The generalized Bolzano-Weierstrass property revisited

TL;DR

This paper explores the conditions under which topological spaces exhibit the Generalized Bolzano-Weierstrass property, where every sequence of subsets has a Kuratowski-convergent subsequence, extending classical compactness concepts.

Contribution

It provides a detailed analysis and characterization of topological spaces with the Generalized Bolzano-Weierstrass property, revisiting and expanding upon existing theories.

Findings

Characterization of spaces with the property

Conditions ensuring Kuratowski convergence

Connections to classical compactness

Abstract

We investigate the question of when a topological space $X$ has the $\textit{Generalized Bolzano-Weierstrass property}$: every sequence of subsets of $X$ has a convergent subsequence (in the sense of Kuratowski).