International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 756376, 7 pages

http://dx.doi.org/10.1155/2014/756376

## Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

^{1}School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia^{2}College of Education, Tikrit University, Iraq^{3}Department of Mathematics, College of Science in Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia^{4}Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

Received 18 April 2014; Revised 26 May 2014; Accepted 27 May 2014; Published 12 June 2014

Academic Editor: Nawab Hussain

Copyright © 2014 Fatimah M. Mohammed 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

The aim of this paper is to introduce the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions in double fuzzy topological spaces. Some interesting properties and characterizations of these functions are introduced and discussed. Furthermore, the relationships among the new concepts are discussed with some necessary examples.

#### 1. Introduction

In 1968, Chang [1] was the first to introduce the concept of fuzzy topological spaces. These spaces and their generalization are later developed by Goguen [2], who replaced the closed interval by more general lattice . On the other hand, by the independent and parallel generalization of Kubiak and Šostak’s [3, 4], made topology itself fuzzy besides their dependence on fuzzy set in 1985.

Various generalizations of the concept of fuzzy set have been done by many authors. In [5–10], Atanassove introduced the notion of intuitionistic fuzzy sets. Later Çoker [11] defined intuitionistic fuzzy topology in Chang’s sense. Then, Mondal and Samanta [12] introduced the intuitionistic gradation of openness of fuzzy sets. Gutiérrez García and Rodabaugh [13], in 2005, replaced the term “intuitionistic” and concluded that the most appropriate work is under the name “double.”

In 1980, Jain [14] introduced the notion of slightly continuous functions. On the other hand, Nour [15] defined slightly semicontinuous functions as a weak form of slight continuity and investigated their properties. In [16], Noiri introduced the concept of slightly -continuous functions. Sudha et al. [17] introduced slightly fuzzy -continuous functions. Also in 2004, Ekici and Caldas [18] introduced the notion of slight -continuity (slight -continuity).

In this paper, the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions are introduced. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the concepts are obtained and established with some interesting counter examples.

#### 2. Preliminaries

Throughout this paper, let be a nonempty set, the unit interval , , and . The family of all fuzzy sets in is denoted by . is the family of all fuzzy points in . By and we denote the smallest and the greatest fuzzy sets on . For a fuzzy set , denotes its complement. Given a function , and defined the direct image and the inverse image of , defined by and for each , , and , respectively. All other notations are standard notations of fuzzy set theory.

*Definition 1 (see [12, 13]). *A double fuzzy topology on is a pair of maps , , which satisfies the following properties: (O1) for each ;(O2) and for each ;(O3) and for each , .

The triplet is called double fuzzy topological spaces (dfts, for short). A fuzzy set is called an -fuzzy open (-fo, for short) if and , is called an -fuzzy closed (-fc, for short) if and only if is an -fo set, and is called -fuzzy clopen (-fco, for short) if and only if is -fo set and -fc set. Let and be two dfts's. A function is said to be a double fuzzy continuous if and only if and for each .

Before starting to present our results, there are two questions that we must ask ourselves. First, what is the difference between classical topology and double fuzzy topology? Secondly, where we can apply our results?

To answer the first question, we should know that double fuzzy sets and hence double fuzzy topological spaces deal with obscurities. In addition to that, we observed that the concept of double fuzzy topological spaces is a generalization of fuzzy topological spaces and classical topology. For example, when the first condition in Definition 1 does not hold, we get the definition of fuzzy topological spaces in Kubiak-Šostak’s sense [3, 4]. Also, in the same definition, when we replace with , we will get results in double gradation fuzzifying topological spaces [19]. Appropriate changes can be made to get results in the classical topological spaces.

With regard to applications, since double fuzzy topology forms an extension of fuzzy topology and general topology, we think that our results can be applied in the fuzzy mathematics, which has many applications in different branches of engineering and ICT. For example, recently double fuzzy topological spaces have been applied to study sensor bias [20] and there exist well-established applications of fuzzy topological spaces in the areas of digital topology [21], image processing [22], and geographic information systems (GIS) problems [23].

