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Seenivasan, V.; Kamala, K.

doi:https://doi.org/10.1155/2016/2587875

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Research Article | Open Access

Volume 2016 | Article ID 2587875 | https://doi.org/10.1155/2016/2587875

Fuzzy - and Fuzzy Completely - Functions via Fuzzy e-Open Sets

V. Seenivasan¹and K. Kamala¹

Academic Editor: Bruno Carpentieri

Received07 Aug 2015

Accepted07 Feb 2016

Published08 Mar 2016

Abstract

We introduced the notions of fuzzy - functions and fuzzy completely - functions via fuzzy e-open sets. Some properties and several characterization of these types of functions are investigated.

1. Introduction

With the introduction of fuzzy sets by Zadeh [1] and fuzzy topology by Chang [2], the theory of fuzzy topological spaces was subsequently developed by several fuzzy topologist based on the concepts of general topology. In 2014, the concept of fuzzy e-open sets and fuzzy e-continuity and separations axioms and their properties were defined by Seenivasan and Kamala [3]. In this paper, we introduce the notion of fuzzy - functions, fuzzy -continuous, fuzzy completely - functions, and fuzzy e-kernel via fuzzy e-open sets and studied their properties and several characterizations of these types of functions are investigated. In this paper, we denote fuzzy e-open, fuzzy e-closed, and fuzzy regular closed as, , , and , respectively.

2. Preliminaries

Throughout this paper, and (or simply and ) represent nonempty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned.

Let be any fuzzy set of . The fuzzy closure of , fuzzy interior of , fuzzy -closure of , and the fuzzy -interior of are denoted by , , , and , respectively. A fuzzy set of is called fuzzy regular open [4] (resp., fuzzy regular closed) if (resp., ).

The fuzzy -interior of fuzzy set of is the union of all fuzzy regular open sets contained in . A fuzzy set is called fuzzy -open [5] if . The complement of fuzzy -open set is called fuzzy -closed (i.e, ). A fuzzy set of is called fuzzy -preopen [6] (resp., fuzzy -semi open [7]) if (resp., ). The complement of a fuzzy -preopen set (resp., fuzzy -semiopen set) is called fuzzy -preclosed (resp., fuzzy -semiclosed).

Definition 1. A fuzzy set of a fuzzy topological space is called fuzzy e-open [3] if . Fuzzy e-closed if .
The intersection of all fuzzy e-closed sets containing is called fuzzy e-closure of and is denoted by - and the union of all fuzzy e-open sets contained in is called fuzzy e-interior of and is denoted by -

Definition 2. A mapping is said to be fuzzy [8] if the image of every fuzzy e-open set in is fuzzy e-open set in .

Definition 3. A function is called fuzzy e-irresolute [3]. is fuzzy e-open in for every fuzzy e-open set of .

Definition 4. A fuzzy set is quasicoincident [9] with a fuzzy set denoted by iff there exist such that . If and are not quasicoincident, then we write and

Definition 5. A fuzzy point is quasicoincident [9] with a fuzzy set denoted by iff there exist such that .

Definition 6. A fuzzy topological space is said to be fuzzy - [3] if for each pair of distinct points and of there exist fuzzy e-open sets and such that and and and .

Definition 7. A fuzzy topological space is said to be fuzzy - [3] if for each pair of distinct points and of there exists disjoint fuzzy e-open sets and such that and .

Definition 8. A fuzzy topological space is said to be fuzzy weakly Hausdorff [10] if each element of is an intersection of fuzzy regular closed sets.

Definition 9. A fuzzy topological space is said to be fuzzy e-normal [3] if for every two disjoint fuzzy closed sets and of there exist two disjoint fuzzy e-open sets and such that and and .

Definition 10. A fuzzy topological space is said to be fuzzy strongly normal [10] if for every two disjoint fuzzy closed sets and of there exist two disjoint fuzzy open sets and such that and .

Definition 11. A fuzzy topological space is said to be fuzzy Urysohn [11] if for every distinct points and in there exist fuzzy open sets and in such that and and

Definition 12. A space is called fuzzy S-closed [2] (resp., fuzzy e-compact [3]) if every fuzzy regular closed (resp., fuzzy e-open) cover of has a finite subcover.

Definition 13. A function is called fuzzy completely continuous [12] if is fuzzy regular open in for every fuzzy open set of .

Definition 14. A fuzzy filter base is said to be fuzzy -convergent [10] to a fuzzy point in if for any fuzzy regular closed set in containing there exists a fuzzy set such that .

Definition 15. A collection of fuzzy subsets of a fuzzy topological spaces is said to form fuzzy filterbases [13] iff for every finite collection , .

