Recovering the boundary path space of a topological graph using pointless topology

TL;DR

This paper extends the concept of boundary path spaces of topological graphs using pointless topology, generalizing previous local compactness assumptions to sober T1 spaces and applying semilattice techniques.

Contribution

It introduces a new approach to recover boundary path spaces of topological graphs via pointless topology, broadening the class of spaces considered.

Findings

Generalized topology from locally compact to sober T1 spaces

Developed semilattice-based method for boundary path space recovery

Extended results to topological graphs

Abstract

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_1$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs.