International Journal of Mathematics and Mathematical Sciences

Volume 2013, Article ID 381068, 4 pages

http://dx.doi.org/10.1155/2013/381068

## On -Continuous Functions in Biminimal Structure Spaces

Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham 44150, Thailand

Received 17 June 2013; Accepted 18 August 2013

Academic Editor: Luc Vrancken

Copyright © 2013 Chawalit Boonpok et al. 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 introduce the notion of -continuous functions and some other forms of continuity in biminimal structure spaces. Some new characterizations and several fundamental properties of -continuous functions are obtained.

#### 1. Introduction

Weak continuity due to Levine [1] is one of the most important weak forms of continuity in topological spaces. Rose [2] has introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. In [3], Baker obtained several properties of subweak continuity which are analogous to results in [4]. Njåstad [5] introduced a weak form of open sets called -sets. In [6], the author showed that connectedness is preserved under weakly -continuous surjections. Mashhour et al. [7] have called strongly semicontinuous -continuous and obtained several properties of such functions. In [8], they stated without proofs that -continuity implies -continuity and is independent of almost continuity in the sense of Singal [9]. On the other hand, in 1980 Maheshwari and Thakur [10] defined -irresolute and obtained several properties of -irresolute functions. Levine [11] defined the notions of semiopen sets and semicontinuity in topological spaces. Maheshwari and Prasad [12] extended the notions of semiopen sets and semicontinuity to the bitopological setting. Bose [13] further investigated many properties of semiopen sets and semicontinuity in bitopological spaces. Mashhour et al. [7] introduced the notions of preopen sets and precontinuity in topological spaces. Jelić [14] generalized the notions of preopen sets and precontinuity to the setting of bitopological spaces. The purpose of the present paper is to introduce the notion of -continuous functions in biminimal structure spaces and investigate the properties of these functions.

#### 2. Preliminaries

*Definition 1 (see [15]). * Let be a nonempty set and the power set of . A subfamily of is called a *minimal structure* (briefly *m-structure*) on if and .

By , we denote a nonempty set with an -structure on and it is called an -space. Each member of is said to be *-open,* and the complement of an -open set is said to be *-closed*.

*Definition 2 (see [16]). *Let be a nonempty set and an -structure on . For a subset of , the -*closure* of and the -*interior* of are defined as follows: (1); (2).

Lemma 3 (see [16]). *Let be a nonempty set and a minimal structure on . For subset and of , the following properties hold: *(1)* and . *(2)*If , then and if , then . *(3)*, , , and . *(4)*If , then and . *(5)* and . *(6)* and . *

*Definition 4 (see [16]). *An -structure on a nonempty set is said to *have property * if the union of any family of subsets belonging to belongs to .

Lemma 5 (see [17]). *Let be a nonempty set and an -structure on satisfying property . For a subset of , the following properties hold:*(1)* if and only if . *(2)* is -closed if and only if . *(3)* and is -closed. *

*Definition 6. *Let be a nonempty set and minimal structures on . The triple is called a *bispace* (briefly bi* m-space*) [18] or *biminimal structure space* (briefly *bimspace*) [19].

Let be a biminimal structure space and a subset of . The -closure of and the -interior of with respect to are denoted by and , respectively, for . Also and .

*Definition 7 (see [20]). *A subset of a biminimal structure space is said to be ---*open* (resp., --*semiopen*, --*preopen*) if (resp., , ).

Lemma 8. *Let be a biminimal structure space and a family of subsets of . *(1)*If is ---open for each , then is ---open. *(2)*If is ---closed for each , then is ---closed. *

*Definition 9. *Let be a biminimal structure space and a subset of . Then the -*α*-closure of and the -*α*-interior of are defined as follows: (1); (2).

