Probabilistic metric structures: Topological classification. (English) Zbl 0587.54049

Hicks, T. L.; Sharma, P. L.

Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 14, No. 1, 35-42 (1984).

The authors show the strong connection between metrizability of a topological space and probabilistic metric spaces. They extract the essence of the condition that a Menger space be metrizable; namely, that \(\sup_{x<1}T(x,x)=1\), and reformulate this condition more directly. The condition that \(\sup_{x<1}T(x,x)=1\) implies (C): For each \(\epsilon >0\) there is a \(\delta >0\) such that if \(1-F_{xy}(\delta)<\delta\) and \(1- F_{yz}(\delta)<\delta\), then \(1-F_{xz}(\epsilon)<\epsilon\). Moreover, once (C) is isolated the authors show in Theorem 2.2 that a topological space (X,\(\tau)\) is metrizable if and only if there is a probabilistic premetric space structure on X which satisfies (C).

Reviewer: R.Tardiff

MSC:

54E35	Metric spaces, metrizability
54A40	Fuzzy topology

Keywords:

weakly first countable topological space; probabilistic metric spaces