Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 973608, 6 pages http://dx.doi.org/10.1155/2013/973608

Research Article Some New Sets and Topologies in Ideal Topological Spaces

R. Manoharan1 and P. Thangavelu2

1 Department of Mathematics, Sathyabama University, Chennai, Tamil Nadu 600119, India 2 Department of Mathematics, Karunya University, Coimbatore, Tamil Nadu 641114, India

Correspondence should be addressed to R. Manoharan; mano [email protected]

Received 30 July 2013; Accepted 16 September 2013

Academic Editors: Y. Fu and C. Wang

Copyright © 2013 R. Manoharan and P. Thangavelu. 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.

An ideal is a triplet (𝑋, 𝜏, I), where 𝑋 is a nonempty set, 𝜏 is a topology on 𝑋,andI is an ideal of subsets of ∗ ∗ ∗ 𝑋.Inthispaper,weintroduce𝐿 -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 defines a Kuratowski closure operator for the topology 𝜏 (I) ∗ (or simply 𝜏 )whichisfinerthan𝜏.AnidealI on a space ∗ The contributions of Hamlett and Jankovic [1–4]inideal (𝑋, 𝜏) is said to be codense ideal if and only if 𝜏∩I ={0}. 𝑋 ∗ topological spaces initiated the generalization of some impor- is always a proper subset of 𝑋.Also,𝑋=𝑋 if and only if the tant properties in via topological ideals. ideal is codense. The properties like decomposition of continuity, separation axioms, connectedness, compactness, and resolvability [5– Lemma 1 (see [3]). Let (𝑋, 𝜏) be a space with I1 and I2 being 9] have been generalized using the concept of ideals in ideals on 𝑋,andlet𝐴 and 𝐵 be two subsets on 𝑋.Then topological spaces. ∗ ∗ By a space (𝑋, 𝜏),wemeanatopologicalspace𝑋 with a (i) 𝐴⊆𝐵⇒𝐴 ⊆𝐵 ; topology 𝜏 defined on 𝑋 on which no separation axioms are I ⊆ I ⇒𝐴∗(I )⊆𝐴∗(I ) assumed unless otherwise explicitly stated. For a given point (ii) 1 2 2 1 ; 𝑥 in a space (𝑋, 𝜏), the system of open neighborhoods of 𝑥 is ∗ ∗ ∗ (iii) 𝐴 =𝑐𝑙(𝐴 ) ⊆ 𝑐𝑙(𝐴) (𝐴 is a closed subset of 𝑐𝑙(𝐴)); denoted by 𝑁(𝑥) = {𝑈 ∈ 𝜏 .: 𝑥∈𝑈} For a given subset 𝐴 of a ∗ ∗ ∗ space (𝑋, 𝜏), cl(𝐴) and int(𝐴) areusedtodenotetheclosureof (iv) (𝐴 ) ⊆𝐴 ; 𝐴 𝐴 and interior of , respectively, with respect to the topology. ∗ ∗ ∗ A nonempty collection of subsets of a set 𝑋 is said to be (v) (𝐴 ∪ 𝐵) =𝐴 ∪𝐵 ; an ideal on 𝑋, if it satisfies the following two conditions: (i) If ∗ ∗ ∗ ∗ ∗ (vi) 𝐴 −𝐵 =(𝐴−𝐵) −𝐵 ⊆(𝐴−𝐵) ; 𝐴∈I and 𝐵⊆𝐴,then𝐵∈I; (ii) If 𝐴∈I and 𝐵∈I,then ∗ ∗ ∗ 𝐴∪𝐵 ∈ I. An ideal topological space (or ideal space) (𝑋, 𝜏, I) (vii) for every 𝐼∈I,(𝐴∪𝐼) =𝐴 =(𝐴−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 Definition 2 (see [3]). Let (𝑋, 𝜏) be a space with an ideal I on ∗ on 𝑋. Then for any subset 𝐴 of 𝑋, 𝐴 (I, 𝜏) = {𝑥 ∈ 𝑋/𝐴∩ 𝑋. One says that the topology 𝜏 is compatible with the ideal I, 𝑈∉I for every 𝑈 ∈ 𝑁(𝑥)} is called the local function of denoted by 𝜏∼I,ifthefollowingholds,forevery𝐴⊆𝑋:if 𝐴 with respect to I and 𝜏.Ifthereisnoambiguity,wewill for every 𝑥∈𝐴, there exists a 𝑈 ∈ 𝑁(𝑥) such that 𝑈∩𝐴 ∈I, ∗ ∗ ∗ ∗ ∗ write 𝐴 (I) or simply 𝐴 for 𝐴 (I,𝜏).Also,cl (𝐴) = 𝐴∪𝐴 then 𝐴∈I. 2 Chinese Journal of Mathematics