Lemma 10. *Let be a biminimal structure space. For a subset of , the following properties hold: *(1)* is ---closed; *(2)* is ---open; *(3)* is ---closed if and only if ; *(4)* is ---open if and only if . *

Lemma 11. *Let be a biminimal structure space and a subset of . Then if and only if for every ---open set containing . *

Lemma 12. *Let be a biminimal structure space and a subset A of *(1)*; *(2)*. *

*Proof. *(1) By Lemma 10, is ---closed. Then is ---open. On the other hand, , and hence . Conversely, let . Then there exists an ---open such that . Then is ---closed and . Since , and hence . Therefore, .

(2) This follows from (1) immediately.

#### 3. Characterizations of -Continuous Functions

*Definition 13. *Let be a biminimal structure space and a bitopological space. A function is said to be -*continuous at point * if for each -open set of containing , there exists an ---open set containing such that .

A function is said to be *-continuous* if has this property at each point of .

Theorem 14. *Let be a biminimal structure space and a bitopological space. For a function , the following properties are equivalent: *(1)* is -continuous at ; *(2)* for every -open set containing ; *(3)* for every subset of with ; *(4)* for every subset of with ; *(5)* for every subset of with ; *(6)* for every -closed set of with . *

*Proof. *: Let containing . Then there exists an ---open set containing such that . Thus, . Hence, .

: Let be any subset of such that , and let containing . Then . There exists an ---open set such that . Since , by Lemma 11, and . Since containing , , and hence .

: Let be any subset of and . Then by (3), . Hence, we have .

: Let be any subset of such that . Then . By (4), we have . Hence, .

: Let be any -closed set of such that . Then . By (5), . Hence, .

: Let and containing . Suppose that . Then . By (6), . Hence, . This contradicts the hypothesis.

: Let containing . Then by (2), , and hence there exists an ---open set containing such that . Therefore, , and hence is -continuous at .

Theorem 15. *Let be a biminimal structure space and a bitopological space. For a function , the following properties are equivalent: *(1)* is -continuous; *(2)* is ---open for every -open set in ; *(3)* for every subset of ; *(4)* for every subset of ; *(5)* for every subset of ;*(6)* is ---closed for every -closed set of . *

*Proof. *: Let be any -open set of and . Then . There exists an ---open set containing such that , and hence . Hence, we have . Therefore, . This shows that is ---open.

: Let be any subset of . Since is -open and by (2), we have . Hence, . Therefore, . Since and hence . Therefore, .

: Let be any subset of . Then by (3), we have . Hence, .

: Let be any subset of and . Then , and hence . Therefore, . By (4), we have , and hence . Therefore, .

: Let be any -closed set of such that . Then . By (5), . Therefore, , and hence . This shows that is ---closed.

: Let be any -closed set of such that . Then by (6), we have . Hence, by Theorem 14(6), is -continuous.

*Definition 16. *Let be a biminimal structure space and a bitopological space. A function is said to be(1)-*continuous* (resp., -*continuous*) if is --semiopen (resp., --semiopen) in for each -open set of ; (2)-*continuous* or --*precontinuous* [21] if is --preopen in for each -open set of .

Lemma 17. *Let be a subset of a biminimal structure space . Then is ---open in if and only if is --semiopen and --preopen in . *

*Proof. *Let be ---open in . By the definition of ---open sets, we have and . Therefore, we obtain is --semiopen and --preopen in .

Conversely, let be --semiopen and --preopen in . Since is --semiopen, , and hence it follows from is --preopen that . Therefore, is ---open in .

Theorem 18. *Let be a biminimal structure space and a bitopological space. A function is -continuous if and only if is -continuous and -continuous.*

*Proof. *This is an immediate consequence of Lemma 17.

*Definition 19. *Let be a biminimal structure space and a bitopological space. A function is said to be -*continuous* if is -continuous and for every such that .

Theorem 20. *Let be a biminimal structure space, and let have property , and let be a bitopological space. A function is -continuous if and only if for every such that *(1)* and *(2)*. *

*Proof. *It is obvious that is -continuous if and only if satisfies (1). We assume that is -continuous and show equality (2). For any , it follows from (1) that . Since the intersection of two --regular open sets is --regular open, we obtain . Hence, equality (2) holds.

#### Acknowledgment

This research was financially supported by the Faculty of Science, Mahasarakham University.

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