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

17 pages, 1 figure. V3 contains no longer a game-theoretical approach to a previous result on the theory of end spaces