Some New Sets and Topologies in Ideal Topological Spaces

Abstract

An ideal topological space is a triplet (𝑋, 𝜏, ℑ), where 𝑋 is a nonempty set, 𝜏 is a topology on 𝑋, and ℑ is an ideal of subsets of 𝑋. In this paper, we introduce 𝐿^∗-perfect, 𝑅^∗-perfect, and 𝐶^∗-perfect sets in ideal spaces and study their properties. We obtained a characterization for compatible ideals via 𝑅^∗-perfect sets. Also, we obtain a generalized topology via ideals which is finer than 𝜏 using 𝑅^∗-perfect sets on a finite set.

1. Introduction and Preliminaries

The contributions of Hamlett and Jankovic [1–4] in ideal topological spaces initiated the generalization of some important properties in general topology via topological ideals. The properties like decomposition of continuity, separation axioms, connectedness, compactness, and resolvability [5–9] have been generalized using the concept of ideals in topological spaces.

By a space (𝑋,𝜏), we mean a topological space 𝑋 with a topology 𝜏 defined on 𝑋 on which no separation axioms are assumed unless otherwise explicitly stated. For a given point 𝑥 in a space (𝑋,𝜏), the system of open neighborhoods of 𝑥 is denoted by 𝑁(𝑥)={𝑈∈𝜏∶𝑥∈𝑈}. For a given subset 𝐴 of a space (𝑋,𝜏),cl(𝐴) and int(𝐴) are used to denote the closure of 𝐴 and interior of 𝐴, respectively, with respect to the topology.

A nonempty collection of subsets of a set 𝑋 is said to be an ideal on 𝑋, if it satisfies the following two conditions: (i) If 𝐴∈ℑ and 𝐵⊆𝐴, then 𝐵∈ℑ; (ii) If 𝐴∈ℑ and 𝐵∈ℑ, then 𝐴∪𝐵∈ℑ. An ideal topological space (or ideal space) (𝑋,𝜏,ℑ) means a topological space (𝑋,𝜏) with an ideal ℑ defined on 𝑋. Let (𝑋,𝜏) be a topological space with an ideal ℑ defined on 𝑋. Then for any subset 𝐴 of 𝑋,𝐴^∗(ℑ,𝜏)={𝑥∈𝑋/𝐴∩𝑈∉ℑ for every 𝑈∈𝑁(𝑥)} is called the local function of 𝐴 with respect to ℑ and 𝜏. If there is no ambiguity, we will write 𝐴^∗(ℑ) or simply 𝐴^∗ for 𝐴^∗(ℑ,𝜏). Also, cl^∗(𝐴)=𝐴∪𝐴^∗ defines a Kuratowski closure operator for the topology 𝜏^∗(ℑ) (or simply 𝜏^∗) which is finer than 𝜏. An ideal ℑ on a space (𝑋,𝜏) is said to be codense ideal if and only if 𝜏∩ℑ={∅}. 𝑋^∗ is always a proper subset of 𝑋. Also, 𝑋=𝑋^∗ if and only if the ideal is codense.

Lemma 1 ([see 12]). Let (𝑋,𝜏) be a space with ℑ₁ and ℑ₂ being ideals on 𝑋, and let 𝐴 and 𝐵 be two subsets on 𝑋. Then(i)𝐴⊆𝐵⇒𝐴^∗⊆𝐵^∗; (ii)ℑ₁⊆ℑ₂⇒𝐴^∗(ℑ₂)⊆𝐴^∗(ℑ₁); (iii)𝐴^∗=𝑐𝑙(𝐴^∗)⊆𝑐𝑙(𝐴) (𝐴^∗ is a closed subset of 𝑐𝑙(𝐴));(iv)(𝐴^∗)^∗⊆𝐴^∗; (v)(𝐴∪𝐵)^∗=𝐴^∗∪𝐵^∗; (vi)𝐴^∗−𝐵^∗=(𝐴−𝐵)^∗−𝐵^∗⊆(𝐴−𝐵)^∗;(vii)for every 𝐼∈ℑ,(𝐴∪𝐼)^∗=𝐴^∗=(𝐴−𝐼)^∗.

Definition 2 (see [3]). Let (𝑋,𝜏) be a space with an ideal ℑ on 𝑋. One says that the topology 𝜏 is compatible with the ideal ℑ, denoted by 𝜏∼ℑ, if the following holds, for every 𝐴⊆𝑋: if for every 𝑥∈𝐴, there exists a 𝑈∈𝑁(𝑥) such that 𝑈∩𝐴∈ℑ, then 𝐴∈ℑ.

