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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 682829, 9 pages
http://dx.doi.org/10.5402/2012/682829
Research Article

πœƒ-ℐ𝑔-Closed Sets

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-ℐ𝑔-Closed sets and discuss their properties. Using these sets, we characterize 𝒯1/2-π‘ π‘π‘Žπ‘π‘’π‘  and 𝒯ℐ-π‘†π‘π‘Žπ‘π‘’π‘ .

1. Introduction and Preliminaries

An ideal ℐ on a topological space (𝑋,𝜏) is a nonempty collection of subsets of 𝑋 which satisfies (i) π΄βˆˆβ„ and π΅βŠ‚π΄ implies π΅βˆˆβ„ and (ii) 𝐴,π΅βˆˆβ„ implies 𝐴βˆͺπ΅βˆˆβ„. 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‹†βˆΆβ„˜(𝑋)β†’β„˜(𝑋) called a local function [1] of 𝐴 with respect to 𝜏 and ℐ is defined as follows: for π΄βŠ‚π‘‹, 𝐴⋆(𝑋,𝜏)={π‘₯βˆˆπ‘‹βˆ£π‘ˆβˆ©π΄βˆ‰β„,foreveryπ‘ˆβˆˆπœ(π‘₯)}, where 𝜏(π‘₯)={π‘ˆβˆˆπœβˆ£π‘₯βˆˆπ‘ˆ}. A Kuratowski closure operator cl⋆(β‹…) for a topology πœβ‹†(ℐ,𝜏) called the ⋆-topology, finer than 𝜏, is defined by cl⋆(𝐴)=𝐴βˆͺ𝐴⋆(ℐ,𝜏) [2]. When there is no confusion we will simply write 𝐴⋆ for 𝐴⋆(ℐ,𝜏) and πœβ‹† for πœβ‹†(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 π΄β‹†βŠ‚π΄. A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is said to be an ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ [4] if π΄β‹†βŠ‚π‘ˆ whenever π΄βŠ‚π‘ˆ and π‘ˆ is open. A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is said to be ℐ𝑔-π‘œπ‘π‘’π‘› if π‘‹βˆ’π΄ is ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. An ideal space (𝑋,𝜏,ℐ) is said to be a 𝒯ℐ-space [4] if every ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘. 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 ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘ set 𝐹 such that 𝐴=π‘ˆβˆ©πΉ. If ℐ={βˆ…}, then ℐ-π‘™π‘œπ‘π‘Žπ‘™π‘™π‘¦β‹†-π‘π‘™π‘œπ‘ π‘’π‘‘ sets coincide with locally closed sets.

By a space, we always mean a topological space (𝑋,𝜏) with no separation properties assumed. If π΄βŠ‚π‘‹,cl(𝐴) and int(𝐴) will, respectively, denote the closure and interior of 𝐴 in (𝑋,𝜏) and int⋆(𝐴) will denote the interior of 𝐴 in (𝑋,πœβ‹†). A subset 𝐴 of a topological space (𝑋,𝜏) is said to be a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set [6] if cl(𝐴)βŠ‚π‘ˆ whenever π΄βŠ‚π‘ˆ and π‘ˆ is open. A subset 𝐴 of a topological space (𝑋,𝜏) is said to be a 𝑔-π‘œπ‘π‘’π‘› set if π‘‹βˆ’π΄ is a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set. A space (𝑋,𝜏) is said to be a 𝒯1/2-π‘ π‘π‘Žπ‘π‘’ [6] if every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is a closed set.

