Several Types of Totally Continuous Functions in Double Fuzzy Topological Spaces

Abstract

We introduce the notions of totally continuous functions, totally semicontinuous functions, and semitotally continuous functions in double fuzzy topological spaces. Their characterizations and the relationship with other already known kinds of functions are introduced and discussed.

1. Introduction

The concept of fuzzy sets was introduced by Zadeh in his classical paper [1]. In 1968, Chang [2] used fuzzy sets to introduce the notion of fuzzy topological spaces. Çoker [3, 4] defined the intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. Later on, Demirci and Çoker [5] defined intuitionistic fuzzy topological spaces in Kubiak-Šostak’s sense as a generalization of Chang’s fuzzy topological spaces and intuitionistic fuzzy topological spaces. Mondal and Samanta [6] succeeded to make the topology itself intuitionistic. The resulting structure is given the new name “intuitionistic gradation of openness.” The name “intuitionistic” did not continue due to some doubts that were thrown about the suitability of this term. These doubts were quickly ended in 2005 by Gutiérrez García and Rodabaugh [7]. They proved that this term is unsuitable in mathematics and applications. Therefore, they replaced the word “intuitionistic” by “double” and renamed its related topologies. The notion of intuitionistic gradation of openness is given the new name “double fuzzy topological spaces.”

The fuzzy type of the notion of topology can be studied in the fuzzy mathematics, which has many applications in different branches of mathematics and physics theory. For example, fuzzy topological spaces can be applied in the modeling of spatial objects such as rivers, roads, trees, and buildings. Since double fuzzy topology forms an extension of fuzzy topology and general topology, we think that our results can be applied in modern physics and GIS Problems.

Jain et al. introduced totally continuous, fuzzy totally continuous, and intuitionistic fuzzy totally continuous functions in topological spaces, respectively (see [8–11]).

In this paper, we introduce the notions of totally continuous, totally semicontinuous, and semitotally continuous functions in double fuzzy topological spaces and investigate some of their characterizations. Also, we study the relationships between these new classes and other classes of functions in double fuzzy topological spaces.

2. Preliminaries

Throughout this paper, let be a nonempty set and let be the closed interval , and . The set of all fuzzy subsets (resp., fuzzy points) of is denoted by (resp., ). For and , a fuzzy point is defined by if and only if . We denote a fuzzy set which is quasicoincident with a fuzzy set by , if there exists such that , otherwise, by . Given a function , and are the direct image and the inverse image of , respectively, and are defined by and for each , , and , respectively.

Definition 1 (see [12, 13]). The pair of functions is called a double fuzzy topology on if it satisfies the following conditions. (O1) for each .(O2) and for each , .(O3) and for any .The triplet is called a double fuzzy topological space (dfts, for short). and may be interpreted as a gradation of openness and gradation of nonopenness for . A function is said to be a double fuzzy continuous (briefly, dfc) if and for each .

Theorem 2 (see [12–14]). Let be a dfts. Then, for each , , and , we define an operator as follows: For , , and , the operator satisfies the following statements. (C1).(C2).(C3).(C4) if and .(C5).

Theorem 3 (see [12–14]). Let be a dfts. Then, for each , , and , we define an operator as follows: For , , and , the operator satisfies the following statements. (I1).(I2).(I3).(I4).(I5) if and .(I6).(I7)If , then .

Definition 4 (see [12, 15]). Let be a dfts, , , and . A fuzzy set is called: (1)an -fuzzy semiopen (for short, -fso) if ). A fuzzy set is called -fuzzy semiclosed (for short, -fsc) if is an -fuzzy semiopen set,(2)an -fuzzy preopen (for short, -fpo) if ). A fuzzy set is called -fuzzy preclosed (for short, -fpc) if is an -fuzzy preopen set.

Definition 5 (see [16]). Let be a dfts, , , , and . A fuzzy set is called -fuzzy -neighborhood of , if , , and .

Definition 6. Let be a function between dfts’s and . Then is called: (1)double fuzzy open [15], if and , for each , , and ;(2)double fuzzy closed [6] if and , for each , , and .

Definition 7 (see [15]). Let be a dfts, , and . The two fuzzy sets , are said to be -fuzzy separated if and only if and . A fuzzy set which cannot be expressed as a union of any two -fuzzy separated sets is said to be -fuzzy connected.

3. Totally Continuous and Totally Semicontinuous Functions in Double Fuzzy Topological Spaces

In this section, some new classes of functions are introduced. Their characterizations and relationship with other functions are introduced.