Definition 3. A subset 𝐴 of an ideal space (𝑋,𝜏,ℑ) is said to be(i)𝜏^∗-closed [3] if 𝐴^∗⊆𝐴,(ii)*-dense-in-itself [10] if 𝐴⊆𝐴^∗,(iii)𝐼-open [11] if 𝐴⊆int(𝐴^∗),(iv)almost 𝐼-open [12] if 𝐴⊆cl(int(𝐴^∗)),(v)𝐼-dense [7] if 𝐴^∗=𝑋,(vi)almost strong 𝛽-𝐼-open [13] if 𝐴⊆cl^∗(int(𝐴^∗)),(vii)*-perfect [10] if 𝐴=𝐴^∗,(viii)regular 𝐼-closed [14] if 𝐴=(int(𝐴))^∗,(ix)an 𝑓_𝐼-set [15] if 𝐴⊆(int(𝐴))^∗.

Theorem 4 ([3]). Let (𝑋,𝜏) be a space with an ideal ℑ on 𝑋. Then the following are equivalent.(i)𝜏∼ℑ.(ii)If 𝐴 has a cover of open sets each of whose intersection with 𝐴 is ℑ, then 𝐴 is in ℑ.(iii)For every 𝐴⊆𝑋, 𝐴∩𝐴^∗=∅⇒𝐴∈ℑ.(iv)For every 𝐴⊆𝑋, 𝐴−𝐴^∗∈ℑ.(v)For every 𝜏^∗-closed subset 𝐴,𝐴−𝐴^∗∈ℑ.(vi)For every 𝐴⊆𝑋, if 𝐴 contains no nonempty subset 𝐵 with 𝐵⊆𝐵^∗, then 𝐴∈ℑ.

2. 𝐿^∗-Perfect, 𝑅^∗-Perfect, and 𝐶^∗-Perfect Sets

In this section, we define three collections of subsets 𝔏, ℜ and ℭ in an ideal space and study some of their properties.

Definition 5. Let (𝑋,𝜏,ℑ) be an ideal topological space. A subset 𝐴 of 𝑋 is said to be(i)𝐿^∗-perfect if 𝐴−𝐴^∗∈ℑ,(ii)𝑅^∗-perfect if 𝐴^∗−𝐴∈ℑ,(iii)𝐶^∗-perfect if 𝐴 is both 𝐿^∗-perfect and 𝑅^∗-perfect. The collection of 𝐿^∗-perfect sets, 𝑅^∗-perfect sets, and 𝐶^∗-perfect sets in (𝑋,𝜏,ℑ) is denoted by 𝔏, ℜ, and ℭ, respectively.

Remark 6. (i) If ℑ={∅}, then 𝔏=𝐴⊆𝑋∶𝐴⊆𝐴^∗=cl(𝐴)=thecollectionofall^∗(𝑋),-dense-in-itselfsets=℘ℜ=𝐴⊆𝑋∶𝐴^∗⊆𝐴=thecollectionofall𝜏^∗=-closedsets𝐴⊆𝑋∶𝐴=𝐴^∗(𝐴()=cl)=thecollectionofallclosedsetsin𝑋,𝜏=thecollectionofall^∗-perfectsets,ℭ=𝐴⊆𝑋∶𝐴^∗⊆𝐴,𝐴⊆𝐴^∗=𝐴𝑋∶𝐴=𝐴^∗(𝐴)()=cl=thecollectionofallclosedsetsin𝑋,𝜏=thecollectionofall^∗-perfectsets. (1)
(ii) If ℑ=𝔓(𝑋), then 𝔏=𝔗=℘(𝑋).
(iii) If 𝜏∼ℑ, then 𝔏=℘(𝑋) (by Theorem 4(iv)).

Remark 7. Every *-perfect set is both 𝑅^∗-perfect and 𝐿^∗-perfect (and hence 𝐶^∗-perfect). In Proposition 15, we proved that every members of an ideal is both 𝑅^∗-perfect and 𝐿^∗-perfect (and hence 𝐶^∗-perfect). But any nonempty member of an ideal is not a *-perfect set. Hence 𝑅^∗-perfect and 𝐿^∗-perfect sets (and hence 𝐶^∗-perfect sets) need not be *-perfect.