For a subset 𝐴 of a space (𝑋,𝜏), the πœƒ-π‘–π‘›π‘‘π‘’π‘Ÿπ‘–π‘œπ‘Ÿ [7] of 𝐴 is the union of all open sets of 𝑋 whose closures contained in 𝐴 and is denoted by intπœƒ(𝐴). The subset 𝐴 is called πœƒ-π‘œπ‘π‘’π‘› if 𝐴=intπœƒ(𝐴). The complement of a πœƒ-π‘œπ‘π‘’π‘› set is called a πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘ set. Equivalently, π΄βŠ‚π‘‹ is called πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘ [7] if 𝐴=clπœƒ(𝐴), where clπœƒ(𝐴)={π‘₯βˆˆπ‘‹βˆ£cl(π‘ˆ)βˆ©π΄β‰ βˆ…forallπ‘ˆβˆˆπœ(π‘₯)}. The family of all πœƒ-π‘œπ‘π‘’π‘› sets of 𝑋 forms a topology [7] on 𝑋, which is coarser than 𝜏 and is denoted by πœπœƒ. A subset 𝐴 of a topological space (𝑋,𝜏) is said to be a πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘π‘ π‘’π‘‘ [8] if clπœƒ(𝐴)βŠ‚π‘ˆ whenever π΄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‘œπ‘π‘’π‘›π‘ π‘’π‘‘ [8] if π‘‹βˆ’π΄ is a πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set. A subset 𝐴 of a space (𝑋,𝜏) is said to be a Ξ›-𝑠𝑒𝑑 [9, 10] if 𝐴=𝐴Λ, where 𝐴Λ=∩{π‘ˆβˆˆπœβˆ£π΄βŠ‚π‘ˆ}.

A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is said to be πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ [11] if clβ‹†πœƒ(𝐴)=𝐴, where clβ‹†πœƒ(𝐴)={π‘₯βˆˆπ‘‹βˆ£π΄βˆ©cl⋆(π‘ˆ)β‰ πœ™forallπ‘ˆβˆˆπœ(π‘₯)}. 𝐴 is said to be πœƒ-ℐ-π‘œπ‘π‘’π‘› if π‘‹βˆ’π΄ is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘. If ℐ={βˆ…}, clβ‹†πœƒ(𝐴)=clπœƒ(𝐴). If ℐ=β„˜(𝑋), clβ‹†πœƒ(𝐴)=cl(𝐴). For a subset 𝐴 of X, intπœƒπΌ(𝐴)=βˆͺ{π‘ˆβˆˆπœβˆ£cl⋆(π‘ˆ)βŠ‚π΄} [11]. We denote this intπœƒπΌ(𝐴) by intβ‹†πœƒ(𝐴). The family of all πœƒ-ℐ-π‘œπ‘π‘’π‘› sets of (𝑋,𝜏,ℐ) is a topology and it is denoted by πœπœƒ-ℐ (see [11, Theorem  1]).

Lemma 1.1 (see [11, Corollary  4 if Theorem  2]). πœπœƒβŠ‚πœπœƒ-β„βŠ‚πœ.

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 πœπœƒ-ℐ=πœπœƒ,(2)if ℐ={πœ™}, then πœ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 ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if every open set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘.

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‘œπ‘π‘’π‘› if and only if intβ‹†πœƒ(𝐴)=𝐴.

Lemma 1.5. Let (𝑋,𝜏,ℐ) be an ideal space and 𝐴 a subset of 𝑋. Then clβ‹†πœƒ(𝐴)={π‘₯βˆˆπ‘‹βˆ£π‘ˆβˆ©cl⋆(𝐴)β‰ πœ™ for all π‘ˆβˆˆπœ(π‘₯)} is closed.

Proof. If π‘₯∈cl(clβ‹†πœƒ(𝐴)) and π‘ˆβˆˆπœ(π‘₯), then π‘ˆβˆ©clβ‹†πœƒ(𝐴)β‰ πœ™. Then, π‘¦βˆˆπ‘ˆβˆ©clβ‹†πœƒ(𝐴) for some π‘¦βˆˆπ‘‹. Since π‘ˆβˆˆπœ(𝑦) and π‘¦βˆˆclβ‹†πœƒ(𝐴), from the definition of clβ‹†πœƒ(𝐴) we have 𝐴∩cl⋆(π‘ˆ)β‰ πœ™. Therefore, π‘₯∈clβ‹†πœƒ(𝐴). So cl(clβ‹†πœƒ(𝐴))βŠ‚clβ‹†πœƒ(𝐴) and hence clβ‹†πœƒ(𝐴) is closed.

Lemma 1.6. Let (𝑋,𝜏,ℐ) be an ideal space and 𝐴 a subset of 𝑋. Then, π‘‹βˆ’clβ‹†πœƒ(π‘‹βˆ’π΄)=intβ‹†πœƒ(𝐴).