∗ Definition 3. Asubset𝐴 of an ideal space (𝑋, 𝜏, I) is said to R ={𝐴⊆𝑋:𝐴 ⊆𝐴} be ∗ = the collection of all 𝜏 -closed sets ∗ ∗ (i) 𝜏 -closed [3]if𝐴 ⊆𝐴, ∗ ={𝐴⊆𝑋:𝐴=𝐴 = cl (𝐴)} ∗ ∗ (ii) -dense-in-itself [10]if𝐴⊆𝐴 , = the collection of all closed sets in (𝑋, 𝜏) ∗ (iii) 𝐼-open [11]if𝐴⊆int(𝐴 ), ∗ = the collection of all -perfect sets, 𝐼 𝐴⊆ ( (𝐴∗)) (iv) almost -open [12]if cl int , C ={𝐴⊆𝑋:𝐴∗ ⊆𝐴,𝐴⊆𝐴∗} ∗ (v) 𝐼-dense [7]if𝐴 =𝑋, ∗ =𝐴𝑋:𝐴=𝐴 = cl (𝐴) ∗ ∗ (vi) almost strong 𝛽-𝐼-open [13]if𝐴⊆cl (int(𝐴 )), = the collection of all closed sets in (𝑋, 𝜏) ∗ ∗ (vii) -perfect [10]if𝐴=𝐴 , ∗ = the collection of all -perfect sets. ∗ (viii) regular 𝐼-closed [14]if𝐴=(int(𝐴)) , (1) ∗ (ix) an 𝑓𝐼-set [15]if𝐴⊆(int(𝐴)) . (ii) If I = P(𝑋),thenL = T = ℘(𝑋). (iii) If 𝜏∼I,thenL = ℘(𝑋) (by Theorem 4(iv)). Theorem 4 ([3]). Let (𝑋, 𝜏) be a space with an ideal I on 𝑋. ∗ ∗ ∗ Then the following are equivalent. Remark 7. Every -perfect set is both 𝑅 -perfect and 𝐿 - ∗ perfect (and hence 𝐶 -perfect). In Proposition 15,weproved ∗ ∗ 𝜏∼I that every members of an ideal is both 𝑅 -perfect and 𝐿 - (i) . ∗ perfect (and hence 𝐶 -perfect). But any nonempty member ∗ ∗ ∗ (ii) If 𝐴 hasacoverofopensetseachofwhoseintersection of an ideal is not a -perfect set. Hence 𝑅 -perfect and 𝐿 - ∗ ∗ with 𝐴 is I,then𝐴 is in I. perfect sets (and hence 𝐶 -perfect sets) need not be -perfect. 𝐴⊆𝑋 𝐴∩𝐴∗ =0⇒𝐴∈I ∗ (iii) For every , . Proposition 8. If a subset 𝐴 of an ideal space (𝑋, 𝜏, I) is 𝐶 - ∗ ∗ perfect, then 𝐴Δ𝐴 ∈ I. (iv) For every 𝐴⊆𝑋, 𝐴−𝐴 ∈ I. ∗ ∗ ∗ ∗ ∗ Proof. Since 𝐴 is both 𝐿 -perfect and 𝑅 -perfect, 𝐴−𝐴 ∈ I (v) For every 𝜏 -closed subset 𝐴, 𝐴 −𝐴 ∈ I. ∗ and 𝐴 −𝐴 ∈ I. By the finite additive property of ideals, ∗ ∗ ∗ (vi) For every 𝐴⊆𝑋,if𝐴 contains no nonempty subset 𝐵 (𝐴 − 𝐴 )∪(𝐴 −𝐴)∈I.Hence𝐴Δ𝐴 ∈ I. 𝐵⊆𝐵∗ 𝐴∈I with ,then . ∗ Proposition 9. In an ideal space (𝑋, 𝜏, I),every𝜏 - ∗ ∗ ∗ 𝑅∗ 2. 𝐿 -Perfect, 𝑅 -Perfect, and 𝐶 -Perfect Sets is -perfect. ∗ ∗ L R Proof. Let 𝐴 be a 𝜏 -closed set. Therefore, 𝐴 ⊆𝐴.Hence In this section, we define three collections of subsets , and 𝐴∗ −𝐴=𝜙=I 𝐴 𝑅∗ C in an ideal space and study some of their properties. . Therefore, is an -perfect set. Corollary 10. (𝑋, 𝜏, I) Definition 5. Let (𝑋, 𝜏, I) be an ideal topological space. A In an ideal space , ∗ subset 𝐴 of 𝑋 is said to be (i) 𝑋 and 𝜙 are 𝑅 -perfect sets, ∗ ∗ ∗ (ii) every 𝜏-closed set is 𝑅 -perfect, (i) 𝐿 -perfect if 𝐴−𝐴 ∈ I, (iii) for any subset 𝐴 of an ideal topological space ∗ ∗ ∗ ∗ ∗ (ii) 𝑅 -perfect if 𝐴 −𝐴∈I, (𝑋, 𝜏, I), 𝑐𝑙(𝐴), 𝐴 ,𝑐𝑙 (𝐴) are 𝑅 -perfect sets, ∗ ∗ ∗ ∗ (iv) every regular-𝐼-closed set is 𝑅 -perfect. (iii) 𝐶 -perfect if 𝐴 is both 𝐿 -perfect and 𝑅 -perfect.

