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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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    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 $1$.2012-01-05

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(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.