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.