∗ ∗ ∗ Proof. The proof follows from Proposition 9. The collection of 𝐿 -perfect sets, 𝑅 -perfect sets, and 𝐶 - perfect sets in (𝑋, 𝜏, I) is denoted by L, R,andC,respec- The following example shows that the converses of Propo- tively. sition 9 and Corollary 10(iv) are not true.

I ={0} Example 11. Let (𝑋, 𝜏, I) be an ideal space with 𝑋 = {𝑎, 𝑏, 𝑐}, Remark 6. (i) If ,then ∗ 𝜏 = {𝜙, 𝑋, {𝑎}},andI = {𝜙, {𝑏}}.Theset{𝑎, 𝑐} is 𝑅 -perfect set ∗ ∗ 𝜏 𝐼 L ={𝐴⊆𝑋:𝐴⊆𝐴 = cl (𝐴)} which is not a -closed set and hence not a regular- -closed set. ∗ = the collection of all -dense-in-itself sets Proposition 12. If a subset 𝐴 of an ideal topological space ∗ =℘(𝑋) , (𝑋, 𝜏, I) is such that 𝐴∈I,then𝐴 is 𝐶 -perfect. Chinese Journal of Mathematics 3

∗ ∗ Proof. Since 𝐴∈I,𝐴 =𝜙.Then𝐴−𝐴 =𝐴∈I and 3. Main Results ∗ ∗ ∗ 𝐴 −𝐴 = 𝜙 ∈ I.Hence𝐴 is both an 𝐿 -perfect and 𝑅 - In this section, we prove that finite union and intersection of perfect set. ∗ ∗ 𝑅 -perfect sets are again 𝑅 -perfect set. Using these results, Corollary 13. 𝐴 (𝑋, 𝜏, I) we obtain a new topology for the finite topological spaces Let be a subset of an ideal space . ∗ which is finer than 𝜏 -topology. Consider the following. ∗ ∗ In Ideal spaces, usually 𝐴⊂𝐵implies 𝐴 ⊂𝐵.We ∗ (i) If 𝐴∈I,theneverysubsetof𝐴 is a 𝐶 -perfect set. observe that there are some sets 𝐴 and 𝐵 such that 𝐴⊂𝐵 ∗ ∗ ∗ ∗ ∗ but 𝐴 =𝐵 . (ii) If 𝐴 is 𝑅 -perfect, then 𝐴 −𝐴is 𝐶 -perfect. ∗ ∗ ∗ (iii) If 𝐴 is an 𝐿 -perfect set, then 𝐴−𝐴 is a 𝐶 -perfect set. 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}, 𝐴 𝐶∗ 𝐴Δ𝐴∗ 𝐶∗ , , (iv) If is -perfect, then is a -perfect set. {𝑐, 𝑑}}.Herethesets𝐴 = {𝑎} and 𝐵 = {𝑎, 𝑏} are such that ∗ ∗ 𝐴⊆𝐵,but𝐴 =𝐵 = {𝑎, 𝑏, 𝑐}. Proof. The proof follows from Proposition 12. Proposition 20. Let (𝑋, 𝜏, I) be an ideal space. Let 𝐴 and 𝐵 Corollary 14. Let (𝑋, 𝜏) be a space with an ideal I on 𝑋 such ∗ ∗ be two subsets of 𝑋 such that 𝐴⊆𝐵and 𝐴 =𝐵 ;then that 𝜏∼I.Thenforevery𝐴⊆𝑋, 𝐵 𝑅∗ 𝐴 𝑅∗ ∗ ∗ (i) is -perfect if is -perfect; (i) if 𝐴∩𝐴 =𝜙,then𝐴 is 𝐶 -perfect; 𝐴 𝐿∗ 𝐵 𝐿∗ ∗ ∗ (ii) is -perfect if is -perfect. (ii) 𝐴−𝐴 is 𝐶 -perfect; ∗ ∗ ∗ Proof. (i) Let 𝐴 be an 𝑅 -perfect set. Then 𝐴 −𝐴∈I.Now, (iii) if 𝐴 contains nonempty subset 𝐵 with 𝐵⊆𝐵,then𝐴 ∗ ∗ ∗ ∗ 𝐵 −𝐵=𝐴 −𝐵⊆𝐴 −𝐴.Byhereditypropertyofideals, is 𝐶 -perfect. ∗ ∗ 𝐵 −𝐵∈I.Hence𝐵 is 𝑅 -perfect. ∗ ∗ (ii) Let 𝐵 be an 𝐿 -perfect set. Then 𝐵−𝐵 ∈ I.Now, Proof. Follows from Theorem 4 and Proposition 12. ∗ ∗ ∗ 𝐴−𝐴 =𝐴−𝐵 ⊆𝐵−𝐵.Byhereditypropertyofideals, ∗ 𝐴−𝐴∗ ∈ I 𝐴 𝐿∗ Proposition 15. In an ideal space (𝑋, 𝜏, I),every -dense-in- .Hence is -perfect. ∗ itself set is an 𝐿 -perfect set. Corollary 21. Let (𝑋, 𝜏, I) be an ideal space. Let 𝐴 and 𝐵 be ∗ ∗ 𝑋 𝐴⊆𝐵⊆𝑐𝑙∗𝐴 Proof. Let 𝐴 be a -dense-in-itself set of 𝑋.Then𝐴⊆𝐴. two subsets of such that ;then ∗ ∗ Therefore, 𝐴−𝐴 =𝜙∈I.Hence𝐴 is an 𝐿 -perfect set. ∗ ∗ (i) 𝐵 is 𝑅 -perfect if 𝐴 is 𝑅 -perfect, ∗ ∗ Corollary 16. In an ideal space (𝑋, 𝜏, I), (ii) 𝐴 is 𝐿 -perfect if 𝐵 is 𝐿 -perfect.

