Abstract

The notion of faintly -continuous function has been introduced. Relationship between this new class of function with similar types of functions has been given. Some characterizations and properties of such function are also being discussed.

1. Introduction

In topology, weak forms of open sets play an important role in the generalization of different forms of continuity. Using different forms of open sets, many authors have introduced and studied various types of continuity. In this paper, a unified version of some types of continuity has been introduced from a generalized topological space to a topological space. Generalized topology was first introduced by Császár (see [15]).

We recall some notions defined in [1]. Let be a nonempty set, and denotes the power set of . We call a class a generalized topology [1], (GT) if and union of elements of belongs to . A set , with a GT on it, is said to be a generalized topological space (GTS) and is denoted by . Let be a topological space. The -closure [6] of a subset of a topological space is defined by , where a subset is called regular open if . A subset of a topological space is called semiopen [7] (resp., preopen [8], -open [9], -open [10], -open [11], -preopen [12], -semiopen [13], and -open [14]) if (resp., , , , , , , and ). A point is in (resp., ) if for each semiopen (resp., preopen) set containing , . A point is called a -cluster [6] (resp., semi--cluster [15], -cluster [16]) point of denoted by (resp., , if (resp., , ) for every open (resp., semiopen, preopen) set containing . A subset is called -closed (resp., semi--closed, -closed) if (resp., , -). The complement of a -closed (resp., semi--closed, -closed) set is called -open (resp., semi--open, -open). The family of all -open sets in a topological space forms a topology which is weaker than the original topology. The finite union of regular open sets is said to be -open [17]. A subset of a topological space is said to be -closed [17] if whenever and is -open. A subset of is called -open [18] if for each there exists an open set containing such that . The family of all -open subsets of a space forms a topology on finer than . For any topological space , the collection of all semiopen (resp., preopen, -open, -open, -open, -preopen, -semiopen, -open, -open, semi--open, -open, -open, and -open) sets are denoted by (resp., , , , , , , , , , , , and ). We note that each of these collections forms a GT on .

For a GTS , the elements of are called -open sets and the complement of -open sets are called -closed sets. For , we denote by the intersection of all -closed sets containing , that is, the smallest -closed set containing , and by the union of all -open sets contained in , that is, the largest -open set contained in (see [1, 2]).

It is easy to observe that and are idempotent and monotonic, where : is said to be idempotent if and only if implies and monotonic if and only if . It is also well known from [2, 3] that if is a GT on and , , then if and only if and .

Hereafter, throughout the paper, we will use to mean a generalized topological space and to be a topological space unless otherwise stated.

Definition 1. A function is said to be faintly -continuous at if for each -open set in containing there exists in containing such that . If is faintly -continuous at each point of , then is called faintly -continuous on .

Definition 2. A function is said to be -continuous [19] (resp., weakly -continuous, almost -continuous) if for each and each open set of containing there exists containing such that (resp., , .

Remark 3. (i) Let and be two topological spaces. If , , , , , , , , , , , , , and , then a faintly -continuous function reduces to a faintly continuous [20], faintly semicontinuous [21], faintly precontinuous [21], faintly -continuous [22, 23], faintly -continuous [21], faintly -continuous [21], faintly -precontinuous [24], faintly -semicontinuous [25], faintly -continuous [26], quasi--continuous [21], faintly semi- continuous [27], faintly pre--continuous [27], faintly -continuous function [28], and faintly -continuous function [29], respectively. On the other hand, every faintly -continuous function [30] is faintly -continuous if is closed under arbitrary union (as is a GT in that case).
(ii) It follows from Definitions 1 and 2 that(a)every -continuous function is faintly -continuous;(b)every -continuous function is almost -continuous;(c)every almost -continuous function is weakly -continuous.
But the converses are false as shown in the next example.

Example 4. (a) Let , let , and let . Consider the function defined by , and . It is easy to check that is faintly -continuous but not -continuous.
(b) Let , let , and let . Consider the identity map . It can be easily checked that is almost -continuous but not -continuous.
(c) Let , let , and let . Consider the identity map . It can be easily checked that is not almost -continuous but weakly -continuous.

Theorem 5. For a function , the following statements are equivalent: (a) is faintly -continuous;(b) is -open for each -open set of ;(c) is -closed for each -closed set of .

Proof. : Let be a -open set of , and let . Since and is faintly -continuous, there exists containing such that . It then follows that . Hence, is -open.
: Let and let be a -open set in containing . Then, by (b), is -open in containing . Let . Then, .
: Let be a -closed set of . Since is -open, by (b), it follows that is -open. This shows that is -closed.
: Let be a -open set in . Then, is -closed in . By (c), is -closed. Thus, is -open.

