Closed Sets

M. Navaneethakrishnan1Β and S. Alwarsamy2

1Department of Mathematics, Kamaraj College, Thoothukudi 628003, India
2Department of Mathematics, RDM Government Arts College, Sivagankai 630561, India

Received 3 November 2011; Accepted 22 December 2011

Academic Editor: D.Β Franco

Copyright Β© 2012 M. Navaneethakrishnan and S. Alwarsamy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define πœƒ - ℐ 𝑔 - Closed sets and discuss their properties. Using these sets, we characterize 𝒯 1 / 2 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 𝑠 and 𝒯 ℐ - 𝑆 𝑝 π‘Ž 𝑐 𝑒 𝑠 .

1. Introduction and Preliminaries

An ideal ℐ on a topological space ( 𝑋 , 𝜏 ) is a nonempty collection of subsets of 𝑋 which satisfies (i) 𝐴 ∈ ℐ and 𝐡 βŠ‚ 𝐴 implies 𝐡 ∈ ℐ and (ii) 𝐴 , 𝐡 ∈ ℐ implies 𝐴 βˆͺ 𝐡 ∈ ℐ . Given a topological space ( 𝑋 , 𝜏 ) with an ideal ℐ on 𝑋 and if β„˜ ( 𝑋 ) is the set of all subsets of 𝑋 , a set operator ( β‹… ) ⋆ ∢ β„˜ ( 𝑋 ) β†’ β„˜ ( 𝑋 ) called a local function [1] of 𝐴 with respect to 𝜏 and ℐ is defined as follows: for 𝐴 βŠ‚ 𝑋 , 𝐴 ⋆ ( 𝑋 , 𝜏 ) = { π‘₯ ∈ 𝑋 ∣ π‘ˆ ∩ 𝐴 βˆ‰ ℐ , f o r e v e r y π‘ˆ ∈ 𝜏 ( π‘₯ ) } , where 𝜏 ( π‘₯ ) = { π‘ˆ ∈ 𝜏 ∣ π‘₯ ∈ π‘ˆ } . A Kuratowski closure operator c l ⋆ ( β‹… ) for a topology 𝜏 ⋆ ( ℐ , 𝜏 ) called the ⋆ - topology, finer than 𝜏 , is defined by c l ⋆ ( 𝐴 ) = 𝐴 βˆͺ 𝐴 ⋆ ( ℐ , 𝜏 ) [2]. When there is no confusion we will simply write 𝐴 ⋆ for 𝐴 ⋆ ( ℐ , 𝜏 ) and 𝜏 ⋆ for 𝜏 ⋆ ( ℐ , 𝜏 ) . If ℐ is an ideal on 𝑋 , then ( 𝑋 , 𝜏 , ℐ ) is called an ideal space. A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be ⋆ - closed [3] if 𝐴 ⋆ βŠ‚ 𝐴 . A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be an ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 [4] if 𝐴 ⋆ βŠ‚ π‘ˆ whenever 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open. A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 if 𝑋 βˆ’ 𝐴 is ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . An ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be a 𝒯 ℐ - space [4] if every ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be ℐ - locally ⋆ - closed [5] if there exist an open set π‘ˆ and a ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set 𝐹 such that 𝐴 = π‘ˆ ∩ 𝐹 . If ℐ = { βˆ… } , then ℐ - 𝑙 π‘œ 𝑐 π‘Ž 𝑙 𝑙 𝑦 ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets coincide with locally closed sets.

By a space, we always mean a topological space ( 𝑋 , 𝜏 ) with no separation properties assumed. If 𝐴 βŠ‚ 𝑋 , c l ( 𝐴 ) and i n t ( 𝐴 ) will, respectively, denote the closure and interior of 𝐴 in ( 𝑋 , 𝜏 ) and i n t ⋆ ( 𝐴 ) will denote the interior of 𝐴 in ( 𝑋 , 𝜏 ⋆ ) . A subset 𝐴 of a topological space ( 𝑋 , 𝜏 ) is said to be a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set [6] if c l ( 𝐴 ) βŠ‚ π‘ˆ whenever 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open. A subset 𝐴 of a topological space ( 𝑋 , 𝜏 ) is said to be a 𝑔 - π‘œ 𝑝 𝑒 𝑛 set if 𝑋 βˆ’ 𝐴 is a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. A space ( 𝑋 , 𝜏 ) is said to be a 𝒯 1 / 2 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 [6] if every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is a closed set.