∗ ∗ ∗ ∗ ∗ ∗ ∗ (i) every 𝐼-dense set is 𝐿 -perfect, Proof. Since 𝐴⊆𝐵⊆cl 𝐴, 𝐴 ⊆𝐵 ⊆(cl 𝐴) =𝐴 .Hence ∗ ∗ ∗ 𝐴 =𝐵. Therefore, the result follows from Proposition 20. (ii) every 𝐼-open set is 𝐿 -perfect, ∗ (iii) every almost strong 𝛽-I-open set is 𝐿 -perfect, ∗ Proposition 22. Let 𝐴 be a subset of an ideal topological space 𝐼 𝐿 ∗ ∗ ∗ (iv) every almost -open set is -perfect, (𝑋, 𝜏, I) such that 𝐴 is 𝐿 -perfect set and 𝐴∩𝐴 is 𝑅 -perfect; ∗ ∗ ∗ (v) every regular-𝐼-closed set is 𝐿 -perfect, then both 𝐴 and 𝐴∩𝐴 are 𝐶 -perfect. ∗ ∗ ∗ (vi) every 𝑓𝐼-set is 𝐿 -perfect. Proof. Since 𝐴 is 𝐿 -perfect, 𝐴−𝐴 ∈ I.ByLemma 1(vii), ∗ ∗ ∗ for every 𝐼∈I, (𝐴∪𝐼) =𝐴 =(𝐴−𝐼) . Therefore, (𝐴∪(𝐴− ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Proof. Since all the above sets are -dense-in-itself, by Propo- 𝐴 )) =𝐴 =(𝐴−(𝐴−𝐴 )) . This implies 𝐴 =(𝐴∩𝐴 ) . 𝐿∗ ∗ ∗ ∗ ∗ sition 15, these sets are -perfect. Therefore, we have 𝐴∩𝐴 ⊆𝐴with (𝐴 ∩ 𝐴 ) =𝐴.By ∗ ∗ ∗ Proposition 20, 𝐴 is 𝑅 -perfect if 𝐴∩𝐴 is 𝑅 -perfect and Remark 17. The members of the ideal of an ideal space ∗ ∗ ∗ ∗ ∗ 𝐴∩𝐴 is 𝐿 -perfect if 𝐴 is 𝐿 -perfect set. Hence 𝐴 is 𝑅 - are 𝐿 -perfect, but the nonempty members of the ideal are ∗ ∗ ∗ perfect and 𝐴∩𝐴 is 𝐿 -perfect. not -dense-in-itself. Therefore, the converses of the above Corollary and Proposition 15 neednottobetrue. 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 Proposition 18. (𝑋, 𝜏, I) ∗ In an ideal space , is 𝐿 -perfect. ∗ 𝐿 ∗ ∗ (i) is an -perfect set, Proof. Since 𝐴 is 𝑅 -perfect, 𝐴 −𝐴 ∈ I.ByLemma 1(vii),for ∗ ∗ ∗ ∗ ∗ 𝑋 𝐿∗ every 𝐼∈I, (𝐴∪𝐼) =𝐴 =(𝐴−𝐼) . Therefore, (𝐴 ∪(𝐴 − (ii) is an -perfect set if the ideal is codense. ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ 𝐴)) =𝐴 =(𝐴 −(𝐴 −𝐴)) . This implies 𝐴 =(𝐴∩𝐴 ) . ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ Proof. (i) Since 𝜙−𝜙 =𝜙∈I, the empty set is an 𝐿 -perfect Therefore, we have 𝐴∩𝐴 ⊆𝐴 with (𝐴 ∩ 𝐴 ) =𝐴 .By ∗ ∗ ∗ ∗ ∗ set. (ii) We know that 𝑋=𝑋 if and only if the ideal I is Proposition 20, 𝐴∩𝐴 is 𝐿 -perfect if 𝐴 is 𝐿 -perfect set. ∗ ∗ ∗ codense. Then 𝑋−𝑋 =𝜙∈I. Hence the result follows. Hence 𝐴∩𝐴 is 𝐿 -perfect. 4 Chinese Journal of Mathematics

