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.