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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.

In 1968, Chang [

Various generalizations of the concept of fuzzy set have been done by many authors. In [

In 1980, Jain [

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.

Throughout this paper, let

A double fuzzy topology

The triplet

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

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 [

Let

Let

If

Let

A fuzzy set

An

Let

A fuzzy set

An

An

Let

slightly double fuzzy continuous (briefly, sdfc) if for every

slightly generalized double fuzzy semicontinuous (briefly, sgdfsc) if for each

Let

A fuzzy set

(1) Let

(2) In (1), let

Let

Let

Now,

But

That is,

But

Let

generalized double fuzzy semiopen (briefly, gdfso) if for each

slightly generalized double fuzzy semiopen (briefly, sgdfso) if for each

somewhat generalized double fuzzy semiopen (briefly, swgdfso) if for each

somewhat slightly generalized double fuzzy semiopen (briefly, swsgdfso) if for each

That is,

Let

Let

Now,

Since

Let

The following implication illustrates the relationships between different functions in Figure

None of these implications is reversible where

Let

Let

In (1),

Let

Let

Let

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

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.