∗ ∗ Proposition 24. If 𝐴 and 𝐵 are 𝑅 -perfect sets, then 𝐴∪𝐵is Proposition 32. In an ideal space (𝑋, 𝜏, I), {𝜏 -𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑒𝑡𝑠}∪ ∗ an 𝑅 -perfect set. I ⊆ R.

∗ ∗ Proof. Let 𝐴 and 𝐵 be 𝑅 -perfect sets. Then 𝐴 −𝐴∈I and Proof. The proof follows from Propositions 9 and 12. ∗ ∗ 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 ∪ {𝜏∗ }∪ ∗ The following example shows that -closed sets 𝐵) −(𝐴∪𝐵)∈I. This proves the result. I ≠ R.

∗ ∗ Corollary 25. Finite union of 𝑅 -perfect sets is an 𝑅 -perfect Example 33. Let (𝑋, 𝜏, I) be an ideal space with 𝑋 = {𝑎, 𝑏, 𝑐}, ∗ set. 𝜏 = {0, 𝑋, ,and{𝑎}} I = {0, {𝑏}}.Thecollectionof𝜏 -closed sets is {0, 𝑋, {𝑏}, {𝑏, 𝑐}} and R = {0, 𝑋, {𝑏}, {𝑏, 𝑐}, {𝑎,𝑐}}.

