Topologies Generated by Nested Collections

Abstract

We study binary relations and topologies induced by means of nested collections of subsets of a given nonempty set. Dually, given a topological space, we characterize properties of the topology in terms of suitable nested families of open subsets. By means of this relationship between nested collections of sets and nested topologies, we study the representability of total preorders via (semi-)continuous realvalued order isomorphisms.Keywords Subbasis of a topology · Nested families of subsets of a set · Topologies induced by binary relations · Nested topologies · Ordinal representability properties of topologies Mathematics Subject Classification Primary: 54A05 · Secondary: 54A10 · 54F05

Weightable quasi-metrics related to fuzzy sets

“…Needless to say, we could have directly proved this more general Theorem 3.10, ignoring the previous Theorem 3. 7. Nevertheless, we have decided to include both of them, because the construction of the quasi-metric in the proof of Theorem 3.10 is a bit more articial than that in Theorem 3.7, much more natural (see e.g.…”

“…Thus, the concept of a fuzzy set can actually be regarded from various alternative points of view, as functional equations (see [6,5]), nested families of sets and nested topologies (see [1,7,20]) or representable total preorders (see [12,6]) among others. In addition, in [8] the concept of a weightable quasi-metric was shown to be closely related to some classical functional equations, of which some were also intimately linked to the denition of a fuzzy set of a universe as proved in [6,5].…”