-Continuity and some Results on -Closure Operator

Abstract

Al-Jarrah et al. defined a new topological operator, namely, -closure operator, and proved that it lies between the -closure operator and the usual closure operator. Al-Ghour et al. defined -closure operator and discussed its properties. In this paper, it is proved that the -closure operator lies between the -closure operator and the usual closure operator. Also, sufficient conditions are given for the equivalence between the -closure operator and the -closure operator. Moreover, we define three new types of continuity, namely, -continuity, --continuity, and almost -continuity, and discuss their properties. It is proved that the concepts of usual continuity and -continuity are independent of each other. In addition, the relationships between different types of continuity have been investigated. Further, some examples and counter examples are given.

1. Introduction

Generating new topologies from old ones is a very interesting and useful concept in topology. Let be a topological space (abbreviated as ts) with no separation axioms assumed. The closure of a subset of is denoted by the symbol , and the interior of a subset is denoted by the symbol . A subset is regular open (regular closed) if . A point in is called a -cluster point [1] of a subset if for every open subset containing , . The collection of all -cluster points of a subset is called -closure of and is denoted by the symbol . A subset is called -closed [1] if and its complement is -open. The topology on of all -open sets is denoted by the symbol . The basis of is the family of all regular open sets. A ts is called semiregular if . The -closure operator and its related concepts have been studied and generalized by many authors [2–6]. All regular open sets, with the exception of the empty set, are included in the class of somewhere dense sets. Somewhere dense sets have been studied by many authors [7, 8]. In 1982, Hdeib [9] classified -closed sets as well as the complement of an -closed set, i.e., an -open set. stands for the collection of all -open sets. The collection forms a topology finer than the usual topology . Further, the -open and -closed sets and their properties have been studied by many authors [10–12]. In paper [13], -regularity has been defined as a generalization of regularity by Al-Ghour. In 2019, Al-Jarrah et al. [14] defined -closure of a set by using -closure as follows: A point in is called -contact point of a subset if the interior of every -closed neighborhood (abbreviated as nbhd) of intersects . The set of all -contact points of is called the -closure of and is denoted by the symbol . The -closure operator is a new topological operator which strictly lies between the usual closure and the -closure operators. Al Ghour et al. [13, 15] have defined new topological operators, namely, -closure operator and -closure operator, and discussed their properties.

This paper is organized as follows: After the introduction, definitions and results that have been already defined and proved are included in preliminaries and will be utilized to support various findings in the following sections. In the next section, we have proved that -closure operator lies between the usual closure operator and -closure operator. Sufficient conditions are given for the equivalence of -closure operator and -closure operator, and also of -closure operator and the usual closure operator. In last section, we define some new types of continuity, namely, -continuity, --continuity, and almost -continuity. We observed that continuity and -continuity are independent of each other. Moreover, we give some sufficient conditions for a -continuous function to be continuous. In addition, we establish the relationships between different types of continuity and discuss some of their properties. Some examples and counter-examples are also provided.

2. Preliminaries

Definition 1 (see [9]). Let be a ts and , then, the set of condensation points of is denoted by the symbol and defined as .
The set is called -closed if and its complement is -open. represents the collection of all -open sets.

Theorem 2 (see [9]). Let be a ts and , then (a) forms a topology on (b) and in generalThe closure of in is denoted by the symbol Also, and in general.

Definition 3 (see [1]). Let be a ts and , then, the set is called the -closure of and denoted by the symbol .
The set is called -closed if and its complement is -open. represents the collection of all -open sets.

Theorem 4 (see [1]). Let be a ts and , then (a) forms a topology on (b) and in general

Theorem 5 (see [1]). A ts is regular if .

Definition 6 (see [13]). Let be a ts and , then, the set is called the -closure of and denoted by the symbol .
The set is said to be -closed if and its complement is -open. represents the collection of all -open sets.

Theorem 7 (see [13]). Let be a ts and , then (a) forms a topology on (b) and in general

Theorem 8 (see [13]). A ts is -regular if .

Definition 9 (see [14]). Let be a ts and , then, the set is called the -closure of and denoted by the symbol .
A set is called -closed if and its complement is -open. represents the collection of all -open sets.

