A topological characterization of end space of infinite graphs via games, subspaces and products

TL;DR

This paper offers a new topological characterization of end spaces of infinite graphs using a special subbase and a topological game, with applications to their Baire property, subspace structure, and product behavior.

Contribution

It introduces an alternative characterization of end spaces employing a topological game, expanding understanding of their structure and properties.

Findings

Every end space is hereditarily Baire.

G_delta subspaces of end spaces are also end spaces.

The product of end spaces is not always an end space.

Abstract

In 1992, Diestel asked which topological spaces could be represented as the end space of some graph. In 2023, Pitz provided a solution to this question by giving a topological characterization of end spaces using a hereditarily complete special subbase. In this paper, we present an alternative topological characterization of end spaces, in which we employ a special subbase and a topological game. Furthermore, we provide several applications of this characterization: we show that every end space is hereditarily Baire, that $G_{\delta}$ subspaces of end spaces are also end spaces, and that the product of end spaces is not always an end space.