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.

3$\begingroup$ 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.) $\endgroup$ – Pete L. Clark Apr 21 '10 at 15:10
Remainder. I agree with that. But I don't find it online. 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.

1$\begingroup$ Remainder (aka StoneČech remainder) does usually mean $\beta X\setminus X$. $\endgroup$ – François G. Dorais♦ Apr 21 '10 at 14:32

1$\begingroup$ 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. $\endgroup$ – François G. Dorais♦ Apr 22 '10 at 16:06
Corona? Ideal boundary?

$\begingroup$ Interesting. Do you have references for these terms? $\endgroup$ – François G. Dorais♦ Apr 21 '10 at 15:17

$\begingroup$ 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, distancelike 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. StoneCech compactification adds the biggest, in some sense, corona space. I am not sure about references, may be start with wik en.wikipedia.org/wiki/Compactification_(mathematics) $\endgroup$ – valeri Apr 21 '10 at 16:30

$\begingroup$ I don't know all of these, but I thought that these were compactifications. Are any of them metric completions? $\endgroup$ – François G. Dorais♦ Apr 21 '10 at 17:59

$\begingroup$ yes, usually they are (I am not sure about all) compactifications. $\endgroup$ – 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" http://link.springer.com/article/10.1007/s1095800891495. It is posiible to use word "boundary" as part of the name of $\widehat{X}\setminus X$ if $X$ is domain in $\mathbb{R}^n$.
Penumbra? Cointerior?

1$\begingroup$ 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. $\endgroup$ – Willie Wong Apr 21 '10 at 18:49

$\begingroup$ Penumbra was taken from the analysis of objectoriented 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. $\endgroup$ – supercooldave Apr 22 '10 at 7:20