For a subset 𝐴 of a space ( 𝑋 , 𝜏 ), the πœƒ - 𝑖 𝑛 𝑑 𝑒 π‘Ÿ 𝑖 π‘œ π‘Ÿ [7] of 𝐴 is the union of all open sets of 𝑋 whose closures contained in 𝐴 and is denoted by i n t πœƒ ( 𝐴 ) . The subset 𝐴 is called πœƒ - π‘œ 𝑝 𝑒 𝑛 if 𝐴 = i n t πœƒ ( 𝐴 ) . The complement of a πœƒ - π‘œ 𝑝 𝑒 𝑛 set is called a πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. Equivalently, 𝐴 βŠ‚ 𝑋 is called πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 [7] if 𝐴 = c l πœƒ ( 𝐴 ) , where c l πœƒ ( 𝐴 ) = { π‘₯ ∈ 𝑋 ∣ c l ( π‘ˆ ) ∩ 𝐴 β‰  βˆ… f o r a l l π‘ˆ ∈ 𝜏 ( π‘₯ ) } . The family of all πœƒ - π‘œ 𝑝 𝑒 𝑛 sets of 𝑋 forms a topology [7] on 𝑋 , which is coarser than 𝜏 and is denoted by 𝜏 πœƒ . A subset 𝐴 of a topological space ( 𝑋 , 𝜏 ) is said to be a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 𝑠 𝑒 𝑑 [8] if c l πœƒ ( 𝐴 ) βŠ‚ π‘ˆ whenever 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open. A subset 𝐴 of a space ( 𝑋 , 𝜏 ) is said to be a πœƒ - 𝑔 - π‘œ 𝑝 𝑒 𝑛 𝑠 𝑒 𝑑 [8] if 𝑋 βˆ’ 𝐴 is a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. A subset 𝐴 of a space ( 𝑋 , 𝜏 ) is said to be a Ξ› - 𝑠 𝑒 𝑑 [9, 10] if 𝐴 = 𝐴 Ξ› , where 𝐴 Ξ› = ∩ { π‘ˆ ∈ 𝜏 ∣ 𝐴 βŠ‚ π‘ˆ } .

A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 [11] if c l ⋆ πœƒ ( 𝐴 ) = 𝐴 , where c l ⋆ πœƒ ( 𝐴 ) = { π‘₯ ∈ 𝑋 ∣ 𝐴 ∩ c l ⋆ ( π‘ˆ ) β‰  πœ™ f o r a l l π‘ˆ ∈ 𝜏 ( π‘₯ ) } . 𝐴 is said to be πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 if 𝑋 βˆ’ 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . If ℐ = { βˆ… } , c l ⋆ πœƒ ( 𝐴 ) = c l πœƒ ( 𝐴 ) . If ℐ = β„˜ ( 𝑋 ) , c l ⋆ πœƒ ( 𝐴 ) = c l ( 𝐴 ) . For a subset 𝐴 of X, i n t πœƒ 𝐼 ( 𝐴 ) = βˆͺ { π‘ˆ ∈ 𝜏 ∣ c l ⋆ ( π‘ˆ ) βŠ‚ 𝐴 } [11]. We denote this i n t πœƒ 𝐼 ( 𝐴 ) by i n t ⋆ πœƒ ( 𝐴 ) . The family of all πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 sets of ( 𝑋 , 𝜏 , ℐ ) is a topology and it is denoted by 𝜏 πœƒ - ℐ (see [11, Theorem  1]).

Lemma 1.1 (see [11, Corollary  4 if Theorem  2]). 𝜏 πœƒ βŠ‚ 𝜏 πœƒ - ℐ βŠ‚ 𝜏 .

Lemma 1.2 (see [11, Proposition  3]). Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space. Then, we have(1)if ℐ = { πœ™ } or ℐ = 𝒩 , where 𝒩 is the ideal of nowhere dense sets of ( 𝑋 , 𝜏 ) , then 𝜏 πœƒ - ℐ = 𝜏 πœƒ ,(2)if ℐ = { πœ™ } , then 𝜏 πœƒ - ℐ = 𝜏 .

Lemma 1.3 (see [5, Theorem  2.13]). Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space. Then every subset of 𝑋 is ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if every open set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Lemma 1.4 (see [11, Proposition  1]). Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 a subset of 𝑋 . Then A is πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 if and only if i n t ⋆ πœƒ ( 𝐴 ) = 𝐴 .

Lemma 1.5. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 a subset of 𝑋 . Then c l ⋆ πœƒ ( 𝐴 ) = { π‘₯ ∈ 𝑋 ∣ π‘ˆ ∩ c l ⋆ ( 𝐴 ) β‰  πœ™ for all π‘ˆ ∈ 𝜏 ( π‘₯ ) } is closed.

Proof. If π‘₯ ∈ c l ( c l ⋆ πœƒ ( 𝐴 ) ) and π‘ˆ ∈ 𝜏 ( π‘₯ ) , then π‘ˆ ∩ c l ⋆ πœƒ ( 𝐴 ) β‰  πœ™ . Then, 𝑦 ∈ π‘ˆ ∩ c l ⋆ πœƒ ( 𝐴 ) for some 𝑦 ∈ 𝑋 . Since π‘ˆ ∈ 𝜏 ( 𝑦 ) and 𝑦 ∈ c l ⋆ πœƒ ( 𝐴 ) , from the definition of c l ⋆ πœƒ ( 𝐴 ) we have 𝐴 ∩ c l ⋆ ( π‘ˆ ) β‰  πœ™ . Therefore, π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) . So c l ( c l ⋆ πœƒ ( 𝐴 ) ) βŠ‚ c l ⋆ πœƒ ( 𝐴 ) and hence c l ⋆ πœƒ ( 𝐴 ) is closed.

Lemma 1.6. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 a subset of 𝑋 . Then, 𝑋 βˆ’ c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) = i n t ⋆ πœƒ ( 𝐴 ) .