Proof. π‘₯βˆˆπ‘‹βˆ’clβ‹†πœƒ(π‘‹βˆ’π΄) if and only if π‘₯βˆ‰clβ‹†πœƒ(π‘‹βˆ’π΄) if and only if there exist π‘ˆβˆˆπœ(π‘₯) such that (π‘‹βˆ’π΄)∩cl⋆(π‘ˆ)=πœ™ if and only if π‘₯βˆˆπ‘ˆ and, cl⋆(π‘ˆ)βŠ‚(𝐴) if and only if π‘₯βˆˆπ‘ˆβŠ‚intβ‹†πœƒ(𝐴).

2. πœƒ-ℐ𝑔-Closed Sets

A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is said to be a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set if clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ whenever π΄βŠ‚π‘ˆ and π‘ˆ is open. Every πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ set is a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set. If ℐ={βˆ…}, then clβ‹†πœƒ(𝐴)=clπœƒ(𝐴) and hence πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets coincide with πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets. If ℐ=β„˜(𝑋), then clβ‹†πœƒ(𝐴)=cl(𝐴) and hence πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets coincide with 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets. Since cl⋆(𝐴)βŠ‚cl(𝐴)βŠ‚clβ‹†πœƒ(𝐴)βŠ‚clπœƒ(𝐴), we have the following inclusion diagram: πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘βŸΆπœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘βŸΆπ‘”-π‘π‘™π‘œπ‘ π‘’π‘‘βŸΆβ„π‘”-π‘π‘™π‘œπ‘ π‘’π‘‘.(2.1)

Example 2.1. shows that a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set needs not to be πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, and Example 2.2 shows that πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set needs not to be a πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ 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-π‘π‘™π‘œπ‘ π‘’π‘‘. But 𝐴 is not πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ because π΄βŠ‚{𝑏,𝑐} and clβ‹†πœƒ(𝐴)=π‘‹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-π‘π‘™π‘œπ‘ π‘’π‘‘ and hence πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. Since π΄βŠ‚{𝑏,𝑐} and clπœƒ(𝐴)=π‘‹βŠ„{𝑏,𝑐}, A is not πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)For all π‘₯∈clβ‹†πœƒ(𝐴), cl({π‘₯})βˆ©π΄β‰ πœ™.(c)clβ‹†πœƒ(𝐴)βˆ’π΄ contains no nonempty closed set.

Proof. (π‘Ž)β‡’(𝑏). Suppose π‘₯∈clβ‹†πœƒ(𝐴). If cl({π‘₯})∩𝐴=πœ™, then π΄βŠ‚π‘‹βˆ’cl({π‘₯}). Since 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(𝐴)βŠ‚π‘‹βˆ’cl({π‘₯}). It is a contradiction to the fact that π‘₯∈clβ‹†πœƒ(𝐴). This proves (b).
(𝑏)β‡’(𝑐). Suppose πΉβŠ‚clβ‹†πœƒ(𝐴)βˆ’π΄, 𝐹 is closed and π‘₯∈𝐹. Since πΉβŠ‚π‘‹βˆ’π΄ and 𝐹 closed, cl({π‘₯})βˆ©π΄βŠ‚cl(𝐹)∩𝐴=𝐹∩𝐴=πœ™. Since π‘₯∈clβ‹†πœƒ(𝐴), by (b), cl({π‘₯})βˆ©π΄β‰ πœ™, a contradiction which proves (c).
(𝑐)β‡’(π‘Ž). Let π‘ˆ be an open set containing 𝐴. Since clβ‹†πœƒ(𝐴) is closed, clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ) is closed and clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ)βŠ‚clβ‹†πœƒ(𝐴)βˆ’π΄. By hypothesis, clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ)=πœ™ and hence clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ. Thus, 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

If we put ℐ={πœ™} in Theorem 2.3, we get Corollary 2.4 which gives characterizations for πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets. If we put ℐ=β„˜(𝑋) in Theorem 2.3, we get Corollary 2.5 which gives characterizations for 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ 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-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)For all π‘₯∈clπœƒ(𝐴), cl({π‘₯})βˆ©π΄β‰ πœ™.(c)clπœƒ(𝐴)βˆ’π΄ 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-𝑐𝑙o𝑠𝑒𝑑.(b)For all π‘₯∈cl(𝐴), cl({π‘₯})βˆ©π΄β‰ πœ™.(c)cl(𝐴)βˆ’π΄ contains no nonempty closed set.

