Dynamic topologies and their applications to crisp topologies, fuzzifications of crisp topologies, and fuzzy topologies on the crisp real line

Abstract

We introduce and develop topologies whose members are dynamic sets à la Zeleny. This development spawns the notions of phase and co-phase spaces, fuzzy dual, dynamic dual, and co-fuzzy dual with the following results being obtained: Result A. The notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of Hutton-Gantner-Steinlage-Warren, Klein, and Lowen. Result B. The dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic. Result C. The fuzzy real lines, re-interpreted as fuzzy duals of R with the usual Euclidean topology T , induce on R , via the notion of co-fuzzy dual, a class of canonical, stratified fuzzy topologies {CO R L }. E.g., if L is a DeMorgan algebra and α ϵ L c -{1}, then the α-compactness of A ⊂ R is equivalent to the ordinary compactness of A w.r.t. T ; and if L is a Hutton algebra, ( R , L, CO R L ) is metrizable in the sense of Hutton and Erceg with Erceg metric d( L) such that d(L) (p r 1, p s 1)=¦r − s¦ (where p r is the crisp point or impulse with support r), i.e., d( L) extends the Euclidean metric on R .