Many Valued Topologies And Borel Probability Measures

Abstract

One of the fundamental problems of the discipline of fuzzy topology is the question of its applicability to those mathematical problems which cannot be solved by standard techniques from general topology. As a simple example we mention the problem to construct a unique continuous extension of the Boolean negation from 2 = {0, 1} to the real unit interval [0, 1]. Since 2 is not dense in [0, 1], it is clear that standard techniques from general topology fail! What is even more surprising is the fact that fuzzy topology¹ as originally defined in [7], also called [0, 1]-topology, is also not able to solve this simple question. The reason for this situation has to do with the way how [0, 1]-topology understands the “intersection axiom” of ordinary topologies. The interpretation of intersection of subsets by the minimum of [0, 1]-valued functions entails the possibility to identify fuzzy topological with ordinary topological spaces. This insight is not new and traces back to an unpublished paper by E. Santos 1977 (cf. [17]) and a remark by R. Lowen 1978 (cf. [14]). Hence it is hopeless to look for problems which are solvable using [0, 1]-topological spaces but unsolvable using ordinary topological spaces.