Proof. π‘₯ ∈ 𝑋 βˆ’ c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) if and only if π‘₯ βˆ‰ c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) if and only if there exist π‘ˆ ∈ 𝜏 ( π‘₯ ) such that ( 𝑋 βˆ’ 𝐴 ) ∩ c l ⋆ ( π‘ˆ ) = πœ™ if and only if π‘₯ ∈ π‘ˆ and, c l ⋆ ( π‘ˆ ) βŠ‚ ( 𝐴 ) if and only if π‘₯ ∈ π‘ˆ βŠ‚ i n t ⋆ πœƒ ( 𝐴 ) .

2. πœƒ - ℐ 𝑔 - Closed Sets

A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set if c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ whenever 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open. Every πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. If ℐ = { βˆ… } , then c l ⋆ πœƒ ( 𝐴 ) = c l πœƒ ( 𝐴 ) and hence πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets coincide with πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets. If ℐ = β„˜ ( 𝑋 ) , then c l ⋆ πœƒ ( 𝐴 ) = c l ( 𝐴 ) and hence πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets coincide with 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets. Since c l ⋆ ( 𝐴 ) βŠ‚ c l ( 𝐴 ) βŠ‚ c l ⋆ πœƒ ( 𝐴 ) βŠ‚ c l πœƒ ( 𝐴 ) , we have the following inclusion diagram: πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 ⟢ πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 ⟢ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 ⟢ ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . ( 2 . 1 )

Example 2.1. shows that a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set needs not to be πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , and Example 2.2 shows that πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set needs not to be a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.

Example 2.1. Let 𝑋 = { π‘Ž , 𝑏 , 𝑐 , 𝑑 } , 𝜏 = { πœ™ , { 𝑏 } , { π‘Ž , 𝑏 } , { 𝑏 , 𝑐 } , { π‘Ž , 𝑏 , 𝑐 } , { π‘Ž , 𝑏 , 𝑑 } , 𝑋 } , and ℐ = { πœ™ , { π‘Ž } , { 𝑐 } , { π‘Ž , 𝑐 } } . Let 𝐴 = { 𝑐 } . Then 𝐴 is closed and hence 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . But 𝐴 is not πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 because 𝐴 βŠ‚ { 𝑏 , 𝑐 } and c l ⋆ πœƒ ( 𝐴 ) = 𝑋 βŠ„ { 𝑏 , 𝑐 } .

Example 2.2. Let 𝑋 and 𝜏 be the same as in Example 2.1. Let ℐ = { πœ™ , { π‘Ž } , { 𝑏 } , { π‘Ž , 𝑏 } } and 𝐴 = { 𝑐 } . Then 𝐴 is a πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and hence πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Since 𝐴 βŠ‚ { 𝑏 , 𝑐 } and c l πœƒ ( 𝐴 ) = 𝑋 βŠ„ { 𝑏 , 𝑐 } , A is not πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Theorem 2.3. If A is a subset of an ideal space ( 𝑋 , 𝜏 , ℐ ) , then the following are equivalent.(a)A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b)For all π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) , c l ( { π‘₯ } ) ∩ 𝐴 β‰  πœ™ .(c) c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 contains no nonempty closed set.

Proof. ( π‘Ž ) β‡’ ( 𝑏 ) . Suppose π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) . If c l ( { π‘₯ } ) ∩ 𝐴 = πœ™ , then 𝐴 βŠ‚ 𝑋 βˆ’ c l ( { π‘₯ } ) . Since 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝐴 ) βŠ‚ 𝑋 βˆ’ c l ( { π‘₯ } ) . It is a contradiction to the fact that π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) . This proves (b).
( 𝑏 ) β‡’ ( 𝑐 ) . Suppose 𝐹 βŠ‚ c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 , 𝐹 is closed and π‘₯ ∈ 𝐹 . Since 𝐹 βŠ‚ 𝑋 βˆ’ 𝐴 and 𝐹 closed, c l ( { π‘₯ } ) ∩ 𝐴 βŠ‚ c l ( 𝐹 ) ∩ 𝐴 = 𝐹 ∩ 𝐴 = πœ™ . Since π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) , by (b), c l ( { π‘₯ } ) ∩ 𝐴 β‰  πœ™ , a contradiction which proves (c).
( 𝑐 ) β‡’ ( π‘Ž ) . Let π‘ˆ be an open set containing 𝐴 . Since c l ⋆ πœƒ ( 𝐴 ) is closed, c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) is closed and c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) βŠ‚ c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 . By hypothesis, c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) = πœ™ and hence c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ . Thus, 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.3, we get Corollary 2.4 which gives characterizations for πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.3, we get Corollary 2.5 which gives characterizations for 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets.

Corollary 2.4. If 𝐴 is a subset of a topological space ( 𝑋 , 𝜏 ) , then the following are equivalent.(a)A is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b)For all π‘₯ ∈ c l πœƒ ( 𝐴 ) , c l ( { π‘₯ } ) ∩ 𝐴 β‰  πœ™ .(c) c l πœƒ ( 𝐴 ) βˆ’ 𝐴 contains no nonempty closed set.

Corollary 2.5 (see [12, Theorem  2.2]). If 𝐴 is a subset of a topological space ( 𝑋 , 𝜏 ) , then the following are equivalent. (a) 𝐴 is 𝑔 - 𝑐 𝑙 o 𝑠 𝑒 𝑑 .(b)For all π‘₯ ∈ c l ( 𝐴 ) , c l ( { π‘₯ } ) ∩ 𝐴 β‰  πœ™ .(c) c l ( 𝐴 ) βˆ’ 𝐴 contains no nonempty closed set.