Proof. The proof follows from Proposition 24. ∗ Now, {𝜏 -closed sets}∪I = {0, 𝑋, {𝑏}, {𝑏, 𝑐}} ≠ R. ∗ Proposition 26. If 𝐴 and 𝐵 are 𝐿 -perfect sets, then 𝐴∪𝐵is ∗ Proposition 34. Let (𝑋, 𝜏, I) be an ideal space and 𝐴⊆𝑋. an 𝐿 -perfect set. ∗ ∗ The set 𝐴 is 𝑅 -perfect if and only if 𝐹⊆𝐴 −𝐴in 𝑋 implies ∗ ∗ 𝐹∈I Proof. Since 𝐴 and 𝐵 are 𝐿 -perfect sets, 𝐴−𝐴 ∈ I and 𝐵− that . ∗ ∗ 𝐵 ∈ I. Hence by finite additive property of ideals, (𝐴−𝐴 )∪ ∗ ∗ ∗ ∗ ∗ ∗ Proof. Assume that 𝐴 is an 𝑅 -perfect set. Then 𝐴 −𝐴∈I. (𝐵−𝐵 )∈I.Since(𝐴∪𝐵)−(𝐴∪𝐵) =(𝐴∪𝐵)−(𝐴 ∪𝐵 )⊆ ∗ ∗ ∗ ∗ By heredity property of ideals, every set 𝐹⊆𝐴 −𝐴in 𝑋 is (𝐴−𝐴 )∪(𝐵−𝐵 ),byheredityproperty(𝐴∪𝐵) − (𝐴∪𝐵) ∈ I. ∗ ∗ also in I. Conversely assume that 𝐹⊆𝐴 −𝐴in 𝑋 implies This proves that 𝐴∪𝐵is an 𝐿 -perfect set. ∗ that 𝐹∈I.Since𝐴 −𝐴is a subset of itself, by assumption ∗ ∗ ∗ ∗ 𝐴 −𝐴∈I.Hence𝐴 is 𝑅 -perfect. Corollary 27. Finite union of 𝐿 -perfect sets is an 𝐿 -perfect sets. 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 Proof. The proof follows from Proposition 26. that 𝐹∈I. ∗ Proposition 28. If 𝐴 and 𝐵 are 𝑅 -perfect sets, then 𝐴∩𝐵is ∗ ∗ ∗ Proof. Assume that 𝐴 is an 𝐿 -perfect set. Then 𝐴−𝐴 ∈ I. an 𝑅 -perfect set. ∗ 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 Proof. Suppose that 𝐴 and 𝐵 are 𝑅 -perfect sets. Then 𝐴 − ∗ ∗ that 𝐹∈I.Since𝐴−𝐴 is a subset of itself, by assumption 𝐴∈I and 𝐵 −𝐵∈I. By finite additive property of ideals, ∗ ∗ ∗ ∗ ∗ ∗ ∗ 𝐴−𝐴 ∈ I 𝐴 𝐿 (𝐴 −𝐴)∪(𝐵 −𝐵)∈I.Since(𝐴 ∩𝐵 ) − (𝐴 ∩ 𝐵) ⊆ (𝐴 − .Hence is -perfect. ∗ ∗ ∗ 𝐴) ∪ (𝐵 −𝐵),byheredityproperty(𝐴 ∩𝐵 )−(𝐴∩𝐵)∈I. ∗ ∗ ∗ Proposition 36. Let (𝑋, 𝜏) be a topological space and 𝐴⊆𝑋. Also (𝐴 ∩ 𝐵) −(𝐴∩𝐵)⊆ (𝐴 ∩𝐵 )−(𝐴∩𝐵)∈ I.This Let I1 and I2 be two ideals on 𝑋 with I1 ⊆ I2.Then𝐴 is proves the result. ∗ ∗ 𝑅 -perfect with respect to I2 if it is 𝑅 -perfect with respect to ∗ ∗ I Corollary 29. Finite intersection of 𝑅 -perfect sets is an 𝑅 - 1. perfect set. ∗ ∗ Proof. Since I1 ⊆ I2,𝐴 (I2)⊆𝐴 (I1) by Lemma 1(ii) Let ∗ ∗ 𝐴 be 𝑅 -perfect with respect to I1.Then𝐴 (I1)−𝐴∈I1. Proof. The proof follows from Proposition 28. ∗ ∗ Also, 𝐴 (I2)−𝐴⊆𝐴 (I1)−𝐴. Hence by heredity property ∗ ∗ 𝐴∗(I )−𝐴∈I ⊆ I 𝐴 𝑅∗ Proposition 30. Finite union of 𝐶 -perfect sets is a 𝐶 -perfect of ideals, 2 1 2. Therefore is -perfect I set. with respect to 2.

∗ Proof. From Corollaries 27 and 29,finiteunionof𝐶 -perfect Theorem 37. Let (𝑋, 𝜏) be a space with an ideal I on 𝑋.Then ∗ sets is a 𝐶 -perfect set. the following are equivalent.

Proposition 31. If (𝑋, 𝜏, I) is an ideal topological space with (i) 𝜏∼I. 𝑋 being finite, then the collection R is a topology which is finer ∗ than the topology of 𝜏 -closed sets. (ii) If 𝐴 hasacoverofopensetseachofwhoseintersection with 𝐴 is I,then𝐴 is in I. ∗ Proof. By Corollary 10, 𝑋 and 𝜙 are 𝑅 -perfect sets. By ∗ ∗ 𝐴⊆𝑋 𝐴∩𝐴∗ =𝜙⇒𝐴∈I Corollary 25,finiteunionof𝑅 -perfect sets is an 𝑅 -perfect (iii) If ,then . ∗ set, and by Corollary 29, finite intersection of 𝑅 -perfect sets ∗ ∗ (iv) If 𝐴⊆𝑋,then𝐴−𝐴 ∈ I. is 𝑅 -perfect. Hence the collection R is a topology if 𝑋 is 𝜏∗ 𝑅∗ ∗ ∗ finite. Also, by Proposition 9 every -closed set is an - (v) If 𝐴⊆𝑋and 𝐴 is 𝑅 -perfect set, then 𝐴Δ𝐴 ∈ I. perfect set. Hence the topology R is finer than the topology ∗ ∗ ∗ of 𝜏 -closed sets if 𝑋 is finite. (vi) For every 𝜏 -closed subset 𝐴, 𝐴 −𝐴 ∈ I. Chinese Journal of Mathematics 5

