(Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set?

Question

I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-type" of a topological space; the context I'm running into it in is the theory of curves/one-dimensional spaces.

A curve $X$ is a one-dimensional topological space, and a curve is rational if there's a basis $\lbrace U_\beta \rbrace$ such that $\partial(U_\beta)$ is countable for all $\beta$. Then they say that a rational curve has rim-type $\alpha$ if every such boundary $\partial(U_\beta)$ has an $\alpha$-th derivative of zero, and $\alpha$ is minimal among ordinals with this property.

So what I really need to know is what is meant by "derivative" here. I do own the Menger book, because at one point I considered translating it as a good deed. My vague impression from looking at this section (pp. 291-297 of Kurventheorie) is that it is just some set theory definition involving transfinite induction.

Is the "derivative" of a countable topological space just what remains after removing its isolated points? Or is it more complicated? If relevant, my spaces will be compact metric spaces, so whatever the nicest definition for that case is would be best. My googling skills weren't good enough to get past all the calculus videos when I tried to search for what the derivative of a countable set is.

Thanks!

Answer 1

I believe (based on e.g. this source's use of "derived set") this refers to the Cantor-Bendixson derivative. This does indeed consist of throwing out the isolated points. We can iterate the CB-derivative through the ordinals by repeatedly throwing out isolated points at successor stages and taking intersections at limit stages.

Note that the countability of the space in question doesn't matter here - the CB-derivative is defined for all subsets of all spaces. That said, the structure of a space can impact the possible behaviors of the iterated derivatives of its subsets, especially with regard to how long it takes for the derivative process to stabilize. For example, if the space is countable we trivially have stabilization before $\omega_1$ for every set in the space. Less trivially, but still easily, we can show that the same is true for second-countable spaces.