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.