This is a notation question:
If $A$ is a set in a topological space and $\bar{A}$ is its closure, is there a (standard) name for $\bar{A}\setminus A$?
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$\begingroup$ Though my subject is not topology but analysis, I have never seen a standard name for it. $\endgroup$– András BátkaiCommented Jan 11, 2013 at 8:41
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$\begingroup$ I would think yours is in general an ill-behaved object. Of course, if your set is additionally open, then that's exactly the definition of boundary of $A$. $\endgroup$– Delio MugnoloCommented Jan 11, 2013 at 9:08
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6$\begingroup$ How about "the set of limit points not in $A$" or "the set of closure points not in $A$"? These both sound familiar, and I can't immediately think of any other standard terminology. $\endgroup$– Mark GrantCommented Jan 11, 2013 at 9:20
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6$\begingroup$ you could call these points the: "external limit points" --- but why create new terminology unless you need to use it several times in the same paper... $\endgroup$– SuvritCommented Jan 11, 2013 at 10:02
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2$\begingroup$ I had a series of papers in set-theoretic topology where sets of this form were critical to analyzing a notion of forcing, and I never came across a standard name... $\endgroup$– Todd EisworthCommented Jan 11, 2013 at 15:32
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3 Answers
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You could use "external boundary"... and "internal boundary" for A minus its interior
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$\begingroup$ I don't think this is good terminology, because the intersection of the exterior of a set with its boundary will always be empty, and therefore needn't equal its external boundary. However the basic idea of modifying the word "boundary" with an adjective is a good one, in my opinion. $\endgroup$ Commented Sep 24, 2013 at 8:22
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The english word for the common french expression used for this is "frontier".
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1$\begingroup$ Hi Adrien, the foniter of A (Fr(A)=(clA\cap cl(X\setminus A)) so is not equal with clA\setminus A. For example let A=[1,2] in R with usual topology. – Ali Taherifar 13 mins ago $\endgroup$– AliCommented Jan 11, 2013 at 15:21
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3$\begingroup$ Yes, this is used in analytic geometry (cf. Hardt, R. M. Stratification of real analytic mappings and images. Invent. Math. 28 (1975), 193–208); in Polish, \Lojasiewicz used the term "skraj" in his lectures (which I attended in mid-1990s). However, the usage is not so common in other areas: some algebraic and manifold topologists use "frontier" to denote what in point-set topology is commonly called the "boundary", because "boundary" means for them a different notion. Earlier, the terms "boundary" and "frontier" (resp. "bord" and "frontiere") seem to have been often used interchangeably. $\endgroup$ Commented Jan 11, 2013 at 18:26
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To summarize here all the answers that people gave in comments:
1) There seems not to be a standard notation/terminology for $\bar{A}\setminus A$ in the literature.
2) Points in $\bar{A}\setminus A$ can be referred to as "limit/closure points not in $A$" or "external limit/closure points".
3) $\bar{A}\setminus A$ is different in general than the boundary/frontier of $A$ which is defined as $\bar{A}\setminus \mathrm{int}(A)$, where $\mathrm{int}(A)$ is the interior of $A$.
answered Jan 12, 2013 at 8:02