Definition 8. Let be a dfts, , , and . A fuzzy set is called: (1)an -fuzzy semiclopen (for short, -fsco) if is an -fso set and -fsc set;(2)an -fuzzy preclopen (for short, -fpco) if is an -fpo set and -fpc set.

Definition 9. Let be a function between dfts’s and . Then is called: (1)double fuzzy totally continuous (briefly, dftc) if is -fco, for each , , and such that and ,(2)double fuzzy totally semicontinuous (briefly, dftsc) if is -fsco, for each , , and such that and ,(3)double fuzzy totally precontinuous (briefly, dftpc) if is -fpco, for each , , and such that and ,(4)double fuzzy semicontinuous (briefly, dfsc) if is -fso, for each , , and such that and ,(5)double fuzzy semiopen if is an -fso in , for each -fo set , , and ,(6)double fuzzy semiclosed if is an -fsc in , for each -fc set , , and .

Remark 10. A fuzzy set in a dfts is -fco if and only if it is an -fsco and -fpco set, where and .

Theorem 11. Let be a function. Then the following are equivalent: (1) is a dftc function,(2) is an -fco set of for each , , and such that and ,(3) and for each , , and (4) and for each , , and .

Proof. : Let and . By using (1), is an -fco set in . Since thus is an -fco set in .
: Let . Then, is -fc set in . By (2), is -fco in . Hence, Again by using ,
: Let . By using , we have Hence, By using (3), we have Hence,
: Let such that . By using (4), we have Hence, ; that is, and . By using (4), we have Hence, ; that is, and . Therefore is an -fco set in . Thus, is dftc function.

Definition 12. Let be a dfts. Then, it is called: (1)double fuzzy semiregular (resp., double fuzzy clopen regular) if, for each -fsc (resp., -fco) set of , , and each fuzzy point , there exist disjoint -fso (resp., -fo) sets and such that and ,(2)double fuzzy s-regular (resp., double fuzzy ultraregular) if, for each -fc set of and each , , and , there exist disjoint -fso (resp., -fco) sets and such that and ,(3)double fuzzy s-normal if, for each pair of nonzero disjoint -fc sets can be separated by disjoint -fso sets, and ,(4)double fuzzy clopen normal if, for each pair of disjoint -fco sets and of , there exist two disjoint -fo sets and , and such that and .

Theorem 13. If is dftc injective semiopen function from a double fuzzy clopen regular space onto a double fuzzy space , then is double fuzzy s-regular.

Proof. Suppose is an -fc set in , , and is dftc; is -fco set in , for each , , and such that and . Put and ; then . Since is double fuzzy clopen regular, then there exist disjoint -fo sets and such that and . This implies But is injective and double fuzzy semiopen, so and and are -fso sets in , , and . Therefore is double fuzzy s-regular.

Theorem 14. If is dftc injective semiopen function from a double fuzzy clopen normal space onto a double fuzzy space , then is double fuzzy s-normal.

Proof. Suppose and are any two disjoint -fc sets in , , and . Since is dftc, and are -fco subsets of , for each and , , and such that and , , and . Put and and is injective, so Now, is double fuzzy clopen normal; then there exist disjoint -fo sets and , and such that and which implies and , also by the hypothesis of being injective double fuzzy semiopen we have and which are disjoint -fso sets. Therefore is double fuzzy s-normal.

Theorem 15. Let be dftc, closed injective function. If is double fuzzy s-regular, then is double fuzzy ultraregular.

Proof. Suppose is an -fc set not containing , , and is double fuzzy closed function, so and , for each , , and such that is -fc set in not containing . But is double fuzzy s-regular; then there exist disjoint -fso sets and such that and . This implies But is dftc, so and are -fco, for each and , , and such that , , , and such that and are -fco sets in . Also is injective; then Therefore, is double fuzzy ultraregular.

Theorem 16. Let be a function. Then the following are equivalent: (1) is a dftsc function,(2) is an -fco set of for each , , and such that and ,(3) and for each , , and (4) and for each , , and .

Proof. This proof is similar to the proof of Theorem 11.

Theorem 17. Let and be dft’s. A function is dftc if and only if is dftsc and dftpc.

Proof. Let , , and such that and ; then is -fsco set and -fpco. From Remark 10, is -fco set in . Therefore, is dftc.
The completion of the proof is straightforward.

Theorem 18. Let be a function. Then one has the following: (1)if is dftc function, then is dftsc;(2)if is dftsc function, then is dfsc.

Proof. (1) Let be a dftc function, , , and such that and . By hypothesis, is -fsco set in . Therefore is a dftsc function.
(2) Let , , and such that and . By the hypothesis, is -fsco set in . Hence is a dfsc function.

