Compactness Property of Fuzzy Soft Metric Space and Fuzzy Soft Continuous Function

      The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics.    
Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.


Introduction
Most of the problems in medical science, engineering, economics, and so on, have different doubts. The problems in method identification include properties that are basically non-probabilistic in nature. To respond to such a situation, a new trend of mathematics was created by Zadeh [1], that is named the fuzzy set theory, to process concepts from the fuzzy perspective using a specific membership. The soft set theory is one of the branches of mathematics, which aims to describe phenomena and concepts of ambiguous, vague, undefined, and imprecise meaning. It has rich potential for applications in several directions. The soft set theory is applicable where there is no clearly defined
A fuzzy generalization of the soft sets was initiated for them to be fuzzy soft sets. Soft sets and fuzzy sets were combined to construct the fuzzy soft sets notion [19]. A pair is said to be a fuzzy soft set defined on a universe set such that is a mapping, is a subset of the parameter set and is the family of all fuzzy subsets of . Roy and Samanta [20] applied the definition of the fuzzy soft set given in [19] on topological spaces, i.e. depending on a fuzzy soft set, a topology will be built. Also, several significant theorems were proved. Depending on basic information of the fuzzy soft set presented in [19] and [20], the concept of fuzzy soft metric space was constructed by Beaula and Raja [21] and many concepts were given, such as the Cauchy sequence, fuzzy soft (open and closed) balls, convergence, and boundedness. Furthermore, some fundamental theorems were proved. Analogously to the ideas in [2] and [19], Varol and Aygun [22] were able to present a new definition of the fuzzy soft set. A pair ̌ is called a fuzzy soft set over a universe set , where ̌ , ̌ is a fuzzy set and is the collection of all fuzzy sets on such that for each ̌ implies . As an abbreviation, a fuzzy soft set is denoted by ̌ ̌ . A fuzzy soft set ̌ on a universe set is a mapping from the parameter set to ; that is ̌: where the empty is fuzzy set on . The family of all fuzzy soft sets over will be denoted by F( , ). Kider [23] relied on the notions of the fuzzy soft set presented in [22] to provide a new definition of fuzzy soft metric space. It is worth mentioning that the fuzzy soft set definition given in [19] is different from the fuzzy soft set given in [22]. Consequently, the notion of the fuzzy soft metric space given in [23] is different from the fuzzy soft metric space given in [21] and also different from other definitions of fuzzy soft metric space presented in [24] and [25]. The first aim of this paper is to define the compactness property of the fuzzy soft metric space given in [23] and to investigate some fundamental theorems about compact fuzzy soft metric space. The second aim is to introduce the concepts of sequentially compact and locally compact fuzzy soft metric spaces and to establish their properties. Finally, compact fuzzy soft continuous functions are studied and some of their properties are examined in the fuzzy soft metric space. The structure of the paper is as in the following. In Section 2, some properties and basic concepts of the fuzzy soft metric space are given. Section 3 is devoted to introducing the concept of compact, sequentially compact, and locally compact fuzzy soft metric spaces and establishing main properties related to these concepts in the fuzzy soft metric space. The compact fuzzy soft continuous functions are introduced in Section 4, where some important properties of the given subject are also investigated. The conclusion of this work is given in Section 5.

Preliminaries
This section gives some main important properties of fuzzy length space on a fuzzy set. In 2012, Varol and Aygun [22] defined the fuzzy product between two fuzzy soft sets as below.  [23] presented the notion of fuzzy soft metric space as follows.

Definition 2.4 [23]:
Let be a universe set and let F( , ) be the family of all soft fuzzy sets over . Let ̌ ̌ F( , ) and put ̌ ̌=

Compact Fuzzy Soft Metric Space
In this section, the notion of compact fuzzy soft metric space is introduced and some important theorems about it are proved. Furthermore, the concepts of sequentially compact and locally compact fuzzy soft metric spaces are introduced and some properties about them in the fuzzy soft metric space will be investigated.
is obtained. That is has no Cauchy subsequence, which is a contradiction. For the converse, let be a fuzzy totally bounded, hence has a finite -fuzzy net for any . We consider to be a sequence in . We choose a finite fuzzy net in . Therefore, infinitely many elements, say the subsequence ( ), of the sequence are contained in one of the soft fuzzy open balls of radius 0.5, with the center in the -fuzzy net. We choose a finite fuzzy net in . Therefore, infinitely many elements, say the subsequence ( ), of the sequence ( ) are contained in one of the soft fuzzy open balls of radius 0.25, with the center in the -fuzzy net. By continuing this path, a sequence which its elements are also sequences are obtained, and each sequence is a subsequence of the previous one. for each . This is written as or simply written as . The limit point is called the fuzzy soft limit of ( ) [23]. The next characterization is provided for the fuzzy soft metric space. be an infinite subset of and let be a sequence in . Thus, includes a convergent subsequence and the limit is a fuzzy soft limit point of because is sequentially compact. Conversely, consider to be a sequence in . Consider that the set will be finite which implies that one of the set's points, named , satisfies , . Hence the sequence ( ) is a subsequence of that converges to the same point . Now, consider that the set is infinite. Then by assumption, it has at least one fuzzy soft limit point Let with

Proof
: Let be compact, then by Theorem 3.10, is fuzzy totally bounded and complete. Suppose that be any sequence in . Since is totally fuzzy bounded, then by using Theorem 3.9, the sequence contains a Cauchy subsequence, say ( ) So, converges to since is complete. Consequently, each sequence in includes a convergent subsequence ( ) proof Suppose that is sequentially compact, then this means that each sequence in contains a convergent subsequence . Now, by using Theorem 3.9, we obtain that is totally fuzzy bounded. It remains to prove that is complete. Let be a Cauchy sequence in . By assumption, includes a subsequence that converges to a point . Now, to prove that converges to . Let be given by Remark ( [ ] which shows that is fuzzy soft uniformly continuous.

Conclusions
This paper introduced many concepts of the fuzzy soft metric space, such as compactness, total boundedness, sequential compactness, and local compactness. The basic properties of the given concepts of the fuzzy soft metric space are examined and essential theorems are proved. Every compact fuzzy soft metric space ( ̌ ̌ ̃) is proved to be a locally compact fuzzy soft metric space. Furthermore, a fuzzy soft metric space ( ̌ ̌ ̃) is proved to be locally compact fuzzy soft metric space if it's sequentially compact fuzzy soft metric space. Finally, the concept of uniformly continuous function in a fuzzy soft metric space is presented to study the main properties of compact fuzzy soft continuous function.