The following Corollary 2.6 shows that in 𝒯 1 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 , πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets are πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , the proof of which follows from Theorem 2.3(c). Corollary 2.7 gives the relation between πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets.

Corollary 2.6. If ( 𝑋 , 𝜏 , ℐ ) is a 𝒯 1 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 and A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 then A is a πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.

Corollary 2.7. If ( 𝑋 , 𝜏 , ℐ ) is an ideal space and A is a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set, then the following are equivalent.(a)A is a πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.(b) c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 is a closed set.

Proof. ( π‘Ž ) β‡’ ( 𝑏 ) . If 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 = πœ™ and so c l ⋆ πœƒ ( 𝐴 ) βˆ’ ( 𝐴 ) is closed.
( 𝑏 ) β‡’ ( π‘Ž ) . If c l ⋆ πœƒ ( 𝐴 ) βˆ’ ( 𝐴 ) is closed, since 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , by Theorem 2.3(c), c l ⋆ πœƒ ( 𝐴 ) βˆ’ ( 𝐴 ) = πœ™ and so 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Corollary 2.7, we get Corollary 2.8. If we put ℐ = β„˜ ( 𝑋 ) in Corollary 2.7, we get Corollary 2.9.

Corollary 2.8. If ( 𝑋 , 𝜏 , ) is a topological space and 𝐴 is a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set, then the following are equivalent.(a)A is a πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.(b) c l πœƒ ( 𝐴 ) βˆ’ 𝐴 is a closed set.

Corollary 2.9 (see [6, Corollary  2.3]). If ( 𝑋 , 𝜏 ) is an topological space and 𝐴 is a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set, then the following are equivalent.(a) 𝐴 is a closed set.(b) c l ( 𝐴 ) βˆ’ 𝐴 is a closed set.

Theorem 2.10. If every open set of an ideal space ( 𝑋 , 𝜏 , ℐ ) is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. Since every open set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ ( π‘ˆ ) = π‘ˆ for every π‘ˆ ∈ 𝜏 . Therefore, for every subset 𝐴 of X, i n t ⋆ πœƒ ( 𝐴 ) = βˆͺ { π‘ˆ ∈ 𝜏 ∣ c l ⋆ ( π‘ˆ ) βŠ‚ 𝐴 } = βˆͺ { π‘ˆ ∈ 𝜏 ∣ π‘ˆ βŠ‚ 𝐴 } = i n t ( 𝐴 ) . So c l ⋆ πœƒ ( 𝐴 ) = c l ( 𝐴 ) for every subset 𝐴 of 𝑋 . This implies that every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.11. If every subset of an ideal space ( 𝑋 , 𝜏 , ℐ ) is ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

The proof follows from Lemma 1.3 and Theorem 2.10.

Theorem 2.12. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space. Then every subset of X is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if every open set is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. Suppose every subset of 𝑋 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . If π‘ˆ is open, then π‘ˆ is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and so c l ⋆ πœƒ ( π‘ˆ ) βŠ‚ π‘ˆ . Hence π‘ˆ is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Conversely, suppose 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open. Since every open set is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ and so 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.12, we get Corollary 2.13. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.12, we get Corollary 2.14.

Corollary 2.13. Let ( 𝑋 , 𝜏 ) be a topological space. Then every subset of X is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if every open set is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.14 (see [6, Theorem  2.10]). Let ( 𝑋 , 𝜏 ) be a topological space. Then every subset of X is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if every open set is closed.

Theorem 2.15. If every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set of an ideal space ( 𝑋 , 𝜏 , ℐ ) is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then ( 𝑋 , 𝜏 ) is a 𝒯 1 𝑠 𝑝 π‘Ž 𝑐 𝑒 .

Proof. Suppose { π‘₯ } is not closed for some π‘₯ ∈ 𝑋 . Then, 𝐡 = 𝑋 βˆ’ { π‘₯ } is not open. So 𝐡 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By hypothesis, B is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Therefore, { π‘₯ } is πœƒ - π‘œ 𝑝 𝑒 𝑛 . So { π‘₯ } is both open and closed, a contradiction. Hence, ( 𝑋 , 𝜏 ) is a 𝒯 1 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 .

If we put ℐ = { πœ™ } in Theorem 2.15, we get Corollary 2.16.

Corollary 2.16. If every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set of a space ( 𝑋 , 𝜏 ) is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then ( 𝑋 , 𝜏 ) is a 𝒯 1 𝑠 𝑝 π‘Ž 𝑐 𝑒 .

Theorem 2.17. Intersection of a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set and a πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is always πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. Let 𝐴 be a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set and 𝐹 a πœƒ - ℐ - 𝑐 l π‘œ 𝑠 𝑒 𝑑 set of an ideal space ( 𝑋 , 𝜏 , ℐ ) . Suppose 𝐴 ∩ 𝐹 βŠ‚ π‘ˆ and π‘ˆ is open in 𝑋 . Then, 𝐴 βŠ‚ π‘ˆ βˆͺ ( 𝑋 βˆ’ 𝐹 ) . Now 𝑋 βˆ’ 𝐹 is πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 and hence open. So π‘ˆ βˆͺ ( 𝑋 βˆ’ 𝐹 ) is an open set containing 𝐴 . Since 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ βˆͺ ( 𝑋 βˆ’ 𝐹 ) . Therefore, c l ⋆ πœƒ ( 𝐴 ) ∩ 𝐹 βŠ‚ π‘ˆ which implies that c l ⋆ πœƒ ( 𝐴 ∩ 𝐹 ) βŠ‚ π‘ˆ . So 𝐴 ∩ 𝐹 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.17, we get Corollary 2.18. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.17, we get Corollary 2.19.

