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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 ?

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In the book Dimension and Extensions by Aarts and Nishiura you will find a theorem giving sufficient conditions that the Freudenthal compactification has the same weight as the starting space. Theorem 3.15 and Corollary 3.16 (pages 276/277): if $X$ is rim-compact and the quasi-component space of $X$ is compact then $w(FX)=w(X)$. In your case, if $X$ is metrizable then it is also locally compact and hence rimcompact, and second-countable too; connectivity says that the quasi-component space consists of one point. Hence $FX$ is second-countable and therefore metrizable.

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