Proposition 8. If a subset 𝐴 of an ideal space (𝑋,𝜏,ℑ) is 𝐶^∗-perfect, then 𝐴Δ𝐴^∗∈ℑ.

Proof. Since 𝐴 is both 𝐿^∗-perfect and 𝑅^∗-perfect, 𝐴−𝐴^∗∈ℑ and 𝐴^∗−𝐴∈ℑ. By the finite additive property of ideals, (𝐴−𝐴^∗)∪(𝐴^∗−𝐴)∈ℑ. Hence 𝐴Δ𝐴^∗∈ℑ.

Proposition 9. In an ideal space (𝑋,𝜏,ℑ), every 𝜏^∗-closed set is 𝑅^∗-perfect.

Proof. Let 𝐴 be a 𝜏^∗-closed set. Therefore, 𝐴^∗⊆𝐴. Hence 𝐴^∗−𝐴=𝜙=ℑ. Therefore, 𝐴 is an 𝑅^∗-perfect set.

Corollary 10. In an ideal space (𝑋,𝜏,ℑ),(i)𝑋 and 𝜙 are 𝑅^∗-perfect sets,(ii)every 𝜏-closed set is 𝑅^∗-perfect,(iii)for any subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ),𝑐𝑙(𝐴),𝐴^∗,𝑐𝑙^∗(𝐴) are 𝑅^∗-perfect sets,(iv)every regular-𝐼-closed set is 𝑅^∗-perfect.

Proof. The proof follows from Proposition 9.

The following example shows that the converses of Proposition 9 and Corollary 10(iv) are not true.

Example 11. Let (𝑋,𝜏,ℑ) be an ideal space with 𝑋={𝑎,𝑏,𝑐}, 𝜏={𝜙,𝑋,{𝑎}}, and ℑ={𝜙,{𝑏}}. The set {𝑎,𝑐} is 𝑅^∗-perfect set which is not a 𝜏^∗-closed set and hence not a regular-𝐼-closed set.

Proposition 12. If a subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ) is such that 𝐴∈ℑ, then 𝐴 is 𝐶^∗-perfect.

Proof. Since 𝐴∈ℑ,𝐴^∗=𝜙. Then 𝐴−𝐴^∗=𝐴∈ℑ and 𝐴^∗−𝐴=𝜙∈ℑ. Hence 𝐴 is both an 𝐿^∗-perfect and 𝑅^∗-perfect set.

Corollary 13. Let 𝐴 be a subset of an ideal space (𝑋,𝜏,ℑ). Consider the following.(i)If 𝐴∈ℑ, then every subset of 𝐴 is a 𝐶^∗-perfect set.(ii)If 𝐴 is 𝑅^∗-perfect, then 𝐴^∗−𝐴 is 𝐶^∗-perfect.(iii)If 𝐴 is an 𝐿^∗-perfect set, then 𝐴−𝐴^∗ is a 𝐶^∗-perfect set.(iv)If 𝐴 is 𝐶^∗-perfect, then 𝐴Δ𝐴^∗ is a 𝐶^∗-perfect set.

Proof. The proof follows from Proposition 12.

Corollary 14. Let (𝑋,𝜏) be a space with an ideal ℑ on 𝑋 such that 𝜏∼ℑ. Then for every 𝐴⊆𝑋,(i)if 𝐴∩𝐴^∗=𝜙, then 𝐴 is 𝐶^∗-perfect;(ii)𝐴−𝐴^∗ is 𝐶^∗-perfect;(iii)if 𝐴 contains nonempty subset 𝐵 with 𝐵⊆𝐵^∗, then 𝐴 is 𝐶^∗-perfect.

Proof. Follows from Theorem 4 and Proposition 12.

Proposition 15. In an ideal space (𝑋,𝜏,ℑ), every *-dense-in-itself set is an 𝐿^∗-perfect set.

Proof. Let 𝐴 be a *-dense-in-itself set of 𝑋. Then 𝐴⊆𝐴^∗. Therefore, 𝐴−𝐴^∗=𝜙∈ℑ. Hence 𝐴 is an 𝐿^∗-perfect set.

Corollary 16. In an ideal space (𝑋,𝜏,ℑ),(i)every 𝐼-dense set is 𝐿^∗-perfect,(ii)every 𝐼-open set is 𝐿^∗-perfect,(iii)every almost strong 𝛽-I-open set is 𝐿^∗-perfect,(iv)every almost 𝐼-open set is 𝐿^∗-perfect,(v)every regular-𝐼-closed set is 𝐿^∗-perfect,(vi)every 𝑓_𝐼-set is 𝐿^∗-perfect.