3. Fuzzy - Functions

In this section, the notion of fuzzy - functions is introduced and some characteristics and properties are studied.

Definition 16. A mapping is called fuzzy - if the inverse image of every fuzzy e-open set of is fuzzy e-closed in .

Remark 17. The concepts of fuzzy - and fuzzy e-irresolute are independent notions as illustrated in the following example.

Example 18. Let and and the fuzzy sets be defined as follows: Let and . Then, the mapping is defined by . Then, is fuzzy - but not fuzzy e-irresolute.

Example 19. Let and and the fuzzy sets are defined as follows:Let and . Then, the mapping is defined by . Then, is fuzzy e-irresolute but not fuzzy -.

Definition 20. A mapping is called fuzzy -continuous if the inverse image of every fuzzy open set of is fuzzy e-closed in .

Remark 21. Every fuzzy - function is fuzzy -continuous, but not conversely from the following example.

Example 22. Let and and the fuzzy sets are defined as follows: Let and . Then, the mapping is defined by , and . Then, is fuzzy -continuous but not fuzzy - as the fuzzy set is fuzzy e-open in but is not fuzzy e-closed set in .

Theorem 23. For a fuzzy function , if , the inverse image of for every fuzzy e-closed set of is fuzzy e-open in iff for any ; if , then -.

Proof. Let be a fuzzy e-closed set and . Then, and, by hypothesis, -. We obtain, -. Converse can be shown easily.

Theorem 24. For a fuzzy function , if , for any fuzzy e-closed set and for any , - iff there exists a fuzzy e-open set such that and .

Proof. Let be any fuzzy e-closed set and let . Then, -. Take - then -, and is fuzzy e-open in and
Conversely, let be any fuzzy e-closed set and let . By hypothesis, there exists fuzzy e-open set such that and . This implies, and then -.

Theorem 25. For a fuzzy function , the following statements are equivalent:(1)is fuzzy -.(2)For every fuzzy e- closed set in , is fuzzy e-open in .(3)For every fuzzy open set , - is fuzzy e-closed.(4)For every fuzzy closed set , - is fuzzy e-open.(5)For each and each fuzzy e-closed set in containing , there exists a fuzzy e-open set in containing such that .(6)For each and each fuzzy e-open set in noncontaining , there exists a fuzzy e-closed set in noncontaining such that .

Proof. (1) (2): let be a fuzzy e-open set in . Then, is fuzzy e-closed. By (2), is fuzzy e-open. Thus, is fuzzy e-closed. Converse can be shown easily.
(1) (3): let be a fuzzy open set. Since - is fuzzy e-open, then by (1) it follows that - is fuzzy e-closed. The converse is easy to prove.
(2) (4): let be a fuzzy closed set. Since - is fuzzy e-closed set, then by (2) it follows that - is fuzzy e-open. The converse is easy to prove.
(2) (5): let be any fuzzy e-closed set in containing . By (2), is fuzzy e-open set in and . Take . Then, . The converse can be shown easily.
(5) (6): let be any fuzzy e-open set in noncontaining . Then, is a fuzzy e-closed set containing . By (5), there exists a fuzzy e-open set in containing such that . Hence, and . Take . We obtain that is a fuzzy e-closed set in noncontaining . The converse can be shown easily.

Theorem 26. Let be a function and let be the fuzzy graph function of , defined by for every . If is fuzzy -, then is fuzzy -.

Proof. Let be a fuzzy e-closed set in ; then, is a fuzzy e-closed set in . Since is fuzzy -, then is fuzzy e-open in . Thus, is fuzzy -.

Theorem 27. Let be a family of product spaces. If a function is fuzzy -, then is fuzzy - for each where is the projection of onto .

Proof. Let be any fuzzy e-open set in . Since is a fuzzy continuous and fuzzy open set, it is a fuzzy e-open set. Now , is a fuzzy e-open in . Therefore, is a fuzzy e-irresolute function. Now , since is fuzzy -. Hence is a fuzzy e-closed set, since is a fuzzy e-open set. Hence, is fuzzy -.

Theorem 28. If the function is fuzzy -, then is fuzzy - for each .

Proof. Let be an arbitrary fixed index and let be any fuzzy e-open set of ; then, is fuzzy e-open in , where . Since is fuzzy - function, then is fuzzy e-closed in and hence is fuzzy e-closed in . This implies is fuzzy -.

Theorem 29. If is fuzzy - and is fuzzy closed set of , then is fuzzy -.

Proof. Let be a fuzzy e-open set of ; then, . Since and are fuzzy closed, hence is fuzzy e-closed in the relative topology of .