(vii) For every 𝐴⊆𝑋,if𝐴 contains no nonempty subset 𝐵 (ii) If 𝑥∈𝑋is an interior point of a subset 𝐴 of 𝑋,then𝑥 ∗ ∗ with 𝐵⊆𝐵 ,then𝐴∈I. is an 𝑅 -interior point of 𝐴. ∗ ∗ (iii) From (ii), we have int(𝐴) ⊆ int (𝐴) ⊆ 𝑅 -int(𝐴), ∗ Proof. Toprovethistheorem,itisenoughtoprove(iv)⇒ where int (𝐴) denotes interior of 𝐴 with respect to the (v) ⇒ (vi).OthersfollowfromTheorem 4.(iv)⇒ (v) follows 𝜏∗ ∗ ∗ topology . from Proposition 8.Supposethat𝐴Δ𝐴 ∈ I.Since𝐴−𝐴 ⊆ ∗ ∗ 𝐴Δ𝐴 ,byheredityproperty𝐴−𝐴 ∈ I. Hence (v) ⇒ (vi). Theorem 46. Let 𝐴 and 𝐵 besubsetsofanidealspace(𝑋, 𝜏, I) with 𝑋 being finite. Then the following properties hold. ∗ ∗ (i) 𝑅 -int(𝐴)=∪{𝑈:𝑈⊆𝐴and 𝑈 is an 𝑅 -open set}. ∗ ∗ ∗ 4. 𝑅 -Topology (ii) 𝑅 -int(𝐴) is the largest 𝑅 -open set of 𝑋 contained in 𝐴. By Corollary 10 and Proposition 28,weobservethatthe 𝐴 𝑅∗ 𝐴=𝑅∗ (𝐴) collection R satisfies the conditions of being a basis for (iii) is -openifandonlyif -int . ∗ ∗ ∗ ∗ some topology and it will be called as 𝑅𝑐 (𝜏, I). We define (iv) 𝑅 -int(𝑅 -int(𝐴)) = 𝑅 -int(𝐴). ∗ ∗ ∗ ∗ 𝑅 (𝜏, I) = {𝐴 ⊆ 𝑋/𝑋 −𝑐 𝐴∈𝑅 (𝜏, I)} on a nonempty 𝐴⊆𝐵 𝑅 (𝐴) ⊆ 𝑅 (𝐵) ∗ (v) If ,then -int -int . 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 - Proof. The proof follows from Definitions 39, 40,and41. open sets. If there is no confusion about the topology 𝜏 and ∗ ∗ the ideal I,thenwecall𝑅 (𝜏, I) as 𝑅 -topology when 𝑋 is Definition 47. Let 𝐴 be a subset of an ideal topological space ∗ finite. (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 Definition 38. Asubset𝐴 of an ideal topological space 𝐴 𝑅∗ (𝐴) ∗ set by -cl . (𝑋, 𝜏, I) is said to be 𝑅 -closed if it is a complement of an ∗ 𝑅 -open set. Remark 48. For any subset 𝐴 of an ideal topological space ∗ ∗ (𝑋, 𝜏, I), 𝑅 -cl(𝐴) ⊆ cl (𝐴) ⊆ cl(𝐴). 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 Theorem 49. Let 𝐴 and 𝐵 be subsets of an ideal space (𝑋, 𝜏, I) ∗ ∗ 𝑅 -open set contained in 𝐴.One will denote 𝑅 -interior of a where 𝑋 is finite. Then the following properties hold: ∗ set 𝐴 by 𝑅 − int(𝐴). ∗ ∗ (i) 𝑅 -𝑐𝑙(𝐴) = ∩{𝐹 :𝐴⊆𝐹 and 𝐹 is 𝑅 -closed set}. ∗ ∗ Definition 40. Let 𝐴 be a subset of an ideal topological space (ii) 𝐴 is 𝑅 -closed if and only if 𝐴=𝑅 -𝑐𝑙(𝐴) ∗ (𝑋, 𝜏, I) 𝑥∈𝐴 𝑅 ∗ ∗ ∗ .Apoint is said to be an -interior point (iii) 𝑅 -𝑐𝑙(𝑅 -𝑐𝑙(𝐴)) =𝑅 -𝑐𝑙(𝐴) 𝐴 𝑅∗ 𝑈 𝑥 of the set if there exists an -open set of such that 𝐴⊆𝐵 𝑅∗ 𝑐𝑙(𝐴) ⊆𝑅∗ 𝑐𝑙(𝐵) 𝑥∈𝑈⊆𝐴. (iv) If ,then - - .

