  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 $\Omega_E(G)$ 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 $\Omega
= \{\Omega(H) : H \text{ graph}\}$.
