Characterising path-, ray- and branch spaces of order trees, and end spaces of infinite graphs

TL;DR

This paper characterizes path, ray, and branch spaces of order trees and end spaces of infinite graphs using topological bases, solving a problem posed by Diestel in 1992.

Contribution

It provides topological characterizations of these spaces and establishes equivalences between being homeomorphic to end spaces of graphs and certain properties of bases in Hausdorff spaces.

Findings

Characterizes end spaces of graphs via nested clopen subbases.

Establishes equivalences for Hausdorff spaces related to order trees.

Solves Diestel's 1992 problem on topological characterizations.

Abstract

We investigate path-, ray- and branch spaces of trees, certain topological spaces naturally associated with order theoretic trees, and provide topological characterisations for these spaces in terms of the existence of certain kinds of (sub-)bases. Our results yield a solution to a problem due to Diestel from 1992, by establishing the following equivalences for any Hausdorff space $X$: (1) $X$ is homeomorphic to the end space of a graph, (2) $X$ is homeomorphic to the ray space of a special order tree, (3) $X$ admits a nested clopen subbase that is noetherian, hereditarily complete and $\sigma$-disjoint.