ISRN GeometryVolume 2012 (2012), Article ID 682829, 9 pagesdoi:10.5402/2012/682829

Research Article

Closed Sets

M. Navaneethakrishnan¹ and S. Alwarsamy²

¹Department of Mathematics, Kamaraj College, Thoothukudi 628003, India²Department of Mathematics, RDM Government Arts College, Sivagankai 630561, India

Received 3 November 2011; Accepted 22 December 2011

Academic Editor: D. Franco

Copyright © 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 Closed sets and discuss their properties. Using these sets, we characterize and .

1. Introduction and Preliminaries

An ideal on a topological space is a nonempty collection of subsets of which satisfies (i) and implies and (ii) implies . Given a topological space with an ideal on and if is the set of all subsets of , a set operator called a local function [1] of with respect to and is defined as follows: for , , where . A Kuratowski closure operator for a topology called the topology, finer than , is defined by [2]. When there is no confusion we will simply write for and for . If is an ideal on , then () is called an ideal space. A subset of an ideal space is said to be closed [3] if . A subset of an ideal space () is said to be an [4] if whenever and is open. A subset of an ideal space () is said to be if is . An ideal space () is said to be a space [4] if every set is . A subset of an ideal space () is said to be locally closed [5] if there exist an open set and a set such that . If , then sets coincide with locally closed sets.

By a space, we always mean a topological space ) with no separation properties assumed. If and will, respectively, denote the closure and interior of in and will denote the interior of in . A subset of a topological space is said to be a set [6] if whenever and is open. A subset of a topological space is said to be a set if is a set. A space is said to be a [6] if every set is a closed set.

For a subset of a space ), the - [7] of is the union of all open sets of whose closures contained in and is denoted by . The subset is called - if . The complement of a set is called a set. Equivalently, is called [7] if , where . The family of all sets of forms a topology [7] on , which is coarser than and is denoted by . A subset of a topological space is said to be a [8] if whenever and is open. A subset of a space is said to be a [8] if is a set. A subset of a space is said to be a [9, 10] if , where .

A subset of an ideal space is said to be [11] if , where . is said to be if is . If , . If , . For a subset of X, [11]. We denote this by . The family of all sets of is a topology and it is denoted by (see [11, Theorem  1]).

Lemma 1.1 (see [11, Corollary  4 if Theorem  2]). .

Lemma 1.2 (see [11, Proposition  3]). Let be an ideal space. Then, we have (1)if or , where is the ideal of nowhere dense sets of , then ,(2)if , then .

Lemma 1.3 (see [5, Theorem  2.13]). Let be an ideal space. Then every subset of is if and only if every open set is .

Lemma 1.4 (see [11, Proposition  1]). Let be an ideal space and a subset of . Then A is if and only if .

Lemma 1.5. Let be an ideal space and a subset of . Then for all is closed.

Proof. If and , then . Then, for some . Since and , from the definition of we have . Therefore, . So and hence is closed.

Lemma 1.6. Let be an ideal space and a subset of . Then, .

Proof. if and only if if and only if there exist such that if and only if and, if and only if .

2. Closed Sets

A subset of an ideal space is said to be a set if whenever and is open. Every set is a set. If , then and hence sets coincide with sets. If , then and hence sets coincide with sets. Since , we have the following inclusion diagram:

Example 2.1. shows that a set needs not to be , and Example 2.2 shows that set needs not to be a set.

Example 2.1. Let , , and . Let . Then is closed and hence . But is not because and .

Example 2.2. Let and be the same as in Example 2.1. Let and . Then is a and hence . Since and , A is not .

Theorem 2.3. If A is a subset of an ideal space , then the following are equivalent. (a)A is .(b)For all , .(c) contains no nonempty closed set.

Proof. . Suppose . If , then . Since is , . It is a contradiction to the fact that . This proves (b).. Suppose , is closed and . Since and closed, . Since , by (b), , a contradiction which proves (c).. Let be an open set containing . Since is closed, is closed and . By hypothesis, and hence . Thus, is .

If we put in Theorem 2.3, we get Corollary 2.4 which gives characterizations for sets. If we put in Theorem 2.3, we get Corollary 2.5 which gives characterizations for sets.

Corollary 2.4. If is a subset of a topological space , then the following are equivalent. (a)A is .(b)For all , .(c) contains no nonempty closed set.

Corollary 2.5 (see [12, Theorem  2.2]). If is a subset of a topological space , then the following are equivalent. (a) is .(b)For all , .(c) contains no nonempty closed set.

The following Corollary 2.6 shows that in , sets are , the proof of which follows from Theorem 2.3(c). Corollary 2.7 gives the relation between and sets.

Corollary 2.6. If is a and A is then A is a set.

Corollary 2.7. If is an ideal space and A is a set, then the following are equivalent. (a)A is a set.(b) is a closed set.

Proof. . If is , then and so is closed.. If is closed, since is , by Theorem 2.3(c), and so is .

If we put in Corollary 2.7, we get Corollary 2.8. If we put in Corollary 2.7, we get Corollary 2.9.

Corollary 2.8. If is a topological space and is a set, then the following are equivalent. (a)A is a set.(b) is a closed set.