The following Corollary 2.6 shows that in 𝒯1-π‘ π‘π‘Žπ‘π‘’, πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ sets are πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘, the proof of which follows from Theorem 2.3(c). Corollary 2.7 gives the relation between πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ and πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ sets.

Corollary 2.6. If (𝑋,𝜏,ℐ) is a 𝒯1-π‘ π‘π‘Žπ‘π‘’ and A is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ then A is a πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ set.

Corollary 2.7. If (𝑋,𝜏,ℐ) is an ideal space and A is a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set, then the following are equivalent.(a)A is a πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ set.(b)clβ‹†πœƒ(𝐴)βˆ’π΄ is a closed set.

Proof. (π‘Ž)β‡’(𝑏). If 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘, then clβ‹†πœƒ(𝐴)βˆ’π΄=πœ™ and so clβ‹†πœƒ(𝐴)βˆ’(𝐴) is closed.
(𝑏)β‡’(π‘Ž). If clβ‹†πœƒ(𝐴)βˆ’(𝐴) is closed, since 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, by Theorem 2.3(c), clβ‹†πœƒ(𝐴)βˆ’(𝐴)=πœ™ and so 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ set, then the following are equivalent.(a)A is a πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘ set.(b)clπœƒ(𝐴)βˆ’π΄ is a closed set.

Corollary 2.9 (see [6, Corollary  2.3]). If (𝑋,𝜏) is an topological space and 𝐴 is a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set, then the following are equivalent.(a)𝐴 is a closed set.(b)cl(𝐴)βˆ’π΄ is a closed set.

Theorem 2.10. If every open set of an ideal space (𝑋,𝜏,ℐ) is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘, then every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. Since every open set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘, cl⋆(π‘ˆ)=π‘ˆ for every π‘ˆβˆˆπœ. Therefore, for every subset 𝐴 of X, intβ‹†πœƒ(𝐴)=βˆͺ{π‘ˆβˆˆπœβˆ£cl⋆(π‘ˆ)βŠ‚π΄}=βˆͺ{π‘ˆβˆˆπœβˆ£π‘ˆβŠ‚π΄}=int(𝐴). So clβ‹†πœƒ(𝐴)=cl(𝐴) for every subset 𝐴 of 𝑋. This implies that every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.11. If every subset of an ideal space (𝑋,𝜏,ℐ) is ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, then every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if every open set is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. Suppose every subset of 𝑋 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. If π‘ˆ is open, then π‘ˆ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ and so clβ‹†πœƒ(π‘ˆ)βŠ‚π‘ˆ. Hence π‘ˆ is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘. Conversely, suppose π΄βŠ‚π‘ˆ and π‘ˆ is open. Since every open set is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ and so 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if every open set is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.14 (see [6, Theorem  2.10]). Let (𝑋,𝜏) be a topological space. Then every subset of X is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if every open set is closed.

Theorem 2.15. If every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set of an ideal space (𝑋,𝜏,ℐ) is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘, then (𝑋,𝜏) is a 𝒯1π‘ π‘π‘Žπ‘π‘’.

Proof. Suppose {π‘₯} is not closed for some π‘₯βˆˆπ‘‹. Then, 𝐡=π‘‹βˆ’{π‘₯} is not open. So 𝐡 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By hypothesis, B is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘. Therefore, {π‘₯} is πœƒ-π‘œπ‘π‘’π‘›. So {π‘₯} is both open and closed, a contradiction. Hence, (𝑋,𝜏) is a 𝒯1-π‘ π‘π‘Žπ‘π‘’.

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

Corollary 2.16. If every πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set of a space (𝑋,𝜏) is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘, then (𝑋,𝜏) is a 𝒯1π‘ π‘π‘Žπ‘π‘’.

