Level topologies of fuzzifying topological linear spaces and their applications

In this paper, some properties of level topologies of fuzzifying topological linear spaces are discussed. Then the characterizations of fuzzy boundedness and fuzzy precompactness are obtained. They may be considered as reply to the problem suggested by Qiu [Fuzzifying topological linear spaces, Fuzzy Sets and Systems 147 (2004) 249–272]. As applications, in Menger probabilistic normed spaces with respect to t-norm minimum, unified fuzzifying topological characterizations of probabilistic bounded, probabilistic semi-bounded, probabilistic unbounded sets, and probabilistic precompact, probabilistic semi-precompact, probabilistic unprecompact sets are proved, respectively.