Theorem 2 (see [24, 25]). *Let be a dfts. Then for each , , and , one defines an operator as follows:
**
For , and , the operator satisfies the following statements: *(C1)* = ;*(C2)*;*(C3)*;*(C4)* if and ;*(C5)*.*

Theorem 3 (see [24, 25]). *Let be a dfts. Then for each , , and , one defines 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 [26]). *Let be a dfts. For each , , and . (1)A fuzzy set is called -fuzzy semiclosed (briefly, -fsc) if . is called -fuzzy semiopen (briefly, -fso) if and only if is an -fuzzy semiclosed set.(2)An -fuzzy semiclosure of is defined by = and is -fsc}.

*Definition 5 (see [26]). *Let be a dfts. For each , and . (1)A fuzzy set is called -generalized fuzzy semiclosed (briefly, -gfsc) if , and , . is called -generalized fuzzy semiopen (briefly, -gfso) if and only if is -gfsc set.(2)An -fuzzy generalized semiclosure of is defined by = and is -gfsc}.(3)An -fuzzy generalized semi-interior of is defined by = and is -gfso}.

*Definition 6 (see [27]). *Let and be dfts's. A function is called (1)slightly double fuzzy continuous (briefly, sdfc) if for every , , , and such that is -fco set and , there exists such that , , , and
(2)slightly generalized double fuzzy semicontinuous (briefly, sgdfsc) if for each , , , and such that is -fco set and , there exists an -gfso set such that and

#### 3. Somewhat Slightly Generalized Double Fuzzy Semicontinuous Functions

*Definition 7. *Let and be dfts’s. A function is called somewhat slightly generalized double fuzzy semicontinuous (briefly, swsgdfsc) if for each fuzzy set , , , and such that and , there exists an -gfso set such that and

*Definition 8. *A fuzzy set in a dfts is called -generalized fuzzy semidense (resp., -fuzzy-) set if there exists no -gfsc (resp., -fco) set , , and such that

*Example 9. *(1) Let . Define and as follows:
And define and as follows:
So, if , , then there exists no -gfsc set in such that . Therefore, is -generalized fuzzy semidense set in .

(2) In (1), let and be defined as follows:
So, if , , then there exists no -fco set in such that . Therefore, is -fuzzy- set in .

*Definition 10. *Let be a dfts. For a fuzzy set , , and , and are defined as follows: (1) and is -fco};(2) and is -fco}.

Proposition 11. *Let and be dfts’s, and let be any function. Then the following are equivalent. *(1) * is swsgdfsc function.*(2)*If* * is an* *-fco set such that* * and* *, for each* , *, and* *, then there exists an* *-gfsc set* * such that **.*(3)*If * * is* *-gfs-dense set in* *, then * * is **-fuzzy-* * set in * * such that every* *-fco set* *, for each* *,* *, and* *.*

*Proof. * Suppose is swsgdfsc function, and let be any -fco set in such that and , for each , , and . Then, is -fco in such that and . Then by the hypothesis, there exists an -gfso set , , and such that and . That is, is an -gfsc set and
Put . Then is an -gfsc set in such that .

Let be an -gfs-dense set in , and suppose that is not a fuzzy- set in , such that each -fco set , for each , , and . Then, there exists an -fco set such that
since

Now, is an -fco set such that and , for each , , and . Then by the hypothesis, there exists an -gfsc set such that .

But

That is, . Therefore, there exists an -gfsc set , , and such that , which is a contradiction. Therefore, is an -fuzzy set in such that
for each and -fco set .

Let be an -fco set such that and , for each , , and . Then, . Now, suppose that and . Then,
That is, is an -gfs-dense in . Then by (3), is an -fuzzy set such that there exists an -fco set , for each , , and .

But
since is an -fco and
That is,
which is a contradiction, since . Therefore, and . So is swsgdfsc.

