On similarity between topologies

Abstract

Let T 1 and T 2 be topologies defined on the same set X and let us say that (X T 1 ) and (X T 2 ) are similar if the families of sets which have nonempty interior with respect to T 1 and T 2 coincide. The aim of the paper is to study how similar topologies are related with each other. MSC:54A10, 54A05, 11B05, 28A05, 28A10, 26A15

On separating axioms and similarity of soft topological spaces

“…To distinguish topologies with such properties, we call them similar. This notion was introduced in Bartoszewicz et al (2014), and there were presented some basic properties of such topologies. It was observed that the similar topologies may differ with compactness, separating and countability axioms.…”

On the position of abstract density topologies in the lattice of all topologies

“…As a nonempty semi open set, G has a nonempty interior in T 2 . The relation of similarity was investigated among others in [2] and [14] (under the name of π-relation). The relation of semi-correspondence introduced by N. Levine, was investigated also by S. G. Crossley and S. K. Hildebrand in [4] and T. R. Hamlett in [7].…”

“…In the further investigation some properties of abstract density topologies will be needed (compare [10], [2]): Proposition 2.3. Let Φ be the lower density operator in the space (X, A, I), and let T Φ be the topology generated by Φ.…”

Resolvability Properties of Similar Topologies

“…The notion of similarity between two topological spaces was introduced in [5], although the same relation was mentioned earlier in [16] as a π-relation. Among other results, [5] shows that every topology is similar to some abstract density topology.…”

“…The relation of similarity was investigated in [5]. In that paper one can find the following useful characterisation.…”