Separation axioms and the fuzzy real lines

Abstract

We show that for each lattice L which is completely distributive and equipped with an order-reversing involution, the canonical topology of the fuzzy real line ?(L) is uniformizable in the sense of Hutton; furthermore, this uniformity induces a pseudometric which induces the canonical topology. Consequently, ?(L) satisfies the higher order separation axioms of Hutton and Sarkar. More generally, we answer most of the open questions concerning ?(L) and its subspaces with respect to each of the separation axiom schemes of Hutton and Sarkar, answer some of these same questions for the stratified fuzzy real line ?c(L) and its subspaces, and analyze ?c(L) and its subspaces with respect to the separation axioms of Rodabaugh.