Theorem 2.17. Intersection of a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set and a πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ set is always πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. Let 𝐴 be a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set and 𝐹 a πœƒ-ℐ-𝑐lπ‘œπ‘ π‘’π‘‘ set of an ideal space (𝑋,𝜏,ℐ). Suppose π΄βˆ©πΉβŠ‚π‘ˆ and π‘ˆ is open in 𝑋. Then, π΄βŠ‚π‘ˆβˆͺ(π‘‹βˆ’πΉ). Now π‘‹βˆ’πΉ is πœƒ-ℐ-π‘œπ‘π‘’π‘› and hence open. So π‘ˆβˆͺ(π‘‹βˆ’πΉ) is an open set containing 𝐴. Since 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(𝐴)βŠ‚π‘ˆβˆͺ(π‘‹βˆ’πΉ). Therefore, clβ‹†πœƒ(𝐴)βˆ©πΉβŠ‚π‘ˆ which implies that clβ‹†πœƒ(𝐴∩𝐹)βŠ‚π‘ˆ. So 𝐴∩𝐹 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ set and a πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘ set is always πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.19 (see [6, Corollary  2.7]). Intersection of a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set and a closed set is always a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set.

Theorem 2.20. A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if clβ‹†πœƒ(𝐴)βŠ‚π΄Ξ›.

Proof. Suppose 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ and π‘₯∈clβ‹†πœƒ(𝐴). If π‘₯βˆ‰π΄Ξ›, then there exists an open set π‘ˆ such that π΄βŠ‚π‘ˆ, but π‘₯βˆ‰π‘ˆ. Since 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ and so π‘₯βˆ‰clβ‹†πœƒ(𝐴), a contradiction. Therefore, clβ‹†πœƒ(𝐴)βŠ‚π΄Ξ›. Conversely, suppose that clβ‹†πœƒ(𝐴)βŠ‚π΄Ξ›. If π΄βŠ‚π‘ˆ and π‘ˆ is open, then π΄Ξ›βŠ‚π‘ˆ and so clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ. Therefore, 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if clπœƒ(𝐴)βŠ‚π΄Ξ›.

Corollary 2.22. A subset A of a space (𝑋,𝜏) is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if cl(𝐴)βŠ‚π΄Ξ›.

Theorem 2.23. Let A be a Ξ›-𝑠𝑒𝑑 of an ideal space (𝑋,𝜏,ℐ). Then A is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. Suppose 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By Theorem 2.20, clβ‹†πœƒ(𝐴)βŠ‚π΄Ξ›=𝐴, since 𝐴 is a Ξ›-𝑠𝑒𝑑. Therefore, 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘. Converse follows from the fact that every πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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›-𝑠𝑒𝑑 of a space (𝑋,𝜏). Then A is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if A is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.25. Let A be a Ξ›-𝑠𝑒𝑑 of a space (𝑋,𝜏). Then A is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if A is closed.

Theorem 2.26. Let (𝑋,𝜏,ℐ) be an ideal space and π΄βŠ‚π‘‹. If 𝐴Λ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, then A is also πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. Suppose that 𝐴Λ is a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set. If π΄βŠ‚π‘ˆ and π‘ˆ is open, then π΄Ξ›βŠ‚π‘ˆ. Since 𝐴Λ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(𝐴Λ)βŠ‚π‘ˆ. But, clβ‹†πœƒ(𝐴)βŠ‚clβ‹†πœƒ(𝐴Λ). Therefore, A is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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 π΄βŠ‚π‘‹. If 𝐴Λ is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, then A is also πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.28. Let (𝑋,𝜏) be a space and π΄βŠ‚π‘‹. If 𝐴Λ is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set, then A is also 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Theorem 2.29. For an ideal space (𝑋,𝜏,ℐ), the following are equivalent.(a)Every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)Every singleton of 𝑋 is closed or πœƒ-ℐ-π‘œπ‘π‘’π‘›.