Proof. Since all the above sets are *-dense-in-itself, by Proposition 15, these sets are 𝐿^∗-perfect.

Remark 17. The members of the ideal of an ideal space are 𝐿^∗-perfect, but the nonempty members of the ideal are not *-dense-in-itself. Therefore, the converses of the above Corollary and Proposition 15 need not to be true.

Proposition 18. In an ideal space (𝑋,𝜏,ℑ),(i)empty set is an  𝐿^∗-perfect set,(ii)𝑋  is an  𝐿^∗-perfect set if the ideal is codense.

Proof. (i) Since 𝜙−𝜙^∗=𝜙∈ℑ, the empty set is an 𝐿^∗-perfect set. (ii) We know that 𝑋=𝑋^∗ if and only if the ideal ℑ is codense. Then 𝑋−𝑋^∗=𝜙∈ℑ. Hence the result follows.

3. Main Results

In this section, we prove that finite union and intersection of 𝑅^∗-perfect sets are again 𝑅^∗-perfect set. Using these results, we obtain a new topology for the finite topological spaces which is finer than 𝜏^∗-topology.

In Ideal spaces, usually 𝐴⊂𝐵 implies 𝐴^∗⊂𝐵^∗. We observe that there are some sets 𝐴 and 𝐵 such that 𝐴⊂𝐵 but 𝐴^∗=𝐵^∗.

Example 19. Let (𝑋,𝜏,ℑ) be an ideal space with 𝑋={𝑎,𝑏,𝑐,𝑑}, 𝜏={𝜙,𝑋,{𝑎,𝑐},{𝑑},{𝑎,𝑐,𝑑}}, ℑ={𝜙,{𝑐},{𝑑},{𝑐,𝑑}}. Here the sets 𝐴={𝑎} and 𝐵={𝑎,𝑏} are such that 𝐴⊆𝐵, but 𝐴^∗=𝐵^∗={𝑎,𝑏,𝑐}.

Proposition 20. Let (𝑋,𝜏,ℑ) be an ideal space. Let 𝐴 and 𝐵 be two subsets of 𝑋 such that 𝐴⊆𝐵 and 𝐴^∗=𝐵^∗; then(i)𝐵  is 𝑅^∗-perfect if 𝐴 is 𝑅^∗-perfect;(ii)𝐴  is 𝐿^∗-perfect if 𝐵 is 𝐿^∗-perfect.

Proof. (i) Let 𝐴 be an 𝑅^∗-perfect set. Then 𝐴^∗−𝐴∈ℑ. Now, 𝐵^∗−𝐵=𝐴^∗−𝐵⊆𝐴^∗−𝐴. By heredity property of ideals, 𝐵^∗−𝐵∈ℑ. Hence 𝐵 is 𝑅^∗-perfect.
(ii) Let 𝐵 be an 𝐿^∗-perfect set. Then 𝐵−𝐵^∗∈ℑ. Now, 𝐴−𝐴^∗=𝐴−𝐵^∗⊆𝐵−𝐵^∗. By heredity property of ideals, 𝐴−𝐴^∗∈ℑ. Hence 𝐴 is 𝐿^∗-perfect.

Corollary 21. Let (𝑋,𝜏,ℑ) be an ideal space. Let 𝐴 and 𝐵 be two subsets of 𝑋 such that 𝐴⊆𝐵⊆𝑐𝑙^∗𝐴; then(i)𝐵 is 𝑅^∗-perfect if 𝐴 is 𝑅^∗-perfect,(ii)𝐴 is 𝐿^∗-perfect if 𝐵 is 𝐿^∗-perfect.

Proof. Since 𝐴⊆𝐵⊆cl^∗𝐴, 𝐴^∗⊆𝐵^∗⊆(cl^∗𝐴)^∗=𝐴^∗. Hence 𝐴^∗=𝐵^∗. Therefore, the result follows from Proposition 20.

Proposition 22. Let 𝐴 be a subset of an ideal topological space (𝑋,𝜏,ℑ) such that 𝐴 is 𝐿^∗-perfect set and 𝐴∩𝐴^∗ is 𝑅^∗-perfect; then both 𝐴 and 𝐴∩𝐴^∗ are 𝐶^∗-perfect.