Theorem 10 (see [14]). Let be a ts and let . Then (a)(b)If is -closed, then is -closed(c)If is -closed, then is closed

Definition 11. A ts is called (a)[16] Locally countable if for each , there exists an open subset such that and is countable(b)[13] Antilocally countable if every nonempty open subset is uncountable(c)[17] Locally indiscrete if every open subset is closed

Lemma 12 (see [13]). (a)If is an antilocally countable ts, then for all , (b)If is a locally countable ts, then is a discrete topology

Definition 13 (see [13]). A ts is called -locally indiscrete if every open subset is -closed.

Theorem 14 (see [13]). (a)Every ts that is locally indiscrete is -locally indiscrete(b)Every ts that is locally countable is -locally indiscrete

Theorem 15 (see [14]). Let be a ts and . Then, if for each , and there is such that .

Theorem 16 (see[13]). Let be an -locally indiscrete ts and let . Then (a)(b)If is closed in , then, is -closed in

Theorem 17 (see [13]). For any ts , the following are equivalent: (a) is -regular(b)For each subset ,

Lemma 18 (see [1]). Let be a ts. Then, for each , .

Theorem 19 (see [13]). Let be a ts. Then, for each , .

Theorem 20 (see [13]). (a)Every open -closed subset in a ts is -open(b)Every countable open subset in a ts is -open

Definition 21 (see [18]). A function is called -continuous if for each and each open nbhd of , there is an open nbhd of such that .

Definition 22 (see [13]). A ts is -regular if for each open nbhd of there exists an open subset such that .

Definition 23 (see [19]). A function is called -continuous if for each and each open nbhd of , there is an open nbhd of such that

Definition 24 (see [20]). A function is called almost open if for every regular open subset of , is open in .

Definition 25 (see [20]). Let and be two topological spaces (abbreviated as tss), and a function is called almost continuous if for each and for each open nbhd of , there exists an open nbhd of such that .

3. Some Results on -Closure Operator

In this section, the relationships between usual closure operator, -closure operator, and -closure operator have been discussed.

Theorem 26. Let be a ts and let . Then (a)(b)If is -closed, then is -closed

Proof. (a)To prove , let and with . Therefore, . We know , and then, . We have (b)Let be -closed, then, . By using part (a) and Theorem 10(a), , and hence, is -closed.The following results give the equivalence conditions for the -closure operator with -closure and -closure operators.

Theorem 27. Let be a ts and . If is -locally indiscrete, then (a)(b) is closed in if is -closed in (c) is -closed in if is -closed in

Proof. (a)By Theorems 26, 10(a) and 16(b)Suppose is closed, then, . By part (a), , and therefore, is -closed. The converse follows from Theorem 10(c).(c)Suppose is -closed, then . By part (a), , and therefore, is -closed. The converse follows from Theorem 26(b).

Corollary 28. Let be a ts and . If is -regular, then (a)(b) is closed in if is -closed in (c) is -closed in if is -closed in

Remark 29. The statements of Theorem 27 hold even if is locally indiscrete or locally countable by using Theorem 14.

Theorem 30. Let be a ts and . If is antilocally countable, then (a)(b) is -closed in if is -closed in

Proof. (a)By Theorem 10(a), we have . Now to prove , let and open subset such that . By definition, . Given is antilocally countable, then by Lemma 12(), , and hence, . Therefore, we get (b)Suppose is -closed, then . By part (a), . Therefore, is -closed. The converse follows from Theorem 10(b).The following result gives some properties of -closure operator.

Theorem 31. Let be a ts and let . Then (a)If , then (b)For every , (c)For every , is a closed subset in (d)For every , (e)For every ,

Proof. (a)Let and with . Since , then . Since , therefore, . Hence, (b)By part (), we get . Now to prove . Let , and there are two open sets containing such that and . Now, we have andHence, we get . (c)To prove . Let , there is an open set containing such that . Therefore, . Hence, we get (d)By Theorems 19, 10, and 26(e)Follows from part () and Lemma 18

Theorem 32 (see [14]). Let be a ts, then, .
The equality in Theorem 32 does not hold, as the following examples show.

Example 1. Let be a ts where is real line and . The regular open subsets of ts are , and then, . Also is antilocally countable ts. Therefore, by Theorem 30, we have . Then, . Hence, we get .

Example 2. Let be a ts where is set of integer and . The regular open subsets of ts are , and then, . Also is locally countable ts. Therefore, by Remark 29, we have . Hence, .
If ts is semiregular, then .

Theorem 33. Let be a ts, then, .