Theorem 6. If a function is weakly -continuous, then it is faintly -continuous.

Proof. Let , and let be a -open set containing . Then, there exists an open set such that . Since is weakly -continuous, there exists a -open set containing such that . Thus, is faintly -continuous.

Example 7. Let , let , and let . The identity function is faintly -continuous but not weakly -continuous.

Definition 8. A function is called slightly -continuous if for each and each clopen set of containing there exists a -open set containing such that .

Theorem 9. If is faintly -continuous, then it is slightly -continuous.

Proof. Let , and let be a clopen set containing . Then, is -open. Since is faintly -continuous, there exists containing such that showing to be slightly -continuous.

Example 10. Let be the set of real numbers. Consider the identity mapping , where and denote the cocountable and usual topology, respectively. It is easy to show that is slightly -continuous but not faintly -continuous.

Remark 11. From Remark 3 and Theorems 6 and 9, we have the following implications:-continuity almost -continuity weakly -continuity faintly -continuity slightly -continuity.

Theorem 12. Let be regular. If a function is faintly -continuous, then it is -continuous.

Proof. Let be an open set of containing . Since is regular, is -open in . Then by Theorem 5, is -open in . Let . Then, is a -open set containing such that . Thus, is -continuous.

Definition 13. A topological space is said to be almost regular [31] if for any regular closed set and any point there exist disjoint open sets and such that and .

Theorem 14. If a function is faintly -continuous and is almost regular, then is almost -continuous.

Proof. Let , and let be an open set in . Then, is a regular open set in so that it is -open in (as in an almost regular space, every regular open set is -open [20]). Hence, by faintly -continuity of , there exists a -open set containing such that showing to be almost -continuous.

The clopen subsets of a topological space form a base for a topology on . This topology is called ultraregularization [32] of and is denoted by . A topological space is said to be ultra-regular [33] if .

Theorem 15. If is ultraregular, then for a function the following are equivalent:(i) is -continuous;(ii) is almost -continuous;(iii) is weakly -continuous;(iv) is faintly -continuous;(v) is slightly -continuous.

Proof. We will only show that in an ultra regular space every slightly -continuous function is -continuous. The rest will follow from Remark 11. Let be an open set in containing . Then, as is ultra regular, there exists a clopen set in containing such that . Since is slightly -continuous, there exists a -open set in containing such that . Thus, is -continuous.

Theorem 16. If and are two GTs on such that and if is faintly -continuous, then is faintly -continuous.

Proof. It follows immediately from Theorem 5.

Observation. Let be a topological space. Then, we have(i); (ii); (iii); (iv); (v), . Thus, from Theorem 16, we can deduce relations between different types of faintly -continuous functions.

3. Properties of Faintly -Continuous Functions

Definition 17. A GTS (resp., a topological space ) is said to be - [34] (resp., - [35]) if for any two distinct points , of there exist two disjoint -open (resp., -open) sets and containing and , respectively.
It is well known from [36] that a topological space is Hausdorff if and only if it is -.

Theorem 18. If is a faintly -continuous injection and is , then is -.

Proof. Let and be two distinct points of . Then, and are two distinct points of . Thus, there exist two disjoint open sets and containing and respectively. Then, by Theorem 5, and are two -open sets in . Clearly, , , and showing to be -.

Definition 19. A topological space is said to be(i)-regular [26] if for each -closed set and each point there exist disjoint -open sets and such that , ;(ii)-normal [26] if for any two disjoint -closed subsets and there exist disjoint -open sets and such that , .

Definition 20. A GTS is said to be(i)-regular [37] if for each -closed set and each point , there exist disjoint -open sets and such that , .(ii)-normal [37] if for any two disjoint -closed subsets and , there exist disjoint -open sets and such that , .

Definition 21. A function is called -open if for each -open set in is -open in .

Theorem 22. If is faintly -continuous, -open bijection, and is -regular, then is -regular.

Proof. Let be a -closed subset of , and let . Let . Since is faintly -continuous, by Theorem 5, is -closed in so that . Let . Then, ; thus, by -regularity of , there exist two disjoint -open sets and such that and . Thus, we have and and . As is -open, and are -open in showing to be -regular.

Theorem 23. If is faintly -continuous, -open surjection, and is -normal, then is -normal.

Proof. Let and be two disjoint -closed subsets of . Since is faintly -continuous, by Theorem 5, and are two -closed subsets in . Let , and let . Then, and are two disjoint -closed subsets of . Since is -normal, there exist two disjoint -open sets and such that and . We thus have and . Also and are two disjoint -open sets in showing to be -normal.