Proof. Since 𝐴 is 𝐿^∗-perfect, 𝐴−𝐴^∗∈ℑ. By Lemma 1(vii), for every 𝐼∈ℑ, (𝐴∪𝐼)^∗=𝐴^∗=(𝐴−𝐼)^∗. Therefore, (𝐴∪(𝐴−𝐴^∗))^∗=𝐴^∗=(𝐴−(𝐴−𝐴^∗))^∗. This implies 𝐴^∗=(𝐴∩𝐴^∗)^∗. Therefore, we have 𝐴∩𝐴^∗⊆𝐴 with (𝐴∩𝐴^∗)^∗=𝐴^∗. By Proposition 20, 𝐴 is 𝑅^∗-perfect if 𝐴∩𝐴^∗ is 𝑅^∗-perfect and 𝐴∩𝐴^∗ is 𝐿^∗-perfect if 𝐴 is 𝐿^∗-perfect set. Hence 𝐴 is 𝑅^∗-perfect and 𝐴∩𝐴^∗ is 𝐿^∗-perfect.

Proposition 23. If a subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ) is 𝑅^∗-perfect set and 𝐴^∗ is 𝐿^∗-perfect, then 𝐴∩𝐴^∗ is 𝐿^∗-perfect.

Proof. Since 𝐴 is 𝑅^∗-perfect, 𝐴^∗−𝐴∈ℑ. By Lemma 1(vii), for every 𝐼∈ℑ, (𝐴∪𝐼)^∗=𝐴^∗=(𝐴−𝐼)^∗. Therefore, (𝐴^∗∪(𝐴^∗−𝐴))^∗=𝐴^∗∗=(𝐴^∗−(𝐴^∗−𝐴))^∗. This implies 𝐴^∗∗=(𝐴∩𝐴^∗)^∗. Therefore, we have 𝐴∩𝐴^∗⊆𝐴^∗ with (𝐴∩𝐴^∗)^∗=𝐴^∗∗. By Proposition 20, 𝐴∩𝐴^∗ is 𝐿^∗-perfect if 𝐴^∗ is 𝐿^∗-perfect set. Hence 𝐴∩𝐴^∗ is 𝐿^∗-perfect.

Proposition 24. If 𝐴 and 𝐵 are 𝑅^∗-perfect sets, then 𝐴∪𝐵 is an 𝑅^∗-perfect set.

Proof. Let 𝐴 and 𝐵 be 𝑅^∗-perfect sets. Then 𝐴^∗−𝐴∈ℑ and 𝐵^∗−𝐵∈ℑ. By finite additive property of ideals, (𝐴^∗−𝐴)∪(𝐵^∗−𝐵)∈ℑ. Since (𝐴^∗∪𝐵^∗)−(𝐴∪𝐵)⊆(𝐴^∗−𝐴)∪(𝐵^∗−𝐵), by heredity property (𝐴^∗∪𝐵^∗)−(𝐴∪𝐵)∈ℑ. Hence (𝐴∪𝐵)^∗−(𝐴∪𝐵)∈ℑ. This proves the result.

Corollary 25. Finite union of 𝑅^∗-perfect sets is an 𝑅^∗-perfect set.

Proof. The proof follows from Proposition 24.

Proposition 26. If 𝐴 and 𝐵 are 𝐿^∗-perfect sets, then 𝐴∪𝐵 is an 𝐿^∗-perfect set.

Proof. Since 𝐴 and 𝐵 are 𝐿^∗-perfect sets, 𝐴−𝐴^∗∈ℑ and 𝐵−𝐵^∗∈ℑ. Hence by finite additive property of ideals, (𝐴−𝐴^∗)∪(𝐵−𝐵^∗)∈ℑ. Since (𝐴∪𝐵)−(𝐴∪𝐵)^∗=(𝐴∪𝐵)−(𝐴^∗∪𝐵^∗)⊆(𝐴−𝐴^∗)∪(𝐵−𝐵^∗), by heredity property (𝐴∪𝐵)−(𝐴∪𝐵)^∗∈ℑ. This proves that 𝐴∪𝐵 is an 𝐿^∗-perfect set.

Corollary 27. Finite union of 𝐿^∗-perfect sets is an 𝐿^∗-perfect sets.

Proof. The proof follows from Proposition 26.

Proposition 28. If 𝐴 and 𝐵 are 𝑅^∗-perfect sets, then 𝐴∩𝐵 is an 𝑅^∗-perfect set.

