The order aspect of the fuzzy real line

Abstract

It is the purpose of this paper to put the “fuzzy real line” in a setting which proves to be advantageous to a more fundamental study of that space. Actually there are three different fuzzy real lines to be found in the literature, mainly defined by U. Hohle in [1], [2] and by B. Hutton in [4], [5], and a fourth one shall be added in this work. The main result of this paper is the fact that three of the four spaces are homeomorphic to fuzzy topological spaces the underlying sets of which are, in each case, the probability measures on ℝ, and the fuzzy (resp. quasi fuzzy and translation-closed fuzzy) topologies of which are determined by the left and right sections of a canonical fuzzy extension of the strict order relation on ℝ. From this it will follow very fundamentally that it is the order of ℝ, and not the topology, which determines the fuzzy real line.