Abstract
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.
Highlights
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 first aim of this paper is to define the compactness property of the fuzzy soft metric space given in
Summary
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 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.
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