Definition 30. The intersection of all fuzzy e-open set of a fuzzy topological space containing is called the fuzzy e-kernel of (briefly, -),
The following properties hold for fuzzy sets of :(1)- iff for any fuzzy e-closed set containing .(2)- and - if is fuzzy e-open in .(3); then, --.

Theorem 31. For a fuzzy function , the following statements are equivalent:(1) is fuzzy -.(2)-- for every fuzzy set of .(3)-- for every fuzzy set of .

Proof. : let and -. There exists a fuzzy e-closed set in , such that and Therefore, . This implies that and - Thus, - and -. Hence, --.
: let ; then, . By hypothesis, ---. Hence, --.
: let be any fuzzy e-open set of ; we have --, since is fuzzy e-open and -. This implies that is fuzzy e-closed in .

Definition 32. The fuzzy e-Frontier of a fuzzy set of a fuzzy topological space is given by ---.

Theorem 33. The fuzzy point such that is not fuzzy - is exactly the union of fuzzy e-Frontier if the inverse image of the fuzzy e-closed set in contains .

Proof. Suppose that is not fuzzy - at the point ; then there exists a fuzzy e-closed set such that and for all fuzzy e-open set such that . It follows that and hence --. Thus, - and hence -.
Conversely, suppose that -, is fuzzy e-closed set of containing , and is fuzzy - at . There exists fuzzy e-open set such that and . Thus, - and hence - for each fuzzy e-closed set of containing , a contradiction. Therefore, is not fuzzy -.

Theorem 34. The following hold for functions and :(a)If is fuzzy - and is fuzzy -continuous then is fuzzy -continuous.(b)If is fuzzy - and is fuzzy e-irresolute then is fuzzy -.

Theorem 35. If is a fuzzy e-irresolute surjective function and is a fuzzy function such that is fuzzy -, then is fuzzy -.

Proof. Let be any fuzzy e-closed set in . Since is fuzzy -, is fuzzy e-open in . Therefore, is fuzzy e-open in . Since is fuzzy e-irresolute, surjection implies is fuzzy e-open in . Thus, is fuzzy -.

Theorem 36. If is a fuzzy surjective function and is a fuzzy function such that is fuzzy -continuous, then is fuzzy -continuous.

Proof. Let be any fuzzy closed set in . Since is fuzzy -continuous, is fuzzy e-open in . Therefore, is fuzzy e-open in . Since is fuzzy , surjection implies is fuzzy e-open in . Thus, is fuzzy -continuous.

4. Fuzzy Completely - Functions

In this section, the notion of fuzzy completely - functions is introduced and the relation between other functions is studied and further some structure preservation properties are investigated.

Definition 37. A mapping is called fuzzy completely - if inverse image of every fuzzy e-open set in is fuzzy regular closed in .

Example 38. Let and the fuzzy sets are defined as follows: Let and . Then, the mapping is defined by Then, is fuzzy completely -.

Remark 39. Every fuzzy completely - function is fuzzy - and fuzzy -continuous, but the converse is not true, which can be seen in the following example.

Example 40. Let and and the fuzzy sets , and are defined as follows: Let and . Then, the mapping is defined by . Then, is fuzzy -continuous and also fuzzy - but not fuzzy completely - as the fuzzy set is fuzzy e-open in but is not fuzzy regular closed set in .

From the above examples, we have the following implications.

None of the these implications is reversible.

Theorem 41. For a fuzzy function , if , the inverse image of every fuzzy e-closed set of is fuzzy -open in iff for any if , then .

Proof. Let be a fuzzy e-closed set and . Then, and, by hypothesis, . From here, . The converse can be shown easily.

Theorem 42. For a fuzzy function , if , for any fuzzy e-closed set and for any , iff there exists a fuzzy -open set such that and .

Proof. Let be any fuzzy e-closed set and let . Then, . Take ; then, ; is fuzzy -open in and
Conversely, let be any fuzzy e-closed set and let . By hypothesis, there exists fuzzy -open set such that and . This implies and then .

Theorem 43. For a fuzzy function , the following statements are equivalent:(1) is fuzzy completely -.(2)For every fuzzy e-closed set in , is fuzzy regular open in .(3)For every fuzzy open set , - is fuzzy regular closed.(4)For every fuzzy closed set , - is fuzzy regular open.(5)For each and each fuzzy e-closed set in containing , there exists a fuzzy regular open set in containing such that .(6)For each and each fuzzy e-open set in non containing , there exists a fuzzy regular closed set in noncontaining such that .