Proof. Suppose that 𝐴 and 𝐵 are 𝑅^∗-perfect sets. Then 𝐴^∗−𝐴∈ℑ and 𝐵^∗−𝐵∈ℑ. By finite additive property of ideals, (𝐴^∗−𝐴)∪(𝐵^∗−𝐵)∈ℑ. Since (𝐴^∗∩𝐵^∗)−(𝐴∩𝐵)⊆(𝐴^∗−𝐴)∪(𝐵^∗−𝐵), by heredity property (𝐴^∗∩𝐵^∗)−(𝐴∩𝐵)∈ℑ. Also (𝐴∩𝐵)^∗−(𝐴∩𝐵)⊆(𝐴^∗∩𝐵^∗)−(𝐴∩𝐵)∈ℑ. This proves the result.

Corollary 29. Finite intersection of 𝑅^∗-perfect sets is an 𝑅^∗-perfect set.

Proof. The proof follows from Proposition 28.

Proposition 30. Finite union of 𝐶^∗-perfect sets is a 𝐶^∗-perfect set.

Proof. From Corollaries 27 and 29, finite union of 𝐶^∗-perfect sets is a 𝐶^∗-perfect set.

Proposition 31. If (𝑋,𝜏,ℑ) is an ideal topological space with 𝑋 being finite, then the collection ℜ is a topology which is finer than the topology of 𝜏^∗-closed sets.

Proof. By Corollary 10, 𝑋 and 𝜙 are 𝑅^∗-perfect sets. By Corollary 25, finite union of 𝑅^∗-perfect sets is an 𝑅^∗-perfect set, and by Corollary 29, finite intersection of 𝑅^∗-perfect sets is 𝑅^∗-perfect. Hence the collection ℜ is a topology if 𝑋 is finite. Also, by Proposition 9 every 𝜏^∗-closed set is an 𝑅^∗-perfect set. Hence the topology ℜ is finer than the topology of 𝜏^∗-closed sets if 𝑋 is finite.

Proposition 32. In an ideal space (𝑋,𝜏,ℑ), {𝜏^∗-𝑐𝑙𝑜𝑠𝑒𝑑𝑠𝑒𝑡𝑠}∪ℑ⊆ℜ.

Proof. The proof follows from Propositions 9 and 12.

The following example shows that {𝜏^∗-closedsets}∪ℑ≠ℜ.

Example 33. Let (𝑋,𝜏,ℑ) be an ideal space with 𝑋={𝑎,𝑏,𝑐}, 𝜏={∅,𝑋,{𝑎}}, and ℑ={∅,{𝑏}}. The collection of 𝜏^∗-closed sets is {∅,𝑋,{𝑏},{𝑏,𝑐}} and ℜ={∅,𝑋,{𝑏},{𝑏,𝑐},{𝑎,𝑐}}.

Now, {𝜏^∗-closed sets}∪ℑ={∅,𝑋,{𝑏},{𝑏,𝑐}}≠ℜ.

Proposition 34. Let (𝑋,𝜏,ℑ) be an ideal space and 𝐴⊆𝑋. The set 𝐴 is 𝑅^∗-perfect if and only if 𝐹⊆𝐴^∗−𝐴 in 𝑋 implies that 𝐹∈ℑ.

Proof. Assume that 𝐴 is an 𝑅^∗-perfect set. Then 𝐴^∗−𝐴∈ℑ. By heredity property of ideals, every set 𝐹⊆𝐴^∗−𝐴 in 𝑋 is also in ℑ. Conversely assume that 𝐹⊆𝐴^∗−𝐴 in 𝑋 implies that 𝐹∈ℑ. Since 𝐴^∗−𝐴 is a subset of itself, by assumption 𝐴^∗−𝐴∈ℑ. Hence 𝐴 is 𝑅^∗-perfect.

Proposition 35. Let (𝑋,𝜏,ℑ) be an ideal space and 𝐴⊆𝑋. The set 𝐴 is 𝐿^∗-perfect if and only if 𝐹⊆𝐴−𝐴^∗ in 𝑋 implies that 𝐹∈ℑ.

Proof. Assume that 𝐴 is an 𝐿^∗-perfect set. Then 𝐴−𝐴^∗∈ℑ. By heredity property of ideals, every set 𝐹⊆𝐴−𝐴^∗ in 𝑋 is also in ℑ. Conversely, assume that 𝐹⊆𝐴−𝐴^∗ in 𝑋 implies that 𝐹∈ℑ. Since 𝐴−𝐴^∗ is a subset of itself, by assumption 𝐴−𝐴^∗∈ℑ. Hence 𝐴 is 𝐿^∗-perfect.