Proof. (π‘Ž)β‡’(𝑏). Let π‘₯βˆˆπ‘‹. If {π‘₯} is not closed, then 𝐴=π‘‹βˆ’{π‘₯}βˆ‰πœ and then 𝐴 is trivially πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By (a), 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘. Hence {π‘₯} is πœƒ-ℐ-π‘œπ‘π‘’π‘›.
(𝑏)β‡’(π‘Ž). Let 𝐴 be a πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set and let π‘₯∈clβ‹†πœƒ(𝐴). We have the following cases.
Case 1. {π‘₯} is closed. By Theorem 2.3, clβ‹†πœƒ(𝐴)βˆ’π΄ does not contain a nonempty closed subset. This shows {π‘₯}∈𝐴.Case 2. {π‘₯} is πœƒ-ℐ-π‘œπ‘π‘’π‘›. Then, {π‘₯}βˆ©π΄β‰ πœ™. Hence, π‘₯∈𝐴.
Thus in both cases π‘₯∈𝐴 and so 𝐴=clβ‹†πœƒ(𝐴), that is, 𝐴 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘, 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-π‘π‘™π‘œπ‘ π‘’π‘‘ set is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)Every singleton of 𝑋 is closed or πœƒ-π‘œπ‘π‘’π‘›.

Corollary 2.31 (see [13, Theorem  2.5]). For an ideal space (𝑋,𝜏), the following are equivalent.(a)Every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is closed.(b)Every singleton of 𝑋 is closed or open.

Theorem 2.32. Let (𝑋,𝜏,ℐ) be an ideal space and π΄βŠ‚π‘‹. Then A is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if 𝐴=πΉβˆ’π‘, where 𝐹 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ and 𝑁 contains no nonempty closed set.

Proof. If 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, then by Theorem 2.3, 𝑁=clβ‹†πœƒ(𝐴)βˆ’π΄ contains no nonempty closed set. If 𝐹=clβ‹†πœƒ(𝐴), then 𝐹 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ such that πΉβˆ’π‘=clβ‹†πœƒ(𝐴)βˆ’(clβ‹†πœƒ(𝐴)βˆ’π΄)=clβ‹†πœƒ(𝐴)∩((π‘‹βˆ’clβ‹†πœƒ(𝐴))βˆͺ𝐴)=𝐴. Conversely, suppose 𝐴=πΉβˆ’π‘, where 𝐹 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ and 𝑁 contains no nonempty closed set. Let π‘ˆ be an open set such that π΄βŠ‚π‘ˆ. Then, πΉβˆ’π‘βŠ‚π‘ˆ which implies that 𝐹∩(π‘‹βˆ’π‘ˆ)βŠ‚π‘. Now, π΄βŠ‚πΉ and 𝐹 is πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ implies that clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ)βŠ‚clβ‹†πœƒ(𝐹)∩(π‘‹βˆ’π‘ˆ)βŠ‚πΉβˆ©(π‘‹βˆ’π‘ˆ)βŠ‚π‘. Since πœƒ-ℐ-π‘π‘™π‘œπ‘ π‘’π‘‘ sets are closed, clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ) is closed. By hypothesis, clβ‹†πœƒ(𝐴)∩(π‘‹βˆ’π‘ˆ)=πœ™ and so clβ‹†πœƒ(𝐴)βŠ‚π‘ˆ, which implies that 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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 π΄βŠ‚π‘‹. Then A is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ subset of X if and only if 𝐴=πΉβˆ’π‘, where F is πœƒ-π‘π‘™π‘œπ‘ π‘’π‘‘ and N contains no nonempty closed set.

Corollary 2.34 (see [12, Corollary  2.3]). Let (𝑋,𝜏) be a space and π΄βŠ‚π‘‹. Then A is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ if and only if 𝐴=πΉβˆ’π‘, 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-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ subset of X and π΄βŠ‚π΅βŠ‚clβ‹†πœƒ(𝐴), then B is also πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. clβ‹†πœƒ(𝐡)βˆ’π΅βŠ‚clβ‹†πœƒ(𝐴)βˆ’π΄, and since clβ‹†πœƒ(𝐴)βˆ’π΄ has no nonempty closed subset, neither does clβ‹†πœƒ(𝐡)βˆ’π΅. By Theorem 2.3, 𝐡 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ subset of X and π΄βŠ‚π΅βŠ‚clπœƒ(𝐴), then 𝐡 is also πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Corollary 2.37 (see [6, Theorem  2.8]). Let (𝑋,𝜏) be a space. If A is a 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ subset of X and π΄βŠ‚π΅βŠ‚cl(𝐴), then B is also 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

A subset 𝐴 of an ideal space (𝑋,𝜏,ℐ) is said to be πœƒ-ℐ𝑔-π‘œπ‘π‘’π‘› if π‘‹βˆ’π΄ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.