Proof. To prove . Let , then, is -closed and by Theorem 26, is -closed. Hence, .
The equality in Theorem 33 does not hold, as the following example shows.

Example 3. Let be a ts where is the real line and . By Example 2.26 of [13], . Now, to prove by using Theorem 15, we have to find for each such that . Let , then, . Now, . So, is true for , and this implies . But . Hence, .
If ts is -regular (-locally indiscrete or locally indiscrete or locally countable), then . If ts is antilocally countable, then . If ts is regular, then . If a ts is antilocally countable and -regular, then . It can be easily seen by Example 2 and Example 3 that and are incomparable.

Theorem 34. (a)Every open -closed subset in a ts is -open(b)Every countable open subset in a ts is -open

Proof. Directly follow from Theorems 33 and 20.

4. -Continuity

In this section, we define some new types of continuity and discuss their relationships.

Definition 35. A function is called -continuous if for each and each open subset of containing , there is an open subset of containing such that .

Theorem 36. Every -continuous function is -continuous.

Proof. Let and be two tss and be a -continuous function. Let be in and let open nbhd of . By assumption, there exists an open nbhd of such that . This implies that is -continuous.
The converse of the Theorem 36 does not hold, as the following example shows.

Example 4. Let a function ( be the set of integers) be defined as , where is indiscrete topology on and is finite complement topology on . Then, is -continuous but not -continuous. Let and such that . Take such that and . Hence, is -continuous. Now to prove is not -continuous, let and . Then, with . Suppose there exists such that and , but open subset in containing is . Take , then is not true because and . This implies that is not -continuous.

The following result gives sufficient criteria for a -continuous to be -continuous.

Theorem 37. Let and be two tss, and let be a -continuous function with antilocally countable, then, is -continuous.

Proof. Let be in and let be any open nbhd of . By assumption, is -continuous, so there exists an open nbhd of such that . Given is antilocally countable, then by Lemma 12, we have , and thus, . This implies that is -continuous.
The independence of continuity and -continuity have been observed by Noiri [18]. In the following Examples 5 and 6, it is shown that the concepts of continuity and -continuity are independent of each other.

Example 5. Let be the real line with the usual topology and the real line with the co-countable topology. Let be the function defined by . The ts is antilocally countable and by Example 4.4 of [18], is -continuous but not continuous, and then, by Theorem 37, is -continuous.

Example 6. Let be the set of natural number with topology , and let a function be defined as . It can be easily seen that is a continuous function. To check that is -continuous, take and , and then, is open and . Also is locally countable, and then, . Now for an open subset with , we have or . In both possibilities, and which is not possible. This implies that is not -continuous.

The following results from 38 to 41 give sufficient criteria for a -continuous function to be continuous.

Theorem 38. Let and be two tss and let be a -continuous function with locally countable, then, is continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is -continuous, so there exists open nbhd of such that . Since is locally countable, then by Lemma 12(b), we have is the discrete topology. Therefore, , and thus, . This implies that is continuous.

Theorem 39. Let and be two ts’s and let be a -continuous function with -locally indiscrete, then is continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is -continuous, so there exists an open nbhd of such that . Since is -locally indiscrete, then is -closed and . Thus . Hence is continuous.

Corollary 40. Let and be two ts’s and let be a -continuous function with locally indiscrete, then, is continuous.

Theorem 41. Let and be two ts’s and let be a -continuous function with -regular, then is continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is an -regular ts, then by Definition 22, there exists of open subset such that . Given is -continuous, there exists an open subset containing such that . Thus, we have . This implies that is continuous.
The next result gives sufficient criteria for a -continuous function to be -continuous.

Theorem 42. If is a -continuous function and almost-open, then is -continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is -continuous, so there exists an open nbhd of such that . Therefore, . Since is almost-open, then and thus . Hence, is -continuous.

Corollary 43. If is a -continuous function and open, then is -continuous.

Definition 44. Let and be two ts’s and a function is called --continuous function if for each and for each open nbhd of , there is an open subset of containing such that .

Theorem 45. Every --continuous function is almost continuous.

Proof. Let and be two ts’s and let be an --continuous function. Let be in and let be an open nbhd of . By assumption, is --continuous, so there is an open nbhd in containing such that . Hence, is almost continuous.
The following results give sufficient criteria for the almost continuous function to be --continuous.