The converse of the above theorem need not be true in general as shown by the following example.

Example 19. (1) Let and . Define fuzzy sets , , and as follows:
Let and be defined as follows:
Then the function is defined by
Since is an -fo set and is an -fsco set not -fco set, then is dftsc but not dftc.
(2) in (1) define and as follows:
So is -fso set in and not -fsco set; that is, is not dftsc.

Definition 20. Let be a function between dfts’s and . Then is called: (1)double fuzzy irresolute (dfir, for short) if is -fso set, for each -fso set , , and .(2)double fuzzy semi-irresolute (dfsir, for short) if is -fsco, for each -fsco set , , and .

Theorem 21. If a function is a dfsir function and is dftc (dftsc) function, then is dftsc function.

Proof. Let , , and such that and . Since is a dftc (dftsc, resp.) function, is -fco (-fsco, resp.) set in . Also, since is a dfsir function, is -fsco set in . Since , is -fsco set in . Therefore, is a dftsc function.

4. Semitotally Continuous Functions in Double Fuzzy Topological Spaces

Now, we introduce the concept of semitotally continuous function which is stronger than totally continuous function in double fuzzy topological spaces, and then we investigate some characteristic properties.

Definition 22. Let be a function between dfts’s and ; then is called double fuzzy semitotally continuous function (briefly, dfstc) if is -fco, for each -fso set , , and .

Theorem 23. Let be a function between dfts’s and . Then the following are equivalent: (1) is a dfstc function,(2)for each and each -fso set , , , and , there exists an -fco set such that and .

Proof. : Suppose is dfstc and is any -fso set in containing , , and so that . Take ; then is an -fco set in and , since is dfstc and is an -fco in . Also This implies .
: Suppose is an -fso set in , , and and let be any fuzzy point; then . Therefore, by hypothesis, there is an -fco set containing such that , so which implies that is an -fco of and and and . Hence it is an -fco in . Therefore, is dfstc function.

Theorem 24. Let be a function. Then the following are equivalent: (1) is a dfstc function,(2) is an -fco set of for each -fsc set , , and ,(3) for each -fso set , , and ,(4) for each -fsc set , , and .

Proof. : Let be -fsc set, , . Then is -fso set. By definition, is -fco set in . This implies that is -fco set.
: Let be an -fsc set, , ; then . Since is an -fco set in hence is an -fso set. Take ; then by using (3),
: Let be an -fsc set, , ; then is an -fso and that is, But is -fsc set; then
: Let be an -fsc set, , ; then But , and is -fco set. Thus, inverse image of every -fso set is -fco in . Therefore, is dfstc function.

Theorem 25. Every dfstc function is a dftc function.

Proof. Suppose is a dfstc function and , , and such that and . Since is -fso set in and is dfstc function, it follows that is -fco in . Thus inverse image of every -fso set in is -fco in . Therefore is dftc function.

The converse of the theorem need not be true in general as shown by the following example.

Example 26. See Example 19 (1) with and take . Clearly is dftc, but not dfstc.

Theorem 27. Every dfstc function is a dftsc function.

Proof. Suppose is a dfstc function, , , and such that and . Since is -fso in , is dfstc function. It follows that is -fco set and hence -fsco in . Thus is dftsc.

The converse of the above need not be true as shown by the following example.

Example 28. Let and define and as follows: Define and on as follows: We defined the function by So is -fsco set but not -fco set; that is, is not dfstc function.

Theorem 29. Let , , and be dfts’s, and let , , and be functions. Then one has the following: (1)if is dfstc and is dfstc, then is dfstc;(2)if is dfstc and is dfir, then is dfstc;(3)if is dfstc and is dfsc, then is dftc;(4)if is dfstc and is any function, then is dfstc if and only if is dfir.

Proof. It is clear.
Let be an -fso set in , , and and is dfir, so is -fso set in . Also since is dfstc function, then is -fco set in . Hence is dfstc function.
It is similar to the proof of .
The proof follows from .
Conversely, let be dfstc and let be an -fso set in , , and . Now, by hypothesis is dfstc; is -fco in . But is dfstc; then is -fso in . Hence is dfir.