Theorem 2.38. A subset A of an ideal space (𝑋,𝜏,ℐ) is πœƒ-ℐ𝑔-π‘œπ‘π‘’π‘› if and only if πΉβŠ‚intβ‹†πœƒ(𝐴) whenever F is closed and πΉβŠ‚π΄.

Proof. Suppose 𝐴 is a πœƒ-ℐ𝑔-π‘œπ‘π‘’π‘› set and 𝐹 is a closed set contained in 𝐴, then π‘‹βˆ’π΄βŠ‚π‘‹βˆ’πΉ and π‘‹βˆ’πΉ is open. Since π‘‹βˆ’π΄ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, clβ‹†πœƒ(π‘‹βˆ’π΄)βŠ‚(π‘‹βˆ’πΉ) and so πΉβŠ‚π‘‹βˆ’clβ‹†πœƒ(π‘‹βˆ’π΄)=intβ‹†πœƒ(𝐴). Conversely, suppose Xβˆ’π΄βŠ‚π‘ˆ and π‘‹βˆ’π‘ˆ is closed. By hypothesis, π‘‹βˆ’π‘ˆβŠ‚intβ‹†πœƒ(𝐴), which implies that clβ‹†πœƒ(π‘‹βˆ’π΄)=π‘‹βˆ’intβ‹†πœƒ(𝐴)βŠ‚π‘ˆ. Therefore, π‘‹βˆ’π΄ is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ and hence 𝐴 is πœƒ-ℐ𝑔-π‘œπ‘π‘’π‘›.

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‘œπ‘π‘’π‘› if and only if πΉβŠ‚intπœƒ(𝐴) whenever F is closed and πΉβŠ‚π΄.

Corollary 2.40 (see [6, Theorem  4.2]). A subset A of a space (𝑋,𝜏) is 𝑔-π‘œπ‘π‘’π‘› if and only if πΉβŠ‚int(𝐴) whenever F is closed and πΉβŠ‚π΄.

Theorem 2.41. Let (𝑋,𝜏,ℐ) be an ideal space and π΄βŠ‚π‘ˆ. Then the following are equivalent.(a)A is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴)) is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(c)clβ‹†πœƒ(𝐴)βˆ’π΄ is πœƒ-ℐ𝑔-π‘œπ‘π‘’π‘›.

Proof. (π‘Ž)β‡’(𝑏). Suppose 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. If π‘ˆ is any open set containing 𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴)), then π‘‹βˆ’π‘ˆβŠ‚π‘‹βˆ’(𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴))=clβ‹†πœƒ(𝐴)βˆ’π΄. Since 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, by Theorem 2.3(c), it follows that π‘‹βˆ’π‘ˆ=πœ™ and so 𝑋=π‘ˆ. Since 𝑋 is the only open set containing 𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴)), 𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴)) is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.
(𝑏)β‡’(π‘Ž). Suppose 𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴)) is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. If 𝐹 is any closed set contained in clβ‹†πœƒ(𝐴)βˆ’π΄, then 𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴))βŠ‚π‘‹βˆ’πΉ and π‘‹βˆ’πΉ is open. Therefore, clβ‹†πœƒ(𝐴βˆͺ(π‘‹βˆ’clβ‹†πœƒ(𝐴))βŠ‚π‘‹βˆ’πΉ, which implies that clβ‹†πœƒ(𝐴)βˆͺclβ‹†πœƒ(π‘‹βˆ’clβ‹†πœƒ(𝐴))βŠ‚π‘‹βˆ’πΉ and so π‘‹βŠ‚π‘‹βˆ’πΉ; it follows that 𝐹=πœ™. Hence 𝐴 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.
The equivalence of (b) and (c) follows from the fact that π‘‹βˆ’(clβ‹†πœƒ(𝐴)βˆ’π΄)=𝐴βˆͺ(π‘‹βˆ’clβ‹†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 π΄βŠ‚π‘ˆ. Then the following are equivalent.(a)A is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)𝐴βˆͺ(π‘‹βˆ’clπœƒ(𝐴)) is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(c)clπœƒ(𝐴)βˆ’π΄ is πœƒ-𝑔-π‘œπ‘π‘’π‘›.