#### 4. Somewhat Slightly Generalized Double Fuzzy Semiopen Functions

*Definition 12. *Let and be dfts’s. A function is called (1)generalized double fuzzy semiopen (briefly, gdfso) if for each -gfso set , , and , is an -gfso in ;(2)slightly generalized double fuzzy semiopen (briefly, sgdfso) if for each -gfso set and each , , and such that , is an -fco set in and
(3)somewhat generalized double fuzzy semiopen (briefly, swgdfso) if for each -gfso set , , and , there exists an -gfso set such that
(4)somewhat slightly generalized double fuzzy semiopen (briefly, swsgdfso) if for each -gfso set such that and for each , , and , there exists an -fco set , such that
That is, , and there exists an -fco set such that and , for each , , and .

Proposition 13. *Let , , and be dfts’s. If and are swsgdfso functions, then is a swsgdfso function.*

*Proof. *Let be an -gfso set and such that , for each fuzzy set , , and . Since is swsgdfso, then there exists an -fco set , and such that .

Now, is an -gfso in such that
for each .

Since is swsgdfso, then there exists an -fco set and such that
But
Thus, there exists an -fco set and
such that
Therefore, is swsgdfso.

Proposition 14. *Let and be dfts’s, and let be a bijective function. Then the following are equivalent. *(1)* is swsgdfso function.*(2)*If * * is an* *-gfsc set in* * such that* * and* * for each* *, then there exists an* *-fco set* *,* *, and* * such that* *.*

*Proof. * Let be an -gfsc set in such that and , for each , , and . Then, is an -gfso set in such that and , for each . So
Since is a swsgdfso, then there exists an -fco set and such that
Now, is an -fco set in such that and such that
Take
so (2) is proved.

Let be any -gfso set in such that , for each . Then, is an -gfsc set in such that and for each and . Now,
For, if , then
Hence by the hypothesis, there exists an -fco set , , such that
That is,
such that
Let . Then, is an -fco set in such that and . Therefore, is swsgdfso function.

#### 5. Interrelations

The following implication illustrates the relationships between different functions in Figure 1.

None of these implications is reversible where represents implies , as shown by the following examples.

*Example 15. *Let .(1)Let be the identity function. Define , , , and as follows:
And define and as follows:
Then, is sgdfsc function but not sdfc.(2)In (1), is swsgdfsc function but not sdfc.(3)Let be a function defined by
Define , , and as follows:
and define and as follows:
Then, is swsgdfsc function but not sgdfsc.(4)Let be the identity function. Define , , and as follows:
And define and as follows:
Then, is swgdfso function but not gdfso.(5)Let be the identity function. Define , , and as follows:
And define and as follows:
Then, is swsgdfso function but not sgdfso.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding 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 their paper.

