On connected subsets of a convergence space

TL;DR

The paper investigates when connected subsets of a convergence space $(X,\xi)$ align with those of its topological modification $\operatorname{T}\xi$. It develops a framework based on adhesion operators, the reciprocal modification $r\xi$, enclosing sets $e(A)$, and $\operatorname{T}$-subspaces to compare $\xi$-connectedness with $\operatorname{T}\xi$-connectedness, both in general and on finite sets. Key results include $e(A)=\operatorname{adh}_{\xi}A\cup\operatorname{adh}_{\xi^{*}}A=\operatorname{adh}_{r\xi}A$ as the maximal enclosure of a connected $A$, and the equivalence of $\xi$- and $r\xi$-connectedness for nonempty connected $A$, along with a detailed stratification of sandwich‐type and $\operatorname{T}$-subspace phenomena. The paper also characterizes when $\operatorname{S}_{0}\xi=\operatorname{T}\xi$ (defect 1) and explores finite-depth convergences via graphs, showing how connectedness behaves under reciprocal modification and enclosure in both finite and general settings. These insights clarify the stability and limitations of connectedness across generalized convergence spaces, with implications for pretopological structures and associated categorical constructs.

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.

Theorems & Definitions (43)

• Remark 1
• Proposition 2
• proof
• Proposition 3
• Proposition 4
• proof
• Proposition 5
• Proposition 6
• proof
• Proposition 7