The aim of this study is to define fuzzy soft topology which will be compatible to the fuzzy soft theory and investigate some of its fundamental properties. Firstly, we recall some basic properties of fuzzy soft sets and then we give the definitions of cartesian product of two fuzzy soft sets and projection mappings. Secondly, we introduce fuzzy soft topology and fuzzy soft continuous mapping. Moreover, we induce a fuzzy soft topology after given the definition of a fuzzy soft base. Also, we obtain an initial fuzzy soft topology and give the definition of product fuzzy soft topology. Finally, we prove that the category of fuzzy soft topological spaces FSTOP is a topological category over SET.
Our main goal with this paper is to construct soft topology and fuzzifying soft topology induced by fuzzy soft metric. For this, we present fuzzy soft metric spaces compatible to soft set theory and studied some of its basic properties. Then we investigate soft topological structures induced by fuzzy soft metrics.
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