Topological remarks on end and edge-end spaces

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