Fuzzifying topological linear spaces

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 T2 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.