Definition 41. Let (𝑋, 𝜏, I) be an ideal space and 𝑥∈𝑋.One Proof. The proof follows from Definition 47. ∗ ∗ defines 𝑅 -neighborhood of 𝑥 as an 𝑅 -open set containing ∗ Theorem 50. Let 𝐴 be a subset of an ideal space (𝑋, 𝜏, I). 𝑥. One denotes the set of all 𝑅 -neighborhoods of 𝑥 by ∗ Then the following properties hold: 𝑅 -𝑁(𝑥). 𝑅∗ (𝑋−𝐴)=𝑋−𝑅∗ 𝑐𝑙(𝐴) ∗ (i) -int - ; Proposition 42. In an ideal space (𝑋, 𝜏, I),every𝜏 -open set ∗ ∗ ∗ (ii) 𝑅 -𝑐𝑙(𝑋−𝐴)=𝑋−𝑅 -int(𝐴). is an 𝑅 -open set. ∗ ∗ Proof. The proof follows from Definitions 38, 39,and47. Proof. Let 𝐴 be a 𝜏 -open set. Therefore, 𝑋−𝐴is a 𝜏 -closed ∗ set. That implies that 𝑋−𝐴is an 𝑅 -closed set. Hence 𝐴 is an ∗ 𝑅 -open set. Conflict of Interests

∗ Corollary 43. The topology 𝑅 (𝜏, I) on a finite set 𝑋 is finer The authors declare that there is no conflict of interests ∗ than the topology 𝜏 (𝜏, I). regarding the publication of this paper.

Proof. The proof follows from Proposition 42. References

Corollary 44. For any subset 𝐴 of an ideal topological space [1] T. R. Hamlett and D. Jankovic, “Ideals in general topology,” ∗ (𝑋, 𝜏, I), 𝑖𝑛𝑡(𝐴) is an 𝑅 -open set. General Topology and Applications, pp. 115–125, 1988. [2] T. R. Hamlett and D. Jankovic, “Ideals in topological spaces and Proof. The proof follows from Proposition 42. the set operatory,” Bollettino dell’Unione Matematica Italiana, ∗ vol. 7, pp. 863–874, 1990. Remark 45. (i) Since every open set is an 𝑅 -open set, every ∗ [3] D. Jankovic and T. R. Hamlett, “New topologies from old via neighborhood 𝑈 of a point 𝑥∈𝑋is an 𝑅 -neighborhood of ideals,” The American Mathematical Monthly,vol.97,pp.295– 𝑥. 310, 1990. 6 Chinese Journal of Mathematics

[4] D. Jankovic and T. R. Hamlett, “Compatible extensions of ideals,” Bollettino della Unione Matematica Italiana,vol.7,no. 6, pp. 453–465, 1992. [5] G. Aslim, A. Caksu Guler, and T. Noiri, “On decompositions of continuity and some weaker forms of continuity via idealiza- tion,” Acta Mathematica Hungarica,vol.109,no.3,pp.183–190, 2005. [6] F. G. Arenas, J. Dontchev, and M. L. Puertas, “Idealization of some weak separation axioms,” Acta Mathematica Hungarica, vol.89,no.1-2,pp.47–53,2000. [7] J. Dontchev, M. Ganster, and D. Rose, “Ideal resolvability,” Topology and Its Applications,vol.93,no.1,pp.1–16,1999. [8] E. Ekici and T. Noiri, “Connectedness in ideal topological spaces,” Novi Sad Journal of Mathematics,vol.38,no.2,pp.65– 70, 2008. [9] R. L. Newcomb, Topologies which are compact modulo an ideal [Ph.D. dissertation], University of California at Santa Barbara, 1967. [10] E. Hayashi, “Topologies defined by local properties,” Mathema- tische Annalen,vol.156,no.3,pp.205–215,1964. [11] M. E. Abd El-Monsef, E. F.Lashien, and A. A. Nasef, “On 𝐼-open sets and 𝐼-continuous functions,” Kyungpook Mathematical Journal,vol.32,no.1,pp.21–30,1992. [12] M. E. Abd El-Monsef, R. A. Mahmoud, and A. A. Nasef, “Almost-𝐼-openness and almost-𝐼-continuity,”Journal of the Egyptian Mathematical Society,vol.7,no.2,pp.191–200,1999. [13]E.Hatir,A.Keskin,andT.Noiri,“Onnewdecompositionof continuity via idealization,” JP Journal of Geometry & Topology, vol. 1, no. 3, pp. 53–64, 2003. [14] A. Keskin, T. Noiri, and S¸. Yuksel,¨ “Idealization of a decomposi- tion theorem,” Acta Mathematica Hungarica,vol.102,no.4,pp. 269–277, 2004. [15] A. Keskin, T. Noiri, and S. Yuksel, “fI-sets and decomposition of RIC-continuity,” Acta Mathematica Hungarica,vol.104,no. 4, pp. 307–313, 2004. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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