#### References

- C. L. Chang, “Fuzzy topological spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 24, no. 1, pp. 182–190, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. A. Goguen, “L-fuzzy sets,”
*Journal of Mathematical Analysis and Applications*, vol. 18, no. 1, pp. 145–174, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Kubiak,
*On fuzzy topologies [Ph.D. thesis]*, A. Mickiewicz, Poznan, Poland, 1985. - A. P. Šostak, “On a fuzzy topological structure,” in
*Proceedings of the 13th Winter School on Abstract Analysis. Section of Topology*, pp. 89–103, Circolo Matematico di Palermo, Palermo, Italy, 1985. - K. Atanassov and S. Stoeva, “Intuitionistic fuzzy sets,” in
*Proceedings of the Polish Symposium on Interval and Fuzzy Mathematics*, pp. 23–26, Poznan, Poland, August 1983. - K. T. Atanassov, “Intuitionistic fuzzy sets,”
*Fuzzy Sets and Systems*, vol. 20, no. 1, pp. 87–96, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. T. Atanassov, “New operations defined over the intuitionistic fuzzy sets,”
*Fuzzy Sets and Systems*, vol. 61, no. 2, pp. 137–142, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. T. Atanassov, “More on intuitionistic fuzzy sets,”
*Fuzzy Sets and Systems*, vol. 33, no. 1, pp. 37–45, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. T. Atanassov, “Remarks on the intuitionistic fuzzy sets-III,”
*Fuzzy Sets and Systems*, vol. 75, no. 3, pp. 401–402, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. Atanassov,
*Intuitionistic Fuzzy Sets*, Physica, New York, NY, USA, 1999. - D. Çoker, “An introduction to intuitionistic fuzzy topological spaces,”
*Fuzzy Sets and Systems*, vol. 88, no. 1, pp. 81–89, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. K. Mondal and S. K. Samanta, “On intuitionistic gradation of openness,”
*Fuzzy Sets and Systems*, vol. 131, no. 3, pp. 323–336, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. Gutiérrez García and S. E. Rodabaugh, “Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies,”
*Fuzzy Sets and Systems*, vol. 156, no. 3, pp. 445–484, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. C. Jain,
*The role of regularly open sets in general topology [Ph.D. thesis]*, Meerut University, Institute of Advenced Studied, Meerut-India, 1990. - T. M. Nour, “Slightly semi-continuous functions,”
*Bulletin of Calcutta Mathematical Society*, vol. 87, pp. 187–195, 1995. View at Google Scholar - T. Noiri, “Slightly
*β*-continuous functions,”*International Journal of Mathematics and Mathematical Sciences*, vol. 28, no. 8, pp. 469–478, 2001. View at Publisher · View at Google Scholar - M. Sudha, E. Roja, and M. K. Uma, “Slightly fuzzy
*ω*-continuous mappings,”*International Journal of Mathematical Analysis*, vol. 5, no. 13–16, pp. 779–787, 2011. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Ekici and M. Caldas, “Slightly
*γ*-continuous functions,”*Boletim da Sociedade Paranaense de Matematica*, vol. 22, no. 2, pp. 63–74, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - R. N. Bhaumik and S. E. Abbas, “(L, M)-intuitionistic fuzzy topological spaces,”
*Bulletin of Kerala Mathematics Association*, vol. 5, pp. 25–47, 2008. View at Google Scholar - K. Dong, H. Wang, and Y. Liu, “Formation track association algorithm in the presence of sensor bias based on double fuzzy topology,”
*Applied Mechanics and Materials*, vol. 536-537, pp. 329–332, 2014. View at Publisher · View at Google Scholar - P. K. Saha, “Fuzzy digital topology and geometry and their applications to medical imaging,” in
*Pattern Recognition and Machine Intelligence*, vol. 8251, pp. 13–29, Springer, Berlin, Germany, 2013. View at Google Scholar - I. Bloch, “On fuzzy distances and their use in image processing under imprecision,”
*Pattern Recognition*, vol. 32, no. 11, pp. 1873–1895, 1999. View at Publisher · View at Google Scholar · View at Scopus - K. Liu and W. Shi, “Quantitative fuzzy topological relations of spatial objects by induced fuzzy topology,”
*International Journal of Applied Earth Observation and Geoinformation*, vol. 11, no. 1, pp. 38–45, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Demirci and D. Çoker, “An introduction to intuitionistic fuzzy topological spaces in Šostak's sense,”
*Busefal*, vol. 67, pp. 67–76, 1996. View at Google Scholar - Y. C. Kim and S. E. Abbas, “Several types of fuzzy regular spaces,”
*Indian Journal of Pure and Applied Mathematics*, vol. 35, no. 4, pp. 481–500, 2004. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. E. Abbas, “Several types of double fuzzy semiclosed sets,”
*The Journal of Fuzzy Mathematics*, vol. 20, pp. 89–102, 2012. View at Google Scholar - F. M. Mohammed, M. S. M. Noorani, and A. Ghareeb, “Slightly double fuzzy continuous functions,”
*Journal of the Egyptian Mathematical Society*, 2014. View at Publisher · View at Google Scholar