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Chinese Journal of Mathematics
Volume 2013 (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

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

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.

Abstract

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

1. Introduction and Preliminaries

The contributions of Hamlett and Jankovic [14] 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 [59] have been generalized using the concept of ideals in topological spaces.

By a space , we mean a topological space with a topology defined on on which no separation axioms are assumed unless otherwise explicitly stated. For a given point in a space , the system of open neighborhoods of is denoted by . For a given subset of a space and are used to denote the closure of and interior of , respectively, with respect to the topology.

A nonempty collection of subsets of a set is said to be an ideal on , if it satisfies the following two conditions: (i) If and , then ; (ii) If and , then . An ideal topological space (or ideal space) means a topological space with an ideal defined on . Let be a topological space with an ideal defined on . Then for any subset of for every is called the local function of with respect to and . If there is no ambiguity, we will write or simply for . Also, defines a Kuratowski closure operator for the topology (or simply ) which is finer than . An ideal on a space is said to be codense ideal if and only if . is always a proper subset of . Also, if and only if the ideal is codense.

Lemma 1 ([see 12]). Let be a space with and being ideals on , and let and be two subsets on . Then(i); (ii); (iii) ( is a closed subset of );(iv); (v); (vi);(vii)for every .

Definition 2 (see [3]). Let be a space with an ideal on . One says that the topology is compatible with the ideal , denoted by , if the following holds, for every : if for every , there exists a such that , then .

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

Theorem 4 ([3]). Let be a space with an ideal on . Then the following are equivalent.(i).(ii)If has a cover of open sets each of whose intersection with is , then is in .(iii)For every , .(iv)For every , .(v)For every -closed subset .(vi)For every , if contains no nonempty subset with , then .

2. -Perfect, -Perfect, and -Perfect Sets

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

Definition 5. Let be an ideal topological space. A subset of is said to be(i)-perfect if ,(ii)-perfect if ,(iii)-perfect if is both -perfect and -perfect. The collection of -perfect sets, -perfect sets, and -perfect sets in is denoted by , , and , respectively.

Remark 6. (i) If , then
(ii) If , then .
(iii) If , then (by Theorem 4(iv)).

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

Proposition 8. If a subset of an ideal space is -perfect, then .

Proof. Since is both -perfect and -perfect, and . By the finite additive property of ideals, . Hence .

Proposition 9. In an ideal space , every -closed set is -perfect.

Proof. Let be a -closed set. Therefore, . Hence . Therefore, is an -perfect set.

Corollary 10. In an ideal space ,(i) and are -perfect sets,(ii)every -closed set is -perfect,(iii)for any subset of an ideal topological space are -perfect sets,(iv)every regular--closed set is -perfect.

Proof. The proof follows from Proposition 9.

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

Example 11. Let be an ideal space with , , and . The set is -perfect set which is not a -closed set and hence not a regular--closed set.

Proposition 12. If a subset of an ideal topological space is such that , then is -perfect.

Proof. Since . Then and . Hence is both an -perfect and -perfect set.

Corollary 13. Let be a subset of an ideal space . Consider the following.(i)If , then every subset of is a -perfect set.(ii)If is -perfect, then is -perfect.(iii)If is an -perfect set, then is a -perfect set.(iv)If is -perfect, then is a -perfect set.

Proof. The proof follows from Proposition 12.

Corollary 14. Let be a space with an ideal on such that . Then for every ,(i)if , then is -perfect;(ii) is -perfect;(iii)if contains nonempty subset with , then is -perfect.

Proof. Follows from Theorem 4 and Proposition 12.

Proposition 15. In an ideal space , every *-dense-in-itself set is an -perfect set.

Proof. Let be a *-dense-in-itself set of . Then . Therefore, . Hence is an -perfect set.

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

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

Remark 17. The members of the ideal of an ideal space are -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 ,(i)empty set is an  -perfect set,(ii)  is an  -perfect set if the ideal is codense.

Proof. (i) Since , the empty set is an -perfect set. (ii) We know that if and only if the ideal is codense. Then . Hence the result follows.

3. Main Results

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

In Ideal spaces, usually implies . We observe that there are some sets and such that but .

Example 19. Let be an ideal space with , , . Here the sets and are such that , but .

Proposition 20. Let be an ideal space. Let and be two subsets of such that and ; then(i)  is -perfect if is -perfect;(ii)  is -perfect if is -perfect.

Proof. (i) Let be an -perfect set. Then . Now, . By heredity property of ideals, . Hence is -perfect.
(ii) Let be an -perfect set. Then . Now, . By heredity property of ideals, . Hence is -perfect.

Corollary 21. Let be an ideal space. Let and be two subsets of such that ; then(i) is -perfect if is -perfect,(ii) is -perfect if is -perfect.

Proof. Since , . Hence . Therefore, the result follows from Proposition 20.

Proposition 22. Let be a subset of an ideal topological space such that is -perfect set and is -perfect; then both and are -perfect.