Corollary 2.9 (see [6, Corollary  2.3]). If is an topological space and is a set, then the following are equivalent. (a) is a closed set.(b) is a closed set.

Theorem 2.10. If every open set of an ideal space is , then every set is .

Proof. Since every open set is , for every . Therefore, for every subset of X, . So for every subset of . This implies that every set is .

Corollary 2.11. If every subset of an ideal space is , then every set is .

The proof follows from Lemma 1.3 and Theorem 2.10.

Theorem 2.12. Let be an ideal space. Then every subset of X is if and only if every open set is .

Proof. Suppose every subset of is . If is open, then is and so . Hence is . Conversely, suppose and is open. Since every open set is , and so is .

If we put in Theorem 2.12, we get Corollary 2.13. If we put in Theorem 2.12, we get Corollary 2.14.

Corollary 2.13. Let be a topological space. Then every subset of X is if and only if every open set is .

Corollary 2.14 (see [6, Theorem  2.10]). Let be a topological space. Then every subset of X is if and only if every open set is closed.

Theorem 2.15. If every set of an ideal space is , then is a .

Proof. Suppose is not closed for some . Then, is not open. So is . By hypothesis, B is . Therefore, is . So is both open and closed, a contradiction. Hence, is a .

If we put in Theorem 2.15, we get Corollary 2.16.

Corollary 2.16. If every set of a space is , then is a .

Theorem 2.17. Intersection of a set and a set is always .

Proof. Let be a set and a set of an ideal space . Suppose and is open in . Then, . Now is and hence open. So is an open set containing . Since is , . Therefore, which implies that . So is .

If we put in Theorem 2.17, we get Corollary 2.18. If we put in Theorem 2.17, we get Corollary 2.19.

Corollary 2.18 (see [8, Proposition 3.11]). Intersection of a set and a set is always .

Corollary 2.19 (see [6, Corollary  2.7]). Intersection of a set and a closed set is always a set.

Theorem 2.20. A subset of an ideal space is if and only if .

Proof. Suppose is and . If , then there exists an open set such that , but . Since is , and so , a contradiction. Therefore, . Conversely, suppose that . If and is open, then and so . Therefore, is .

If we put in Theorem 2.20, we get Corollary 2.21. If we put in Theorem 2.20, we get Corollary 2.22.

Corollary 2.21. A subset A of a space is if and only if .

Corollary 2.22. A subset A of a space is if and only if .

Theorem 2.23. Let A be a of an ideal space . Then A is if and only if is .

Proof. Suppose is . By Theorem 2.20, , since is a . Therefore, is . Converse follows from the fact that every is .

If we put in Theorem 2.23, we get Corollary 2.24. If we put in Theorem 2.23, we get Corollary 2.25.

Corollary 2.24. Let A be a of a space . Then A is if and only if A is .

Corollary 2.25. Let A be a of a space . Then A is if and only if A is closed.

Theorem 2.26. Let be an ideal space and . If is , then A is also .

Proof. Suppose that is a set. If and is open, then . Since is , . But, . Therefore, A is .

If we put in Theorem 2.26, we get Corollary 2.27. If we put in Theorem 2.26, we get Corollary 2.28.

Corollary 2.27. Let be a topological space and . If is , then A is also .

Corollary 2.28. Let be a space and . If is set, then A is also .

Theorem 2.29. For an ideal space , the following are equivalent. (a)Every set is .(b)Every singleton of is closed or .

Proof. . Let . If is not closed, then and then is trivially . By (a), is . Hence is .. Let be a set and let . We have the following cases.Case 1. is closed. By Theorem 2.3, does not contain a nonempty closed subset. This shows .Case 2. is . Then, . Hence, .Thus in both cases and so , that is, is , which proves (a).

If we put in Theorem 2.29, we get Corollary 2.30. If we put in Theorem 2.29, we get Corollary 2.31.

Corollary 2.30. For an ideal space , the following are equivalent. (a)Every set is .(b)Every singleton of is closed or .

Corollary 2.31 (see [13, Theorem  2.5]). For an ideal space , the following are equivalent. (a)Every set is closed.(b)Every singleton of is closed or open.

Theorem 2.32. Let be an ideal space and . Then A is if and only if , where is and contains no nonempty closed set.

Proof. If is , then by Theorem 2.3, contains no nonempty closed set. If , then is such that . Conversely, suppose , where is and contains no nonempty closed set. Let be an open set such that . Then, which implies that . Now, and is implies that . Since sets are closed, is closed. By hypothesis, and so , which implies that is .

If we put in Theorem 2.32, we get Corollary 2.33. If we put in Theorem 2.32, we get Corollary 2.34.

Corollary 2.33. Let be a space and . Then A is subset of X if and only if , where F is and N contains no nonempty closed set.

Corollary 2.34 (see [12, Corollary  2.3]). Let be a space and . Then A is if and only if , where F is closed and N contains no nonempty closed set.

Theorem 2.35. Let be an ideal space. If A is a subset of X and , then B is also .

Proof. , and since has no nonempty closed subset, neither does . By Theorem 2.3, is .