Definition 24. A GTS is said to be -connected [38] if cannot be written as union of two nonempty -open sets.

Theorem 25. If is a faintly -continuous surjection and is -connected, then is connected.

Proof. Let us assume that be not connected. Then, there exist nonempty disjoint open sets and such that . Hence, we have and . Since is surjective, and are nonempty. Since and are clopen, they are -open in . Thus by Theorem 5, and are -open. Therefore is not -connected—a contradiction.

Definition 26. A GTS is called -compact [39] if every -open cover of has a finite subcover. A subset of is said to be -compact relative to if every cover of by -open sets of has a finite subcover.

Definition 27. A subset of a topological space is called -compact relative to [40] if every cover of by -open sets of has a finite subcover. A topological space is called -compact if is -compact relative to .

Theorem 28. If is a faintly -continuous function and is -compact relative to , then is -compact relative to .

Proof. Let be a cover of by -open sets of . Then, for each , there exists such . Since is a faintly -continuous function, there exists containing such that . Then, the family is a cover of by -open sets of . Since is -compact relative to , there exists a finite number of points, say, , such that . Therefore, we have . This shows that is -compact relative to .

Definition 29. For any subset of a GTS , the -frontier of is denoted by and defined by .

Theorem 30. The set of all points at which the function is not faintly -continuous is identical with the union of -frontier of the inverse images of -open sets of containing .

Proof. Suppose that is not faintly -continuous at . Then, there exists a -open set of containing such that is not a subset of for each -open set containing . Hence, we have for each containing . So, . On the other hand, we have . Hence .
Conversely, suppose that is not faintly -continuous at , and let be any -open set containing . Then, by Theorem 5, . Therefore, for each -open set containing . This completes the proof.

4. Faintly -Closed Graph

Definition 31. A function is said to have a faintly -closed graph if for each there exist containing and a -open set in containing such that .

Lemma 32. The graph of a function is faintly -closed if and only if for each there exist containing and a -open set in containing such that .

Theorem 33. Let a function have a faintly -closed graph . If is a faintly -continuous injection, then is -.

Proof. Let and be two distinct points of . Then, since is an injection, we have . Then, we have . Thus, by Lemma 32, there exist a -open set containing in and a -open set in containing such that . Hence, , and . Since is faintly -continuous, there exists a -open set containing such that . Therefore, we have . Since is injective, we obtain showing to be -.

Definition 34. A function is said to have a faintly strong -closed graph if for each there exist containing and an open set in containing such that .

Lemma 35. The graph of a function is faintly strong -closed if and only if for each there exist containing and an open set in containing such that .

Theorem 36. If a function is faintly -continuous and is , then is faintly strong -closed.

Proof. Let , and let . Then, . Then, by Lemma 32, there exist a -open set in containing and a -open set in containing such that . Since is -open, there exists an open set in containing such that . Thus . Thus, by Lemma 35, is faintly strong -closed.

Theorem 37. If a function is a surjective function with faintly strong -closed graph , then is .

Proof. Let and be any two distinct points of . Then, since is surjective, there exists such that ; hence, . Since is faintly strong -closed, there exist a -open set in containing and an open set of containing such that . Therefore, we have . Hence, there exists an open set of such that and . Moreover, we have , and is open in . This shows that is .

Theorem 38. If a function has a faintly -closed graph, then it has also a faintly strong -closed graph.

Proof. Let , and let . Then, . Then there exist in containing in and a -open set in containing such that . Since is -open, there exists an open set such that . So . Thus, by Lemma 35, has a faintly strong -closed graph.

The converse of the above theorem is not true in general as shown in Example 3 of [20].

Theorem 39. If has a faintly -closed graph, then is closed in for each subset which is -compact relative to .

Proof. Suppose that . Then, for each , . Since is faintly -closed, by Lemma 32, there exist a -open set in containing and a -open set in containing such that . Then, the family is a cover of by -open sets in . So, there exists a finite subfamily of such that . Set . Then, is -open (hence open) in containing . Therefore, . It then follows that . Thus, is closed in .

5. Conclusion

Similar types of faintly continuous functions can be defined from a topological space to another topological space from the definition of faintly -continuous function by taking different GTs on . In fact, different results on weak forms of faintly continuous functions can be derived from faintly -continuous functions by replacing by the corresponding GTs on (see [2029]).

Conflict of Interests

The author Bishwambhar Roy declares that the paper does not have any financial relation with any commercial identity that might lead to conflict of interests.

Acknowledgment

The author is thankful to the referee for his/her comments to improve the paper.