Fuzzifying topological linear spaces

Abstract In this paper, we establish a fundamental framework of fuzzifying topological linear spaces (abbreviated to ftls). After providing some fuzzy logical notations and some basic properties of fuzzifying topological spaces, we define a number of equivalent ftls, and discuss some of their basic properties. Then we deal with the bases for fuzzifying neighborhood systems, and particularly verify a characterization of ftls by using the bases of fuzzifying neighborhood systems of original point 0. Also, we demonstrate two characterizations of T 2 separation axiom. After that, fuzzy boundedness and fuzzy complete boundedness are defined and some related properties are demonstrated; in particular, we prove an equivalent characterization of fuzzy boundedness. Furthermore, we study fuzzy compactness in ftls, and demonstrate that a subset A is compact if and only if it is completely bounded and any Cauchy net S in A converges at some point in A .

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