If we put in Theorem 2.35, we get Corollary 2.36. If we put in Theorem 2.35, we get Corollary 2.37.

Corollary 2.36. Let be a space. If is a subset of X and , then is also .

Corollary 2.37 (see [6, Theorem  2.8]). Let be a space. If A is a subset of X and , then B is also .

A subset of an ideal space is said to be if is .

Theorem 2.38. A subset A of an ideal space is if and only if whenever F is closed and .

Proof. Suppose is a set and is a closed set contained in , then and is open. Since is , and so . Conversely, suppose and is closed. By hypothesis, , which implies that . Therefore, is and hence is .

If we put in Theorem 2.38, we get Corollary 2.39. If we put in Theorem 2.38, we get Corollary 2.40.

Corollary 2.39. A subset A of a space is if and only if whenever F is closed and .

Corollary 2.40 (see [6, Theorem  4.2]). A subset A of a space is if and only if whenever F is closed and .

Theorem 2.41. Let be an ideal space and . Then the following are equivalent. (a)A is .(b) is .(c) is .

Proof. . Suppose is . If is any open set containing , then . Since is , by Theorem 2.3(c), it follows that and so . Since is the only open set containing , is .. Suppose is . If is any closed set contained in , then and is open. Therefore, , which implies that and so ; it follows that . Hence is .The equivalence of (b) and (c) follows from the fact that .

If we put in Theorem 2.41, we get Corollary 2.42. If we put in Theorem 2.41, we get Corollary 2.43.

Corollary 2.42. Let be a space and . Then the following are equivalent. (a)A is .(b) is .(c) is .

Corollary 2.43. Let be an ideal space and . Then the following are equivalent. (a)A is .(b) is .(c) is .

3. Characterization of and -Space

Theorem 3.1. In an ideal space , the following are equivalent. (a)Every set is closed.(b) is a .(c)Every set is closed.

Proof. . Equivalence of (a) and (b) follows from Theorem  4.1 of [8].. Let be a set. Since every set is , A is . By hypothesis, A is closed.. Let . If is not closed, then is not open. So is . By hypothesis, B is closed and so is open. By Corollary 2.31, is a .

Theorem 3.2. In an ideal space the following, are equivalent. (a)Every set is .(b) is a .(c)Every set is .

Proof. . Let . If is not closed, then is the only open set containing and so is . By hypothesis, is . Equivalently is . Thus, every singleton set in is either closed or . By Theorem  3.3 of [4], is a .. The proof follows from the fact that every set is .. The proof follows from the fact that every set is .. Let . If is not closed, then is the only open set containing and so is . By hypothesis, is . Thus, is . Therefore, every singleton set in is either or closed. By Theorem of  3.3 [4], is a .

The proof of the Corollary 3.3 follows from Theorem  3.2 and Theorem  3.10 of [5].

If we put in Corollary 3.3, we get Corollary 3.4.

Corollary 3.3. In an ideal space , the following are equivalent. (a)Every set is .(b)Every set is .(c)Every set is an set.

Corollary 3.4. In a topological space , the following are equivalent. (a)Every set is closed.(b)Every set is a locally closed set.

References

K. Kuratowski, Topology, vol. 1, Academic Press, New York, NY, USA, 1966.

R. Vaidyanathaswamy, Set Topology, Chelsea Publishing, New York, NY, USA, 1946.

D. Janković and T. R. Hamlett, “New topologies from old via ideals,” The American Mathematical Monthly, vol. 97, no. 4, pp. 295–310, 1990.

J. Dontchev, M. Ganster, and T. Noiri, “Unified operation approach of generalized closed sets via topological ideals,” Mathematica Japonica, vol. 49, no. 3, pp. 395–401, 1999.

M. Navaneethakrishnan and D. Sivaraj, “Generalized locally closed sets in ideal topological spaces,” Bulletin of the Allahabad Mathematical Society, vol. 24, no. 1, pp. 13–19, 2009.

N. Levine, “Generalized closed sets in topology,” Rendiconti del Circolo Matematico di Palermo, vol. 19, no. 2, pp. 89–96, 1970.

N. V. Veličko, “H-closed topological spaces,” Matematicheskii Sbornik, vol. 70, no. 112, pp. 98–112, 1966.

J. Dontchev and H. Maki, “On θ-generalized closed sets,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 2, pp. 239–249, 1999.

H. Maki, J. Umehara, and K. Yamamura, “Characterizations of spaces using generalized V-sets,” Indian Journal of Pure and Applied Mathematics, vol. 19, no. 7, pp. 634–640, 1988.

M. Mršević, “On pairwise ℛ and pairwise bitopological spaces,” Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, vol. 30(78), no. 2, pp. 141–148, 1986.

M. Akdag, “open sets,” Kochi Journal of Mathematics, vol. 3, pp. 217–229, 2008.

W. Dunham and N. Levine, “Further results on generalized closed sets in topology,” Kyungpook Mathematical Journal, vol. 20, no. 2, pp. 169–175, 1980.

W. Dunham, “spaces,” Kyungpook Mathematical Journal, vol. 17, no. 2, pp. 161–169, 1977.