Topological remarks on end and edge-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 $Ω_E(G)$ of $G$, in which we can endow a natural (edge-)end topology. For every graph $G$, this paper proves that $Ω_E(G)$ is homeomorphic to $Ω(H)$ for some possibly another graph $H$, where $Ω(H)$ denotes its usual end space. However, we also show that the converse statement does not hold: there is a graph $H$ such that $Ω(H)$ is not homeomorphic to $Ω_E(G)$ for any other graph $G$. In other words, as a main result, we conclude that the class of topological spaces $Ω_E = \{Ω_E(G) : G \text{ graph}\}$ is strictly contained in $Ω= \{Ω(H) : H \text{ graph}\}$.