Corollary 2.18 (see [8, Proposition 3.11]). Intersection of a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set and a πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is always πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.19 (see [6, Corollary  2.7]). Intersection of a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set and a closed set is always a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.

Theorem 2.20. A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if c l ⋆ πœƒ ( 𝐴 ) βŠ‚ 𝐴 Ξ› .

Proof. Suppose 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) . If π‘₯ βˆ‰ 𝐴 Ξ› , then there exists an open set π‘ˆ such that 𝐴 βŠ‚ π‘ˆ , but π‘₯ βˆ‰ π‘ˆ . Since 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ and so π‘₯ βˆ‰ c l ⋆ πœƒ ( 𝐴 ) , a contradiction. Therefore, c l ⋆ πœƒ ( 𝐴 ) βŠ‚ 𝐴 Ξ› . Conversely, suppose that c l ⋆ πœƒ ( 𝐴 ) βŠ‚ 𝐴 Ξ› . If 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open, then 𝐴 Ξ› βŠ‚ π‘ˆ and so c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ . Therefore, 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.20, we get Corollary 2.21. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.20, we get Corollary 2.22.

Corollary 2.21. A subset A of a space ( 𝑋 , 𝜏 ) is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if c l πœƒ ( 𝐴 ) βŠ‚ 𝐴 Ξ› .

Corollary 2.22. A subset A of a space ( 𝑋 , 𝜏 ) is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if c l ( 𝐴 ) βŠ‚ 𝐴 Ξ› .

Theorem 2.23. Let A be a Ξ› - 𝑠 𝑒 𝑑 of an ideal space ( 𝑋 , 𝜏 , ℐ ) . Then A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. Suppose 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By Theorem 2.20, c l ⋆ πœƒ ( 𝐴 ) βŠ‚ 𝐴 Ξ› = 𝐴 , since 𝐴 is a Ξ› - 𝑠 𝑒 𝑑 . Therefore, 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Converse follows from the fact that every πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.23, we get Corollary 2.24. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.23, we get Corollary 2.25.

Corollary 2.24. Let A be a Ξ› - 𝑠 𝑒 𝑑 of a space ( 𝑋 , 𝜏 ) . Then A is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if A is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.25. Let A be a Ξ› - 𝑠 𝑒 𝑑 of a space ( 𝑋 , 𝜏 ) . Then A is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if A is closed.

Theorem 2.26. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 βŠ‚ 𝑋 . If 𝐴 Ξ› is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then A is also πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. Suppose that 𝐴 Ξ› is a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. If 𝐴 βŠ‚ π‘ˆ and π‘ˆ is open, then 𝐴 Ξ› βŠ‚ π‘ˆ . Since 𝐴 Ξ› is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝐴 Ξ› ) βŠ‚ π‘ˆ . But, c l ⋆ πœƒ ( 𝐴 ) βŠ‚ c l ⋆ πœƒ ( 𝐴 Ξ› ) . Therefore, A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.26, we get Corollary 2.27. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.26, we get Corollary 2.28.

Corollary 2.27. Let ( 𝑋 , 𝜏 ) be a topological space and 𝐴 βŠ‚ 𝑋 . If 𝐴 Ξ› is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then A is also πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.28. Let ( 𝑋 , 𝜏 ) be a space and 𝐴 βŠ‚ 𝑋 . If 𝐴 Ξ› is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set, then A is also 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Theorem 2.29. For an ideal space ( 𝑋 , 𝜏 , ℐ ) , the following are equivalent.(a)Every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b)Every singleton of 𝑋 is closed or πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 .

Proof. ( π‘Ž ) β‡’ ( 𝑏 ) . Let π‘₯ ∈ 𝑋 . If { π‘₯ } is not closed, then 𝐴 = 𝑋 βˆ’ { π‘₯ } βˆ‰ 𝜏 and then 𝐴 is trivially πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By (a), 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Hence { π‘₯ } is πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 .
( 𝑏 ) β‡’ ( π‘Ž ) . Let 𝐴 be a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set and let π‘₯ ∈ c l ⋆ πœƒ ( 𝐴 ) . We have the following cases.
Case 1. { π‘₯ } is closed. By Theorem 2.3, c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 does not contain a nonempty closed subset. This shows { π‘₯ } ∈ 𝐴 .Case 2. { π‘₯ } is πœƒ - ℐ - π‘œ 𝑝 𝑒 𝑛 . Then, { π‘₯ } ∩ 𝐴 β‰  πœ™ . Hence, π‘₯ ∈ 𝐴 .
Thus in both cases π‘₯ ∈ 𝐴 and so 𝐴 = c l ⋆ πœƒ ( 𝐴 ) , that is, 𝐴 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , which proves (a).

