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