Abstract
Some of the properties of the completely regular fuzzifying topological spaces are investigated. It is shown that a fuzzifying topology τ is completely regular if and only if it is induced by some fuzzy uniformity or equivalently by some fuzzifying proximity. Also, τ is completely regular if and only if it is generated by a family of probabilistic pseudometrics.
Highlights
The concept of a fuzzifying topology was given in [1] under the name L-fuzzy topology
A classical topology is a special case of a fuzzifying topology
In [6], we studied the level classical topologies τθ, 0 ≤ θ < 1, corresponding to a fuzzifying topology τ
Summary
The concept of a fuzzifying topology was given in [1] under the name L-fuzzy topology. A classical topology is a special case of a fuzzifying topology. In [6], we studied the level classical topologies τθ, 0 ≤ θ < 1, corresponding to a fuzzifying topology τ. We studied connectedness and local connectedness in fuzzifying topological spaces as well as the so-called sequential fuzzifying topologies. 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. Many Theorems on classical topologies follow as special cases of results obtained in the paper.
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