Given a Tychonoff (topological) space $X$, we know that $X$ admits a Hausdorff compactification $cX$. We say the remainder of $X$ in this compactification is $rX=cX\setminus X$. One natural question, which I believe is due to Arhangle'skii, is the following.

Given such a space $X$, what can we say about its remainder in a Hausdorff compactification?

I will assume all compactifications are Hausdorff from now on. I recently attended a talk of Arhangle'skii's, in which he asked a more more focussed version of the above questions.

Can we give necessary and sufficient conditions for a topological space to be the remainder of a compactification of a metric space?

For example, a theorem due Henriksen and Isbell gives us that, given a metrizable space $X$, any remainder $rX$ must be Lindelöf and hence paracompact. If we restrict ourselves further to locally compact spaces, we see that such spaces are open in their compactifications, and hence their remainders are compact. So in this case, the question becomes more tractable.

This leads us to another natural restriction, to completely metrizable spaces. We know that a metric space $X$ is completely metrizable if and only if it is Cech-complete, if and only if it is $G_\delta$ in all of its compactifications. So, their remainders are (completely regular) $F_\sigma$ subsets of compact spaces, which are going to be $\sigma$-compact. I was curious if more was known about this specific class of spaces. That is, my question is as follows:

Given a completely metrizable space $X$, what can we say about the remainders $rX$ of $X$?

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    $\begingroup$ Why does the question become much less interesting for locally compact spaces? I understand that compactness then becomes a necessary condition on the remainder, but I see no reason it would also be sufficient; I'd imagine characterizing remainders even in this case would be nontrivial. $\endgroup$ – Todd Trimble Dec 14 '14 at 3:10
  • $\begingroup$ You're right. I can't think of a particular reason off hand, outside of the fact that compactness is a nice property, that would make it much less interesting. I was mainly going off of my notes from Arhangle'skii's talk, and he didn't seem particularly interested in the case of locally compact spaces. Thanks for the input, I'll edit my question accordingly. $\endgroup$ – Shehzad Ahmed Dec 14 '14 at 3:23

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