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I don’t understand why I have never seen topological perimeter defined anywhere in the literature. Is it not a useful/interesting notion?

Let’s consider the following example. Suppose that $M$ is the set of irrational numbers between $0$ and $1$. Now, $\frac 1e$ is a boundary point of $M$ (as, indeed, are all points of $M$), but $\frac 1e$ seems materially different (to me) from the boundary point of $0$ (or $1$), in that it is “inside” $M$, whereas $0$ is not. It is easy to make this precise: the perimeter is the boundary of the closure of $M$. This seems to be a natural generalization of the ordinary notion of “perimeter”, and I’m at a loss why it’s (apparently) never given.

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So... you are just wondering why there isn't a commonly assigned name for $\partial \bar{M}$ where $M\subset X$ a topological space? –  Willie Wong Jan 5 '12 at 11:01
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One problem: this example uses a (complete) metric space, which is a really "nice" topological space. In more general spaces, this notion of "inside" and "outside" start to quickly lose meaning. Another potential problem: what would the perimeter of a non-connected space mean? Another potential problem: What about something like $[0,1]\cup [2,3]$ in ${\mathbb R}$; are the points "1" and "2" different from "0" and "3" because they are now "outside"? In this case, the "inside outside" argument says "0" and "3" are the perimeter; Willie's characterization above gives you 0,1,2, and 3. –  james Jan 5 '12 at 12:07
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I can’t say that one is never interested in $\operatorname{bdry cl}A$, but it’s certainly not something that comes up often enough to need a name; $\operatorname{int cl}A$ is much more likely to arise, and it doesn’t have a name. –  Brian M. Scott Jan 5 '12 at 13:53
    
@Willie Wong: Yes, I was just wondering. However, what james says in his comment shows me how problematical it is, so if you want to close this question as no longer being relevant, that would be ok, or if james wants to post his comment as an answer, I will be glad to accept it. –  Hexagon Tiling Jan 5 '12 at 13:55
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There is a measure-theoretic notion that might be of interest. Google the phrases "essential boundary" and "Lebesgue density" together. Given a measurable set $E$ (in ${\mathbb R}^{n}$, to be less abstract), the essential boundary of $E$ is the set of all points $x$ such that the Lebesgue density of $E$ at $x$ is strictly between $0$ and $1$. –  Dave L. Renfro Jan 5 '12 at 15:55

1 Answer 1

(Reposted just to not close the question.)

One problem: this example uses a (complete) metric space, which is a really "nice" topological space. In more general spaces, this notion of "inside" and "outside" start to quickly lose meaning. Another potential problem: what would the perimeter of a non-connected space mean? Another potential problem: What about something like [0,1]∪[2,3] in R; are the points "1" and "2" different from "0" and "3" because they are now "outside"? In this case, the "inside outside" argument says "0" and "3" are the perimeter; Willie's characterization (in the comments above) above gives you 0,1,2, and 3.

This isn't to say that such a notion isn't possible; it's just that there are a number of different ways to look at this idea, as Willie points out above.

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