Topological remarks on end and edge-end spaces

TL;DR

This paper explores a new topological space called the edge-end space of a graph, showing it is homeomorphic to some usual end space but not vice versa, revealing a strict subset relationship.

Contribution

It introduces the edge-end space topology and proves its relationship to traditional end spaces, highlighting differences and limitations in their topological equivalences.

Findings

Edge-end space is homeomorphic to some usual end space.

Not all usual end spaces are homeomorphic to an edge-end space.

The class of edge-end spaces is strictly contained within the class of all end spaces.

Abstract

The notion of ends in an infinite graph $G$ might be modified if we consider them as equivalence classes of infinitely edge-connected rays, rather than equivalence classes of infinitely (vertex-)connected ones. This alternative definition yields the edge-end space $\Omega_E(G)$ of $G$, in which we can endow a natural (edge-)end topology. For every graph $G$, this paper proves that $\Omega_E(G)$ is homeomorphic to $\Omega(H)$ for some possibly another graph $H$, where $\Omega(H)$ denotes its usual end space. However, we also show that the converse statement does not hold: there is a graph $H$ such that $\Omega(H)$ is not homeomorphic to $\Omega_E(G)$ for any other graph $G$. In other words, as a main result, we conclude that the class of topological spaces $\Omega_E = \{\Omega_E(G) : G \text{ graph}\}$ is strictly contained in $\Omega = \{\Omega(H) : H \text{ graph}\}$.