If we put ℐ = { πœ™ } in Theorem 2.29, we get Corollary 2.30. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.29, we get Corollary 2.31.

Corollary 2.30. For an ideal space ( 𝑋 , 𝜏 ) , the following are equivalent.(a)Every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b)Every singleton of 𝑋 is closed or πœƒ - π‘œ 𝑝 𝑒 𝑛 .

Corollary 2.31 (see [13, Theorem  2.5]). For an ideal space ( 𝑋 , 𝜏 ) , the following are equivalent.(a)Every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is closed.(b)Every singleton of 𝑋 is closed or open.

Theorem 2.32. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 βŠ‚ 𝑋 . Then A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if 𝐴 = 𝐹 βˆ’ 𝑁 , where 𝐹 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and 𝑁 contains no nonempty closed set.

Proof. If 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , then by Theorem 2.3, 𝑁 = c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 contains no nonempty closed set. If 𝐹 = c l ⋆ πœƒ ( 𝐴 ) , then 𝐹 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 such that 𝐹 βˆ’ 𝑁 = c l ⋆ πœƒ ( 𝐴 ) βˆ’ ( c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 ) = c l ⋆ πœƒ ( 𝐴 ) ∩ ( ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) βˆͺ 𝐴 ) = 𝐴 . Conversely, suppose 𝐴 = 𝐹 βˆ’ 𝑁 , where 𝐹 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and 𝑁 contains no nonempty closed set. Let π‘ˆ be an open set such that 𝐴 βŠ‚ π‘ˆ . Then, 𝐹 βˆ’ 𝑁 βŠ‚ π‘ˆ which implies that 𝐹 ∩ ( 𝑋 βˆ’ π‘ˆ ) βŠ‚ 𝑁 . Now, 𝐴 βŠ‚ 𝐹 and 𝐹 is πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 implies that c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) βŠ‚ c l ⋆ πœƒ ( 𝐹 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) βŠ‚ 𝐹 ∩ ( 𝑋 βˆ’ π‘ˆ ) βŠ‚ 𝑁 . Since πœƒ - ℐ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 sets are closed, c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) is closed. By hypothesis, c l ⋆ πœƒ ( 𝐴 ) ∩ ( 𝑋 βˆ’ π‘ˆ ) = πœ™ and so c l ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ , which implies that 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.32, we get Corollary 2.33. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.32, we get Corollary 2.34.

Corollary 2.33. Let ( 𝑋 , 𝜏 ) be a space and 𝐴 βŠ‚ 𝑋 . Then A is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 subset of X if and only if 𝐴 = 𝐹 βˆ’ 𝑁 , where F is πœƒ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and N contains no nonempty closed set.

Corollary 2.34 (see [12, Corollary  2.3]). Let ( 𝑋 , 𝜏 ) be a space and 𝐴 βŠ‚ 𝑋 . Then A is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 if and only if 𝐴 = 𝐹 βˆ’ 𝑁 , where F is closed and N contains no nonempty closed set.

Theorem 2.35. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space. If A is a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 subset of X and 𝐴 βŠ‚ 𝐡 βŠ‚ c l ⋆ πœƒ ( 𝐴 ) , then B is also πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. c l ⋆ πœƒ ( 𝐡 ) βˆ’ 𝐡 βŠ‚ c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 , and since c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 has no nonempty closed subset, neither does c l ⋆ πœƒ ( 𝐡 ) βˆ’ 𝐡 . By Theorem 2.3, 𝐡 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

If we put ℐ = { πœ™ } in Theorem 2.35, we get Corollary 2.36. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.35, we get Corollary 2.37.

Corollary 2.36. Let ( 𝑋 , 𝜏 ) be a space. If 𝐴 is a πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 subset of X and 𝐴 βŠ‚ 𝐡 βŠ‚ c l πœƒ ( 𝐴 ) , then 𝐡 is also πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Corollary 2.37 (see [6, Theorem  2.8]). Let ( 𝑋 , 𝜏 ) be a space. If A is a 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 subset of X and 𝐴 βŠ‚ 𝐡 βŠ‚ c l ( 𝐴 ) , then B is also 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

A subset 𝐴 of an ideal space ( 𝑋 , 𝜏 , ℐ ) is said to be πœƒ - ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 if 𝑋 βˆ’ 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Theorem 2.38. A subset A of an ideal space ( 𝑋 , 𝜏 , ℐ ) is πœƒ - ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 if and only if 𝐹 βŠ‚ i n t ⋆ πœƒ ( 𝐴 ) whenever F is closed and 𝐹 βŠ‚ 𝐴 .

Proof. Suppose 𝐴 is a πœƒ - ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 set and 𝐹 is a closed set contained in 𝐴 , then 𝑋 βˆ’ 𝐴 βŠ‚ 𝑋 βˆ’ 𝐹 and 𝑋 βˆ’ 𝐹 is open. Since 𝑋 βˆ’ 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) βŠ‚ ( 𝑋 βˆ’ 𝐹 ) and so 𝐹 βŠ‚ 𝑋 βˆ’ c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) = i n t ⋆ πœƒ ( 𝐴 ) . Conversely, suppose X βˆ’ 𝐴 βŠ‚ π‘ˆ and 𝑋 βˆ’ π‘ˆ is closed. By hypothesis, 𝑋 βˆ’ π‘ˆ βŠ‚ i n t ⋆ πœƒ ( 𝐴 ) , which implies that c l ⋆ πœƒ ( 𝑋 βˆ’ 𝐴 ) = 𝑋 βˆ’ i n t ⋆ πœƒ ( 𝐴 ) βŠ‚ π‘ˆ . Therefore, 𝑋 βˆ’ 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 and hence 𝐴 is πœƒ - ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 .