Proposition 36. Let (𝑋,𝜏) be a topological space and 𝐴⊆𝑋. Let ℑ₁ and ℑ₂ be two ideals on 𝑋 with ℑ₁⊆ℑ₂. Then 𝐴 is 𝑅^∗-perfect with respect to ℑ₂ if it is 𝑅^∗-perfect with respect to ℑ₁.

Proof. Since ℑ₁⊆ℑ₂,𝐴^∗(ℑ₂)⊆𝐴^∗(ℑ₁) by Lemma 1(ii) Let 𝐴 be 𝑅^∗-perfect with respect to ℑ₁. Then 𝐴^∗(ℑ₁)−𝐴∈ℑ₁. Also, 𝐴^∗(ℑ₂)−𝐴⊆𝐴^∗(ℑ₁)−𝐴. Hence by heredity property of ideals, 𝐴^∗(ℑ₂)−𝐴∈ℑ₁⊆ℑ₂. Therefore 𝐴 is 𝑅^∗-perfect with respect to ℑ₂.

Theorem 37. Let (𝑋,𝜏) be a space with an ideal ℑ on 𝑋. Then the following are equivalent.(i)𝜏∼ℑ. (ii)If 𝐴 has a cover of open sets each of whose intersection with 𝐴 is ℑ, then 𝐴 is in ℑ.(iii)If 𝐴⊆𝑋, then 𝐴∩𝐴^∗=𝜙⇒𝐴∈ℑ.(iv)If 𝐴⊆𝑋, then 𝐴−𝐴^∗∈ℑ.(v)If 𝐴⊆𝑋 and 𝐴 is 𝑅^∗-perfect set, then 𝐴Δ𝐴^∗∈ℑ.(vi)For every 𝜏^∗-closed subset 𝐴,𝐴−𝐴^∗∈ℑ.(vii)For every 𝐴⊆𝑋, if 𝐴 contains no nonempty subset 𝐵 with 𝐵⊆𝐵^∗, then 𝐴∈ℑ.

Proof. To prove this theorem, it is enough to prove (iv) ⇒ (v) ⇒ (vi). Others follow from Theorem 4. (iv) ⇒ (v) follows from Proposition 8. Suppose that 𝐴Δ𝐴^∗∈ℑ. Since 𝐴−𝐴^∗⊆𝐴Δ𝐴^∗, by heredity property 𝐴−𝐴^∗∈ℑ. Hence (v) ⇒ (vi).

4. 𝑅^∗-Topology

By Corollary 10 and Proposition 28, we observe that the collection ℜ satisfies the conditions of being a basis for some topology and it will be called as 𝑅^∗_𝑐(𝜏,ℑ). We define 𝑅^∗(𝜏,ℑ)={𝐴⊆𝑋/𝑋−𝐴∈𝑅^∗_𝑐(𝜏,ℑ)} on a nonempty set 𝑋. Clearly, 𝑅^∗(𝜏,ℑ) is a topology if the set 𝑋 is finite. The members of the collection 𝑅^∗(𝜏,ℑ) will be called 𝑅^∗-open sets. If there is no confusion about the topology 𝜏 and the ideal ℑ, then we call 𝑅^∗(𝜏,ℑ) as 𝑅^∗-topology when 𝑋 is finite.

Definition 38. A subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ) is said to be 𝑅^∗-closed if it is a complement of an 𝑅^∗-open set.

Definition 39. Let 𝐴 be a subset of an ideal topological space (𝑋,𝜏,ℑ). One defines 𝑅^∗-interior of the set 𝐴 as the largest 𝑅^∗-open set contained in 𝐴.One will denote 𝑅^∗-interior of a set 𝐴 by 𝑅^∗−int(𝐴).

Definition 40. Let 𝐴 be a subset of an ideal topological space (𝑋,𝜏,ℑ). A point 𝑥∈𝐴 is said to be an 𝑅^∗-interior point of the set 𝐴 if there exists an 𝑅^∗-open set 𝑈 of 𝑥 such that 𝑥∈𝑈⊆𝐴.

Definition 41. Let (𝑋,𝜏,ℑ) be an ideal space and 𝑥∈𝑋. One defines 𝑅^∗-neighborhood of 𝑥 as an 𝑅^∗-open set containing 𝑥. One denotes the set of all 𝑅^∗-neighborhoods of 𝑥 by 𝑅^∗-𝑁(𝑥).

