# Is there a (standard) name for $\bar{A}\setminus A$?

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 $\bar{A}\setminus A$?

• Though my subject is not topology but analysis, I have never seen a standard name for it. – András Bátkai Jan 11 '13 at 8:41
• 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 $A$. – Delio Mugnolo Jan 11 '13 at 9:08
• 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. – Mark Grant Jan 11 '13 at 9:20
• 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... – Suvrit Jan 11 '13 at 10:02
• 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... – Todd Eisworth Jan 11 '13 at 15:32

## 3 Answers

You could use "external boundary"... and "internal boundary" for A minus its interior

• 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. – goblin GONE Sep 24 '13 at 8:22

The english word for the common french expression used for this is "frontier".

• 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 – Ali Jan 11 '13 at 15:21
• 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. – Margaret Friedland Jan 11 '13 at 18:26

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 $A$.