Corollary 2.43. Let (𝑋,𝜏) be an ideal space and π΄βŠ‚π‘ˆ. Then the following are equivalent.(a)A is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)𝐴βˆͺ(π‘‹βˆ’cl(𝐴)) is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.(c)cl(𝐴)βˆ’π΄ is 𝑔-π‘œπ‘π‘’π‘›.

3. Characterization of 𝒯1/2 and 𝒯ℐ-Space

Theorem 3.1. In an ideal space (𝑋,𝜏,ℐ), the following are equivalent.(a)Every πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is closed.(b)(𝑋,𝜏) is a 𝒯1/2-π‘ π‘π‘Žπ‘π‘’.(c)Every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ 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-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set. Since every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘, A is 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By hypothesis, A is closed.
(𝑐)β‡’(𝑏). Let π‘₯βˆˆπ‘‹. If {π‘₯} is not closed, then 𝐡=π‘‹βˆ’{π‘₯} is not open. So 𝐡 is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By hypothesis, B is closed and so {π‘₯} is open. By Corollary 2.31, (𝑋,𝜏) is a 𝒯1/2-π‘ π‘π‘Žπ‘π‘’.

Theorem 3.2. In an ideal space (𝑋,𝜏,ℐ) the following, are equivalent.(a)Every πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)(𝑋,𝜏,ℐ) is a 𝒯ℐ-π‘†π‘π‘Žπ‘π‘’.(c)Every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘.

Proof. (π‘Ž)β‡’(𝑏). Let π‘₯βˆˆπ‘‹. If {π‘₯} is not closed, then 𝑋 is the only open set containing π‘‹βˆ’{π‘₯} and so π‘‹βˆ’{π‘₯} is πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By hypothesis, π‘‹βˆ’{π‘₯} is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘. Equivalently {π‘₯} is ⋆-π‘œπ‘π‘’π‘›. Thus, every singleton set in 𝑋 is either closed or ⋆-π‘œπ‘π‘’π‘›. By Theorem  3.3 of [4], (𝑋,𝜏,ℐ) is a 𝒯ℐ-π‘†π‘π‘Žπ‘π‘’.
(𝑏)β‡’(π‘Ž). The proof follows from the fact that every πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘.
(𝑏)β‡’(𝑐). The proof follows from the fact that every set is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘β„π‘”-π‘π‘™π‘œπ‘ π‘’π‘‘.
(𝑐)β‡’(𝑏). Let π‘₯βˆˆπ‘‹. If {π‘₯} is not closed, then 𝑋 is the only open set containing π‘₯βˆ’{π‘₯} and so π‘₯βˆ’{π‘₯} is πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘. By hypothesis, π‘‹βˆ’{π‘₯} is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘. Thus, {π‘₯} is ⋆-π‘œπ‘π‘’π‘›. Therefore, every singleton set in 𝑋 is either ⋆-π‘œπ‘π‘’π‘› or closed. By Theorem of  3.3 [4], (𝑋,𝜏,ℐ) is a 𝒯ℐ-π‘†π‘π‘Žπ‘π‘’.

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-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘.(b)Every πœƒ-ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is ⋆-π‘π‘™π‘œπ‘ π‘’π‘‘.(c)Every ℐ𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is an ℐ-π‘™π‘œπ‘π‘Žπ‘™π‘™π‘¦β‹†-π‘π‘™π‘œπ‘ π‘’π‘‘ set.

Corollary 3.4. In a topological space (𝑋,𝜏), the following are equivalent.(a)Every πœƒ-𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is closed.(b)Every 𝑔-π‘π‘™π‘œπ‘ π‘’π‘‘ set is a locally closed set.

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