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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.

The concept of fuzzy sets was introduced by Zadeh in his classical paper [

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 [

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.

Throughout this paper, let

The pair of functions

Let

Let

If

Let

an

an

Let

Let

double fuzzy open [

double fuzzy closed [

Let

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

Let

an

an

Let

double fuzzy totally continuous (briefly, dftc) if

double fuzzy totally semicontinuous (briefly, dftsc) if

double fuzzy totally precontinuous (briefly, dftpc) if

double fuzzy semicontinuous (briefly, dfsc) if

double fuzzy semiopen if

double fuzzy semiclosed if

A fuzzy set

Let

Let

double fuzzy semiregular (resp., double fuzzy clopen regular) if, for each

double fuzzy s-regular (resp., double fuzzy ultraregular) if, for each

double fuzzy s-normal if, for each pair of nonzero disjoint

double fuzzy clopen normal if, for each pair of disjoint

If

Suppose

If

Suppose

Let

Suppose

Let

This proof is similar to the proof of Theorem

Let

Let

The completion of the proof is straightforward.

Let

if

if

(1) Let

(2) Let

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

(1) Let

Let

Then the function

Since

(2) in (1) define

So

Let

double fuzzy irresolute (dfir, for short) if

double fuzzy semi-irresolute (dfsir, for short) if

If a function

Let

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.

Let

Let

for each

Let

Every dfstc function is a dftc function.

Suppose

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

See Example

Every dfstc function is a dftsc function.

Suppose

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

Let

Let

if

if

if

if

Conversely, let

Let

double fuzzy semi-

double fuzzy semi-

double fuzzy semi-