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.
4 Answers
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 $\beta X\setminus X$. $\endgroup$ Apr 21, 2010 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. $\endgroup$ Apr 22, 2010 at 16:06
Corona? Ideal boundary?

$\begingroup$ Interesting. Do you have references for these terms? $\endgroup$ Apr 21, 2010 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) $\endgroup$– valeriApr 21, 2010 at 16:30

$\begingroup$ I don't know all of these, but I thought that these were compactifications. Are any of them metric completions? $\endgroup$ Apr 21, 2010 at 17:59

$\begingroup$ yes, usually they are (I am not sure about all) compactifications. $\endgroup$– valeriApr 21, 2010 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. $\endgroup$ Apr 21, 2010 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. $\endgroup$ Apr 22, 2010 at 7:20