A subbase property for describing edge-end spaces

TL;DR

This paper provides a topological characterization of edge-end spaces in infinite graphs by combining intersection properties with existing solutions, refining the understanding of their structure.

Contribution

It introduces a new intersection property that, together with Pitz's approach, characterizes edge-end spaces topologically.

Findings

Edge-end spaces form a proper subfamily of end spaces.

A topological description for edge-end spaces is established.

The characterization refines previous understandings of infinite graph ends.

Abstract

In a previous joint work with Aurichi and Magalh\~aes Jr., we showed that the topological spaces arising from the edge-end structure of infinite graphs define a proper subfamily of those obtained through the well-known (vertex-)ends. This result was later recovered by a more general approach due to Pitz, who also stated the problem of finding a purely topological characterization for the class of edge-end spaces. His question reads as an edge-related version of a similar conjecture posed by Diestel in 1992, but there regarding the usual end structure of infinite graphs and which was recently answered also by Pitz via the existence of a suitable clopen subbase. This paper shows how an extra intersection property can be combined with his solution in order to restrict it to the edge-end spaces, hence stating a topological description for this later family as well.