Fuzzifying topological linear spaces. (English) Zbl 1065.46060
Summary: 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 the origin 0. Also, we demonstrate two characterizations of the \(T_2\) 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 \(A\).
MSC:
| 46S40 | Fuzzy functional analysis |
Keywords:
fuzzy topology; fuzzy topological linear spaces; fuzzy boundedness; fuzzy compactness; fuzzy neighbourhood; fuzzy logic; \(T_2\) separation axiomReferences:
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