What question did this study set out to answer?

The aim is to explore properties related to τ-open sets and e-density in topological spaces, including an analogy to Arkhangelskii's theorem.

February 9, 2026Filomat1 citationsOpen Access

On τ-base and e-density of topological spaces

RBR.B. BeshimovNational University of Uzbekistan RJR.M. JuraevNational University of Uzbekistan RMR.Z. ManasipovaNational University of Uzbekistan

Key Points

The aim is to explore properties related to τ-open sets and e-density in topological spaces, including an analogy to Arkhangelskii's theorem.
Proved properties of families of τ-open sets
Studied e-density as a cardinal function
Investigated e-density in relation to existing theorems
Provided an example of a space contrasting e-density with density
Demonstrated that a base is not necessarily a τ-base
Outlined properties of e-density and its implications
Presented a specific example of a topological space where e-density differs from density

Abstract

In this paper, we prove some properties of families of ?-open sets, study the properties of the space of ?-continuous mappings, as well as the properties of the e-density of topological spaces. We prove an analogue of A.V.Arkhangelskii?s theorem for ?-base. Also we showed that the base of a topological space is not always a ?-base. We introduce the notion of e-density of topological spaces. As well as, we investigate some properties of this cardinal function. It is given an example of a topological space, e-density of which is not equal to its density.

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