Definition 30. Let be a dfts. Then it is called: (1)double fuzzy semi-, if, for each pair of distinct fuzzy points in , there exists an -fso set containing one fuzzy point but not the other, and .(2)double fuzzy semi- (resp., double fuzzy clopen if, for each pair of distinct fuzzy points and of , , and , there exist -fso (resp., -fco) sets and containing and , respectively, such that , and , .(3)double fuzzy semi- (resp., double fuzzy ultra-) if, for each disjoint points and of can be separated by disjoint -fso (resp., -fco) sets, and .(4)double fuzzy ultranormal if, for each pair of nonzero disjoint -fc sets can be separated by disjoint -fco sets, and .(5)double fuzzy seminormal if, for each pair of disjoint -fsc sets and of , there exist two disjoint -fso sets and , and such that and .(6)double fuzzy s-connected if is not the union of two disjoint nonzero -fso subsets of , , and .

Theorem 31. Let and be dfts’s and be a function. Then one has the following. (1)If is dfstc injection and is double fuzzy semi-, then is double fuzzy clopen-.(2)If is dfstc injection and is double fuzzy semi-, then is double fuzzy ultra-.(3)If is dfstc injection and is double fuzzy semi-, then is double fuzzy ultra-.(4)If is dfstc injective double fuzzy semiopen function from a double fuzzy clopen regular space onto , then is double fuzzy semiregular.(5)If is dfstc injective double fuzzy semiopen function from a double fuzzy clopen normal space onto , then is double fuzzy seminormal.

Proof. (1) Suppose fuzzy points and are in such that . Since is injective, then in . Also is double fuzzy semi- so, there exist and which are -fso sets in , , and such that , , , and ; that is, , , , and . Since is dfstc, then and are -fco subsets of . That is, is double fuzzy clopen .
(2) Suppose fuzzy points and such that . Since is injective, then in . Also is double fuzzy semi- so there exists an -fso set , , such that but ; that is, , . But is dfstc; is -fco set in . This implies that every pair of distinct points of can be separated by disjoint -fco sets in . Therefore is double fuzzy ultra-.
(3) Suppose fuzzy points and such that . Since is injective, then in . But is double fuzzy semi- so there exist and which are -fso sets in , , and such that , , and ; that is, and . But is dfstc; and are -fco sets in such that . Therefore is double fuzzy ultra-.
(4) Suppose is an -fsc set in and . Assume and . Since is dftsc function, then is an -fco set in such that . But is double fuzzy clopen regular; then there exist disjoint -fo sets and such that and , which implies Also, by hypothesis is injective double fuzzy semiopen; then and are -fso sets such that Therefore is double fuzzy semiregular.
(5) Assume that and are any two disjoint -fsc sets in . Take and . Since is dfstc injective, then and are -fco subsets of such that But is double fuzzy clopen normal; then there exist disjoint -fo sets and such that and which implies Also, since is injective double fuzzy semiopen, and are disjoint -fso sets. Therefore is double fuzzy seminormal.

Theorem 32. Let and be dfts’s and let be a function. Then one has the following.(1)If is dfstc, double fuzzy closed injection and is double fuzzy s-normal, then is double fuzzy ultranormal.(2)If is dfstc, double fuzzy semiclosed injection and is double fuzzy s-regular, then is double fuzzy ultraregular.

Proof. (1) Suppose , are disjoint -fc subsets in , , and . Since is double fuzzy closed and injective, then , , , and , for each , , , and such that and are disjoint -fc subsets in , so and are separated by disjoint -fso sets , , respectively, because is double fuzzy s-normal space; then and . Also is dfstc, and and are -fco sets in such that Therefore is double fuzzy ultranormal.
(2) Assume that an -fc set does not contain fuzzy point and let be a double fuzzy semiclosed function; then is an -fsc set in such that does not contain . Since is double fuzzy semiregular, then there exist disjoint -fso sets and such that and ; that is and . Since is dfstc function, then and are -fco sets in . But is injective; then Therefore, is double fuzzy ultraregular.

Theorem 33. If is dfstc surjection and is double fuzzy connected, then is double fuzzy s-connected.

Proof. Assume is not double fuzzy s-connected and and are -fso sets in , , and such that and . Since is dfstc, Also and are nonzero -fco sets in such that Then is not double fuzzy connected, so it is a contradiction. Hence is double fuzzy s-connected.

5. Conclusion

In this paper, we introduced and characterized the notions of totally continuous functions, totally semicontinuous functions, and semitotally continuous functions in double fuzzy topological spaces. The relationship with other kinds of functions is studied. We could know that double fuzzy topological spaces are a generalization of some other kinds of topological spaces; therefore, our results can be considered as a generalization of the same results in other kinds of topological spaces. Also, it is possible to study this topic for a completely distributive DeMorgan algebra.

Conflict of Interests

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

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant GUP-2013-040. The authors also wish to gratefully acknowledge all those who have generously given their time to referee our paper.