On connected subsets of a convergence space

TL;DR

This paper investigates the relationship between connected subsets in convergence spaces and their topological modifications, identifying conditions under which connectedness properties are equivalent and characterizing the largest connected set containing a given subset.

Contribution

It introduces a characterization of connected subsets in convergence spaces using adherence in reciprocal modifications, clarifying when connectedness is preserved.

Findings

Connectedness in convergence spaces relates to their topological modifications.

The largest set containing a connected subset with all intermediate subsets connected is the adherence in the reciprocal convergence.

Connectedness properties can differ between convergence spaces and their topological counterparts.

Abstract

Though a convergence space is connected if and only if its topological modification is connected, connected subsets differ for the convergence and for its topological modification. We explore for what subsets connectedness for the convergence or for the topological modification turn out to be equivalent. In particular, we show that the largest set containing a given connected set for which all subsets in between are connected is the adherence of the connected set for the reciprocal modification of the convergence.