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Is there a common name for the complement $\widehat{X} \setminus X$ of a metric space $X$ in its metric completion $\widehat{X}$? Since $X$ is not necessarily open in $\widehat{X}$, the term boundary is out of the question (without additional qualifiers). Metric remainder seems appropriate but I did not find it in the literature.

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I don't know that there is any standard term for this, but I approve of metric remainder. (Of course, whatever terminology you use should be defined the first time you use it.) – Pete L. Clark Apr 21 '10 at 15:10
up vote 4 down vote accepted

Remainder. I agree with that. But I don't find it on-line. Maybe "remainder" is primarily used for $\beta X \setminus X$ ? But it should be OK in your setting if you say the first time you use it: "the remainder of $X$ in its completion" or something.

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Remainder (aka Stone-Čech remainder) does usually mean $\beta X\setminus X$. – François G. Dorais Apr 21 '10 at 14:32
Gerald and Pete, I decided to go with metric remainder. It still feels strange that this object hasn't been found to be deserving of a name. – François G. Dorais Apr 22 '10 at 16:06

Corona? Ideal boundary?

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Interesting. Do you have references for these terms? – François G. Dorais Apr 21 '10 at 15:17
I believe, "ideal boundary" pretty much common - see ideal boundaries of Hadamard manifolds (Tits boundary, Gromov ideal boundary etc, there are also ideal boundaries of horofunctions, Buseman functions, distance-like functions); or equivalent notions from the geometric group theory. Also there are some "functional" ideal boundaries usualy called corona (spaces) - Martin boundary, Furstenberg boundary, etc. Stone-Cech compactification adds the biggest, in some sense, corona space. I am not sure about references, may be start with wik – valeri Apr 21 '10 at 16:30
I don't know all of these, but I thought that these were compactifications. Are any of them metric completions? – François G. Dorais Apr 21 '10 at 17:59
yes, usually they are (I am not sure about all) compactifications. – valeri Apr 21 '10 at 19:50

Hausdorff boundary A.P. Kopylov "On unique determination of domains in Euclidean spaces" section 6 "Domains with Hausdorff Boundaries" It is posiible to use word "boundary" as part of the name of $\widehat{X}\setminus X$ if $X$ is domain in $\mathbb{R}^n$.

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Penumbra? Cointerior?

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Any references for these? I am only familiar with penumbra in the setting of convex analysis, and I don't think the concept there can be taken over here directly. – Willie Wong Apr 21 '10 at 18:49
Penumbra was taken from the analysis of object-oriented programs (The Ins and Outs of Objects – Potter, Noble, Clarke, ASWEC 98). I thought cointerior was from mathematical morphology, but I could not find the name again when scanning through some literature. – supercooldave Apr 22 '10 at 7:20

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