Proposition 42. In an ideal space (𝑋,𝜏,ℑ), every 𝜏^∗-open set is an 𝑅^∗-open set.

Proof. Let 𝐴 be a 𝜏^∗-open set. Therefore, 𝑋−𝐴 is a 𝜏^∗-closed set. That implies that 𝑋−𝐴 is an 𝑅^∗-closed set. Hence 𝐴 is an 𝑅^∗-open set.

Corollary 43. The topology 𝑅^∗(𝜏,ℑ) on a finite set 𝑋 is finer than the topology 𝜏^∗(𝜏,ℑ).

Proof. The proof follows from Proposition 42.

Corollary 44. For any subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ), 𝑖𝑛𝑡(𝐴) is an 𝑅^∗-open set.

Proof. The proof follows from Proposition 42.

Remark 45. (i) Since every open set is an 𝑅^∗-open set, every neighborhood 𝑈 of a point 𝑥∈𝑋 is an 𝑅^∗-neighborhood of 𝑥.
(ii) If 𝑥∈𝑋 is an interior point of a subset 𝐴 of 𝑋, then 𝑥 is an 𝑅^∗-interior point of 𝐴.
(iii) From (ii), we have int(𝐴)⊆int^∗(𝐴)⊆𝑅^∗-int(𝐴), where int^∗(𝐴) denotes interior of 𝐴 with respect to the topology 𝜏^∗.

Theorem 46. Let 𝐴 and 𝐵 be subsets of an ideal space (𝑋,𝜏,ℑ) with 𝑋 being finite. Then the following properties hold.(i)𝑅^∗-𝑖𝑛𝑡(𝐴)=∪{𝑈∶𝑈⊆𝐴 and 𝑈 is an 𝑅^∗-open set}.(ii)𝑅^∗-𝑖𝑛𝑡(𝐴) is the largest 𝑅^∗-open set of 𝑋 contained in 𝐴.(iii)𝐴 is 𝑅^∗-open if and only if 𝐴=𝑅^∗-𝑖𝑛𝑡(𝐴).(iv)𝑅^∗-𝑖𝑛𝑡(𝑅^∗-𝑖𝑛𝑡(𝐴))=𝑅^∗-𝑖𝑛𝑡(𝐴). (v)If 𝐴⊆𝐵, then 𝑅^∗-𝑖𝑛𝑡(𝐴)⊆𝑅^∗-𝑖𝑛𝑡(𝐵).

Proof. The proof follows from Definitions 39, 40, and 41.

Definition 47. Let 𝐴 be a subset of an ideal topological space (𝑋,𝜏,ℑ). One defines 𝑅^∗-closure of the set 𝐴 as the smallest 𝑅^∗-closed set containing 𝐴. One will denote 𝑅^∗-closure of a set 𝐴 by 𝑅^∗-cl(𝐴).

Remark 48. For any subset 𝐴 of an ideal topological space (𝑋,𝜏,ℑ), 𝑅^∗-cl(𝐴)⊆cl^∗(𝐴)⊆cl(𝐴).

Theorem 49. Let 𝐴 and 𝐵 be subsets of an ideal space (𝑋,𝜏,ℑ) where 𝑋 is finite. Then the following properties hold:(i)𝑅^∗-𝑐𝑙(𝐴)=∩{𝐹∶𝐴⊆𝐹 and 𝐹 is 𝑅^∗-closed set}.(ii)𝐴 is 𝑅^∗-closed if and only if 𝐴=𝑅^∗-𝑐𝑙(𝐴)(iii)𝑅^∗-𝑐𝑙(𝑅^∗-𝑐𝑙(𝐴))=𝑅^∗-𝑐𝑙(𝐴)(iv)If 𝐴⊆𝐵, then 𝑅^∗-𝑐𝑙(𝐴)⊆𝑅^∗-𝑐𝑙(𝐵).

Proof. The proof follows from Definition 47.

Theorem 50. Let 𝐴 be a subset of an ideal space (𝑋,𝜏,ℑ). Then the following properties hold:(i)𝑅^∗-𝑖𝑛𝑡(𝑋−𝐴)=𝑋−𝑅^∗-𝑐𝑙(𝐴);(ii)𝑅^∗-𝑐𝑙(𝑋−𝐴)=𝑋−𝑅^∗-𝑖𝑛𝑡(𝐴).

Proof. The proof follows from Definitions 38, 39, and 47.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.