Theorem 46. Let and be two ts’s and let be almost continuous with -locally indiscrete, then is --continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is almost continuous, so there is an open nbhd of such that . Since is -locally indiscrete, then is -closed and . Thus, . Hence, is --continuous.
The following corollaries can be easily proved by Theorems 14 and 46.

Corollary 47. Let and be two ts’s and let be almost continuous with locally indiscrete, then is --continuous.

Corollary 48. Let and be two ts’s and let be almost continuous with locally countable, then is --continuous.

Theorem 49. Let and be two ts’s and let be almost continuous with -regular, then is --continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is almost continuous, so there is an open nbhd of such that . Since is -regular, then there is an open subset in containing such that . Therefore, . Hence, is --continuous.

Theorem 50. Every -continuous function is --continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is -continuous, so there is an open nbhd of such that . Thus, . Hence, is --continuous.
The converse of the Theorem 50 does not hold, as the following example shows.

Example 7. The function defined in Example 4.5 of [18], shows that is continuous but not -continuous, and then, is almost continuous by definition. Also, is locally countable, so by Corollary 48, is --continuous.

The following result gives sufficient criteria for a --continuous function to be -continuous.

Theorem 51. Let and be two ts’s and let be --continuous with antilocally countable, then is -continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is --continuous, so there is an open nbhd of such that . Since is antilocally countable, then by Lemma 12, . Therefore, . Hence, is -continuous.

Definition 52. A function is called almost -continuous if for each and each open subset of containing , there is an open subset of containing such that .

Theorem 53. Every almost -continuous function is almost continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is almost -continuous, so there is an open nbhd of such that . By Theorem 2, , and therefore, . Hence, is almost continuous.
The converse of the Theorem 53 does not hold, as the following example shows.

Example 8. Consider a function defined by , where is indiscrete topology and is finite complement topology on . To check is almost continuous, let and such that . Then, for every , . Choose , then and . Now, to check is not almost -continuous, let and . Since is locally countable, then, . So containing is , then, which is not true. Hence, is not almost -continuous.

The following result gives sufficient criteria for the almost continuous function to be almost -continuous:

Theorem 54. Let and be two ts’s and let be almost continuous with antilocally countable, then is almost -continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is almost continuous, so there is an open nbhd of such that . Since is antilocally countable, then . Therefore, . Hence, is almost -continuous.

Theorem 55. Every continuous function is almost -continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is continuous, so there is an open nbhd of such that . Thus, . Hence, is almost -continuous.
The converse of the Theorem 55 does not hold, as the following example shows.

Example 9. Consider a function defined by , where is the usual topology and is the countable complement topology on . It is easy to check that is discontinuous. To check, is almost -continuous. Let and such that . Since is antilocally countable, then, . Therefore, for any open subset, such that . Take such that and . Hence, is almost -continuous.

The following result gives sufficient criteria for the almost -continuous function to be continuous.

Theorem 56. Let and be two ts’s and let be almost -continuous with -locally indiscrete, then is continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is almost -continuous, so there is an open nbhd of such that . Since is -locally indiscrete, then . Therefore, . Hence, is continuous.
The following corollaries can be easily proved by Theorems 14 and 56.

Corollary 57. Let and be two ts’s and let be almost -continuous with locally indiscrete, then is continuous.

Corollary 58. Let and be two ts’s and let be almost -continuous with locally countable, then is continuous.

Theorem 59. Every -continuous function is almost -continuous.

Proof. Let be in and let be an open nbhd of . By assumption, is -continuous, so there is an open nbhd in containing such that . Thus, . Hence, is almost -continuous.
The converse of the Theorem 59 does not hold, as the following example shows the same.

Example 10. In Example 6, is continuous but not -continuous. Also, by Theorem 55, every continuous function is almost -continuous.

5. Conclusion

In this work, we continue the research on -closure operator, and some properties of -closure operator are discussed. Also, we give some sufficient conditions for the equivalence of -closure and -closure operators. Further, we define some new types of continuity, namely, -continuity, --continuity, and almost -continuity. Also, some examples and counter examples are given.

Data Availability

No data were used for this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are very grateful to the referees for their valuable comments and suggestions. One of the authors (MS) is grateful to the University Grants Commission, New Delhi, for providing financial support in the form of a Senior Research Fellowship through Grant No. F.16-6(DEC.2016)/2017(NET)(404363) to complete this study.