Proof. (1) (2): let be a fuzzy e-open set in . Then, is fuzzy e-closed. By (2), is fuzzy regular open. Thus, is fuzzy regular closed. Thus, is fuzzy completely -.The converse can be shown easily.
(1) (3): let be a fuzzy open set. Since - is fuzzy e-open, then by (1) it follows that - is fuzzy regular closed. The converse is easy to prove.
(2) (4): let be a fuzzy closed set. Since - is fuzzy e-closed set, then by (2) it follows that - is fuzzy regular open. The converse is easy to prove.
(2) (5): let be any fuzzy e-closed set in containing . By (2), is fuzzy regular open set in and . Take . Then, . The converse can be shown easily.
(5) (6): let be any fuzzy e-open set in noncontaining . Then, is a fuzzy e-closed set containing . By (5), there exists a fuzzy regular open set in containing such that . Hence, and . Take . We obtain that is a fuzzy regular closed set in noncontaining . The converse can be shown easily.

Theorem 44. Let be a function and let be the fuzzy graph function of , defined by for every . If is fuzzy completely -, then is fuzzy completely -.

Proof. Let be a fuzzy e-closed set in ; then, is a fuzzy e-closed set in . Since is fuzzy completely -, then is fuzzy regular open in . Thus, is fuzzy completely -.

Theorem 45. The following holds for functions and : (a)If is fuzzy - and is fuzzy completely -, then is fuzzy e-irresolute.(b)If is fuzzy completely - and is fuzzy -continuous, then is fuzzy completely continuous.

Definition 46. A fuzzy filter base is said to be fuzzy e-convergent to a fuzzy point in if for any fuzzy e-open set in containing there exists a fuzzy set such that .

Theorem 47. If a fuzzy function is fuzzy completely - for each fuzzy point and each fuzzy filter base in is fuzzy -convergent to , then the fuzzy filter base is fuzzy e-convergent to .

Proof. Let and let be any fuzzy filter base in which is fuzzy -converging to . Since is fuzzy completely -, then for any fuzzy e-open set in containing , there exists a fuzzy regular closed set in containing such that . Since is fuzzy -converging to , there exists a such that . This means that and therefore the fuzzy filter base is fuzzy e-convergent to

Theorem 48. If is a fuzzy completely - surjection and is fuzzy S-closed, then is fuzzy e-compact.

Proof. Suppose that is a fuzzy completely - surjection and is fuzzy S-closed. Let be a fuzzy e-open cover of . Since is a fuzzy completely -, then is fuzzy regular closed cover of and hence there exists finite set of such that . Therefore, we have and is fuzzy e-compact.

Theorem 49. If is a fuzzy completely - injection and is fuzzy -, then is fuzzy weakly Hausdorff.

Proof. Suppose is fuzzy -. For any distinct fuzzy points and in , there exist fuzzy e-open sets and in . Since is injective, , and . Since is fuzzy completely -, and are fuzzy regular closed sets of such that , and . This shows that is fuzzy weakly Hausdorff.

Theorem 50. If is a fuzzy completely - injection and is fuzzy e-normal, then is fuzzy strongly normal.

Proof. Let and be disjoint nonempty fuzzy closed sets of . Since is injective, and are disjoint fuzzy closed sets. Since is fuzzy e-normal, there exist fuzzy e-open sets and such that and and . This implies that - and - are fuzzy e-closed sets in . Then, since is fuzzy completely -, - and - are fuzzy regular open sets. Then, - and - and - and - are disjoint; by definition is fuzzy strongly normal.

Definition 51. A fuzzy topological space is said to be fuzzy -(- [14]) if for every fuzzy set of can be written in the form , where are fuzzy e-open (fuzzy regular open) or fuzzy e-closed (fuzzy regular closed) sets of .

Theorem 52. If is a fuzzy completely - injection and is fuzzy -, then is fuzzy -.

Proof. Let be a any fuzzy set of . Since is fuzzy -, is fuzzy e-open set of . Then, , where are fuzzy e-open set or fuzzy e-closed sets of . Since is completely - injection we have , where are fuzzy regular open sets or fuzzy regular closed sets of . Thus, is fuzzy .

Theorem 53. If is a fuzzy completely - injection and is fuzzy -, then is fuzzy Urysohn.

Proof. Let and be any two distinct fuzzy points in . Since is injective, in . Since is fuzzy -, there exist fuzzy e-open sets and in such that and and . This implies that - and - are fuzzy e-closed sets in . Then, since is fuzzy completely -, there exists fuzzy regular open sets and in containing and , respectively, such that - and -. This implies that - and -; we have that - and - are disjoint and hence ; by definition, is fuzzy Urysohn.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright

Copyright © 2016 V. Seenivasan and K. Kamala. 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.

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