Proof. Since is -perfect, . By Lemma 1(vii), for every , . Therefore, . This implies . Therefore, we have with . By Proposition 20, is -perfect if is -perfect and is -perfect if is -perfect set. Hence is -perfect and is -perfect.

Proposition 23. If a subset of an ideal topological space is -perfect set and is -perfect, then is -perfect.

Proof. Since is -perfect, . By Lemma 1, for every , . Therefore, . This implies . Therefore, we have with . By Proposition 20, is -perfect if is -perfect set. Hence is -perfect.

Proposition 24. If and are -perfect sets, then is an -perfect set.

Proof. Let and be -perfect sets. Then and . By finite additive property of ideals, . Since , by heredity property . Hence . This proves the result.

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

Proof. The proof follows from Proposition 24.

Proposition 26. If and are -perfect sets, then is an -perfect set.

Proof. Since and are -perfect sets, and . Hence by finite additive property of ideals, . Since , by heredity property . This proves that is an -perfect set.

Corollary 27. Finite union of -perfect sets is an -perfect sets.

Proof. The proof follows from Proposition 26.

Proposition 28. If and are -perfect sets, then is an -perfect set.

Proof. Suppose that and are -perfect sets. Then and . By finite additive property of ideals, . Since , by heredity property . Also . This proves the result.

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

Proof. The proof follows from Proposition 28.

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

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

Proposition 31. If is an ideal topological space with being finite, then the collection is a topology which is finer than the topology of -closed sets.

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

Proposition 32. In an ideal space , .

Proof. The proof follows from Propositions 9 and 12.

The following example shows that .

Example 33. Let be an ideal space with , , and . The collection of -closed sets is and .

Now, -closed sets.

Proposition 34. Let be an ideal space and . The set is -perfect if and only if in implies that .

Proof. Assume that is an -perfect set. Then . By heredity property of ideals, every set in is also in . Conversely assume that in implies that . Since is a subset of itself, by assumption . Hence is -perfect.

Proposition 35. Let be an ideal space and . The set is -perfect if and only if in implies that .

Proof. Assume that is an -perfect set. Then . By heredity property of ideals, every set in is also in . Conversely, assume that in implies that . Since is a subset of itself, by assumption . Hence is -perfect.

Proposition 36. Let be a topological space and . Let and be two ideals on with . Then is -perfect with respect to if it is -perfect with respect to .

Proof. Since by Lemma 1(ii) Let be -perfect with respect to . Then . Also, . Hence by heredity property of ideals, . Therefore is -perfect with respect to .

Theorem 37. Let be a space with an ideal on . Then the following are equivalent.(i). (ii)If has a cover of open sets each of whose intersection with is , then is in .(iii)If , then .(iv)If , then .(v)If and is -perfect set, then .(vi)For every -closed subset .(vii)For every , if contains no nonempty subset with , then .

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 . Since , by heredity property . Hence (v) (vi).

4. -Topology

By Corollary 10 and Proposition 28, we observe that the collection satisfies the conditions of being a basis for some topology and it will be called as . We define on a nonempty set . Clearly, is a topology if the set is finite. The members of the collection will be called -open sets. If there is no confusion about the topology and the ideal , then we call as -topology when is finite.

Definition 38. A subset of an ideal topological space is said to be -closed if it is a complement of an -open set.

Definition 39. Let be a subset of an ideal topological space . One defines -interior of the set as the largest -open set contained in .One will denote -interior of a set by .

Definition 40. Let be a subset of an ideal topological space . A point is said to be an -interior point of the set if there exists an -open set of such that .

Definition 41. Let be an ideal space and . One defines -neighborhood of as an -open set containing . One denotes the set of all -neighborhoods of by .

Proposition 42. In an ideal space , every -open set is an -open set.

Proof. Let be a -open set. Therefore, is a -closed set. That implies that is an -closed set. Hence is an -open set.

Corollary 43. The topology on a finite set is finer than the topology .

Proof. The proof follows from Proposition 42.

Corollary 44. For any subset of an ideal topological space , is an -open set.

Proof. The proof follows from Proposition 42.

Remark 45. (i) Since every open set is an -open set, every neighborhood of a point is an -neighborhood of .
(ii) If is an interior point of a subset of , then is an -interior point of .
(iii) From (ii), we have , where denotes interior of with respect to the topology .

Theorem 46. Let and be subsets of an ideal space with being finite. Then the following properties hold.(i) and is an -open set.(ii) is the largest -open set of contained in .(iii) is -open if and only if .(iv). (v)If , then .

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

Definition 47. Let be a subset of an ideal topological space . One defines -closure of the set as the smallest -closed set containing . One will denote -closure of a set by .

Remark 48. For any subset of an ideal topological space , .

Theorem 49. Let and be subsets of an ideal space where is finite. Then the following properties hold:(i) and is -closed set.(ii) is -closed if and only if (iii)(iv)If , then .

Proof. The proof follows from Definition 47.

Theorem 50. Let be a subset of an ideal space . Then the following properties hold:(i);(ii).

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.

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