If we put ℐ = { πœ™ } in Theorem 2.38, we get Corollary 2.39. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.38, we get Corollary 2.40.

Corollary 2.39. A subset A of a space ( 𝑋 , 𝜏 ) is πœƒ - 𝑔 - π‘œ 𝑝 𝑒 𝑛 if and only if 𝐹 βŠ‚ i n t πœƒ ( 𝐴 ) whenever F is closed and 𝐹 βŠ‚ 𝐴 .

Corollary 2.40 (see [6, Theorem  4.2]). A subset A of a space ( 𝑋 , 𝜏 ) is 𝑔 - π‘œ 𝑝 𝑒 𝑛 if and only if 𝐹 βŠ‚ i n t ( 𝐴 ) whenever F is closed and 𝐹 βŠ‚ 𝐴 .

Theorem 2.41. Let ( 𝑋 , 𝜏 , ℐ ) be an ideal space and 𝐴 βŠ‚ π‘ˆ . Then the following are equivalent.(a)A is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b) 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(c) c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 is πœƒ - ℐ 𝑔 - π‘œ 𝑝 𝑒 𝑛 .

Proof. ( π‘Ž ) β‡’ ( 𝑏 ) . Suppose 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . If π‘ˆ is any open set containing 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) , then 𝑋 βˆ’ π‘ˆ βŠ‚ 𝑋 βˆ’ ( 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) = c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 . Since 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , by Theorem 2.3(c), it follows that 𝑋 βˆ’ π‘ˆ = πœ™ and so 𝑋 = π‘ˆ . Since 𝑋 is the only open set containing 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) , 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .
( 𝑏 ) β‡’ ( π‘Ž ) . Suppose 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . If 𝐹 is any closed set contained in c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 , then 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) βŠ‚ 𝑋 βˆ’ 𝐹 and 𝑋 βˆ’ 𝐹 is open. Therefore, c l ⋆ πœƒ ( 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) βŠ‚ 𝑋 βˆ’ 𝐹 , which implies that c l ⋆ πœƒ ( 𝐴 ) βˆͺ c l ⋆ πœƒ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) βŠ‚ 𝑋 βˆ’ 𝐹 and so 𝑋 βŠ‚ 𝑋 βˆ’ 𝐹 ; it follows that 𝐹 = πœ™ . Hence 𝐴 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .
The equivalence of (b) and (c) follows from the fact that 𝑋 βˆ’ ( c l ⋆ πœƒ ( 𝐴 ) βˆ’ 𝐴 ) = 𝐴 βˆͺ ( 𝑋 βˆ’ c l ⋆ πœƒ ( 𝐴 ) ) .

If we put ℐ = { πœ™ } in Theorem 2.41, we get Corollary 2.42. If we put ℐ = β„˜ ( 𝑋 ) in Theorem 2.41, we get Corollary 2.43.

Corollary 2.42. Let ( 𝑋 , 𝜏 ) be a space and 𝐴 βŠ‚ π‘ˆ . Then the following are equivalent.(a)A is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b) 𝐴 βˆͺ ( 𝑋 βˆ’ c l πœƒ ( 𝐴 ) ) is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(c) c l πœƒ ( 𝐴 ) βˆ’ 𝐴 is πœƒ - 𝑔 - π‘œ 𝑝 𝑒 𝑛 .

Corollary 2.43. Let ( 𝑋 , 𝜏 ) be an ideal space and 𝐴 βŠ‚ π‘ˆ . Then the following are equivalent.(a)A is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b) 𝐴 βˆͺ ( 𝑋 βˆ’ c l ( 𝐴 ) ) is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(c) c l ( 𝐴 ) βˆ’ 𝐴 is 𝑔 - π‘œ 𝑝 𝑒 𝑛 .

3. Characterization of 𝒯 1 / 2 and 𝒯 ℐ -Space

Theorem 3.1. In an ideal space ( 𝑋 , 𝜏 , ℐ ) , the following are equivalent.(a)Every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is closed.(b) ( 𝑋 , 𝜏 ) is a 𝒯 1 / 2 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 .(c)Every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is closed.

Proof. ( π‘Ž ) ⇔ ( 𝑏 ) . Equivalence of (a) and (b) follows from Theorem  4.1 of [8].
( 𝑏 ) β‡’ ( 𝑐 ) . Let 𝐴 be a πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set. Since every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 , A is 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By hypothesis, A is closed.
( 𝑐 ) β‡’ ( 𝑏 ) . Let π‘₯ ∈ 𝑋 . If { π‘₯ } is not closed, then 𝐡 = 𝑋 βˆ’ { π‘₯ } is not open. So 𝐡 is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By hypothesis, B is closed and so { π‘₯ } is open. By Corollary 2.31, ( 𝑋 , 𝜏 ) is a 𝒯 1 / 2 - 𝑠 𝑝 π‘Ž 𝑐 𝑒 .

