Suppose that $X$ is a hemicompact space, connected and locally connected. In that case, it seems that it is possible to define a "End-compactification" of $X$ (in the sense of Freudenthal).
Suppose also that $X$ is metrizable. Under what condition on $X$, we will have the End-compactification metrizable ? Is it enough if $X$ is second-countable ?