Completely regular fuzzifying topological spaces

The concept of a fuzzifying topology was given in [1] under the name L-fuzzy topology. Ying studied in [9, 10, 11] the fuzzifying topologies in the case of L = [0,1]. A classical topology is a special case of a fuzzifying topology. In a fuzzifying topology τ on a set X , every subset A of X has a degree τ(A) of belonging to τ, 0 ≤ τ(A) ≤ 1. In [4], we defined the degrees of compactness, of local compactness, Hausdorffnes, and so forth in a fuzzifying topological space (X ,τ). We also introduced the fuzzifying proximities. Every fuzzifying proximity δ induces a fuzzifying topology τδ . In [6], we studied the level classical topologies τθ , 0 ≤ θ < 1, corresponding to a fuzzifying topology τ. In the same paper, we studied connectedness and local connectedness in fuzzifying topological spaces as well as the so-called sequential fuzzifying topologies. In [5], we introduced the fuzzifying syntopogenous structures. We also proved that every fuzzy uniformity , as it is defined by Lowen in [7], induces a fuzzifying proximity δ , and that for every fuzzifying proximity δ, there exists at least one fuzzy uniformity with δ = δ . Some of the results contained in papers [4, 6] are closely related to those which appeared in the papers [12, 13]. In this paper, we continue with the investigation of fuzzifying topologies. In particular, we study the completely regular fuzzifying topologies, that is, those fuzzifying topologies τ for which each level topology τθ is completely regular. As in the classical case, we prove that for a fuzzifying topology τ on X , the following properties are equivalent: (1) τ is completely regular; (2) τ is uniformizable, that is, it is induced by some fuzzy uniformity; (3) τ is proximizable, that is, it is induced by some fuzzifying proximity; and (4) τ is generated by a family of so-called probabilistic pseudometrics on X . We also give a characterization of completely regular fuzzifying spaces in terms of continuous functions. Many Theorems on classical topologies follow as special cases of results obtained in the paper.