Theorem 3.2. In an ideal space ( 𝑋 , 𝜏 , ℐ ) the following, are equivalent.(a)Every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b) ( 𝑋 , 𝜏 , ℐ ) is a 𝒯 ℐ - 𝑆 𝑝 π‘Ž 𝑐 𝑒 .(c)Every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .

Proof. ( π‘Ž ) β‡’ ( 𝑏 ) . Let π‘₯ ∈ 𝑋 . If { π‘₯ } is not closed, then 𝑋 is the only open set containing 𝑋 βˆ’ { π‘₯ } and so 𝑋 βˆ’ { π‘₯ } is πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By hypothesis, 𝑋 βˆ’ { π‘₯ } is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Equivalently { π‘₯ } is ⋆ - π‘œ 𝑝 𝑒 𝑛 . Thus, every singleton set in 𝑋 is either closed or ⋆ - π‘œ 𝑝 𝑒 𝑛 . By Theorem  3.3 of [4], ( 𝑋 , 𝜏 , ℐ ) is a 𝒯 ℐ - 𝑆 𝑝 π‘Ž 𝑐 𝑒 .
( 𝑏 ) β‡’ ( π‘Ž ) . The proof follows from the fact that every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .
( 𝑏 ) β‡’ ( 𝑐 ) . The proof follows from the fact that every set is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .
( 𝑐 ) β‡’ ( 𝑏 ) . Let π‘₯ ∈ 𝑋 . If { π‘₯ } is not closed, then 𝑋 is the only open set containing π‘₯ βˆ’ { π‘₯ } and so π‘₯ βˆ’ { π‘₯ } is πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . By hypothesis, 𝑋 βˆ’ { π‘₯ } is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 . Thus, { π‘₯ } is ⋆ - π‘œ 𝑝 𝑒 𝑛 . Therefore, every singleton set in 𝑋 is either ⋆ - π‘œ 𝑝 𝑒 𝑛 or closed. By Theorem of  3.3 [4], ( 𝑋 , 𝜏 , ℐ ) is a 𝒯 ℐ - 𝑆 𝑝 π‘Ž 𝑐 𝑒 .

The proof of the Corollary 3.3 follows from Theorem  3.2 and Theorem  3.10 of [5].

If we put ℐ = { πœ™ } in Corollary 3.3, we get Corollary 3.4.

Corollary 3.3. In an ideal space ( 𝑋 , 𝜏 , ℐ ) , the following are equivalent.(a)Every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(b)Every πœƒ - ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 .(c)Every ℐ 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is an ℐ - 𝑙 π‘œ 𝑐 π‘Ž 𝑙 𝑙 𝑦 ⋆ - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set.

Corollary 3.4. In a topological space ( 𝑋 , 𝜏 ) , the following are equivalent.(a)Every πœƒ - 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is closed.(b)Every 𝑔 - 𝑐 𝑙 π‘œ 𝑠 𝑒 𝑑 set is a locally closed set.

References

  1. K. Kuratowski, Topology, vol. 1, Academic Press, New York, NY, USA, 1966.
  2. R. Vaidyanathaswamy, Set Topology, Chelsea Publishing, New York, NY, USA, 1946.
  3. D. Janković and T. R. Hamlett, β€œNew topologies from old via ideals,” The American Mathematical Monthly, vol. 97, no. 4, pp. 295–310, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. J. Dontchev, M. Ganster, and T. Noiri, β€œUnified operation approach of generalized closed sets via topological ideals,” Mathematica Japonica, vol. 49, no. 3, pp. 395–401, 1999. View at Zentralblatt MATH
  5. M. Navaneethakrishnan and D. Sivaraj, β€œGeneralized locally closed sets in ideal topological spaces,” Bulletin of the Allahabad Mathematical Society, vol. 24, no. 1, pp. 13–19, 2009. View at Zentralblatt MATH
  6. N. Levine, β€œGeneralized closed sets in topology,” Rendiconti del Circolo Matematico di Palermo, vol. 19, no. 2, pp. 89–96, 1970. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. N. V. Veličko, β€œH-closed topological spaces,” Matematicheskii Sbornik, vol. 70, no. 112, pp. 98–112, 1966.
  8. J. Dontchev and H. Maki, β€œOn θ-generalized closed sets,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 2, pp. 239–249, 1999. View at Publisher Β· View at Google Scholar
  9. H. Maki, J. Umehara, and K. Yamamura, β€œCharacterizations of T1/2−spaces using generalized V-sets,” Indian Journal of Pure and Applied Mathematics, vol. 19, no. 7, pp. 634–640, 1988.
  10. M. Mršević, β€œOn pairwise ℛ and pairwise ℛ∞ bitopological spaces,” Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, vol. 30(78), no. 2, pp. 141–148, 1986. View at Zentralblatt MATH
  11. M. Akdag, β€œθ−I−open sets,” Kochi Journal of Mathematics, vol. 3, pp. 217–229, 2008.
  12. W. Dunham and N. Levine, β€œFurther results on generalized closed sets in topology,” Kyungpook Mathematical Journal, vol. 20, no. 2, pp. 169–175, 1980. View at Zentralblatt MATH
  13. W. Dunham, β€œT1/2−spaces,” Kyungpook Mathematical Journal, vol. 17, no. 2, pp. 161–169, 1977. View at Zentralblatt MATH