Resolvability Properties of Similar Topologies

Abstract

We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.2010 Mathematics subject classification: primary 54A10; secondary 54A05, 11B05, 28A05.

On separating axioms and similarity of soft topological spaces

“…In (Filipczak and Terepeta 2016), there was presented the problem of continuity of functions with similar topologies on the domain or the range. Recently, in Lindner (2015) the idea of resolvability of similar topologies was also discussed. One of the most known examples of such pair of topologies is the natural topology and the Sorgenfrey topology on reals (generated by the base consisting of intervals of the form [a; b), a < b and a, b ∈ R; this topology is also called lower limit topology).…”

Resolvability of Measurable Spaces

“…It is obvious that there may exist different topologies (X, τ 1 ) and (X, τ 2 ) with ND(τ 1 ) = ND(τ 2 ) and NB(τ 1 ) = NB(τ 2 ). Topologies with these properties are called similar and they have already been studied in the context of resolvability in [5,9,12].…”

“…This is a straightforward consequence of [12,Theorem 6], which says that for any topological space H(NB) = {A ∪ B :…”

“…Due to the fact that having the hull property is a necessary condition for (A, I) to be topological, the last theorem is a strengthening of [12,Proposition 20].…”

“…For a topological space (with the hereditarily irresolvable part in the Hewitt decomposition denoted by G), we have H(NB) = {A ∪ N : A ⊂ G ∧ N ∈ ND} (see [12,Theorem 6]) and consequently a topological space is resolvable if and only if H(NB) = ND (Theorem 2.1). We consider this property for a measurable space.…”