# Vanishing Cantor-Bendixson derivative if and only if scattered

In the question asked here, it is claimed in a comment and answer of Brian M. Scott that a subset of $\mathbb{R}$ is scattered if and only if it has vanishing Cantor-Bendixson derivative.

I thought that this should be true in general for a separable metric space, but I could only prove it for closed sets. Indeed I found a scattered subset of $\mathbb{R}^2$ which has uncountable closure and hence nonvanishing derivative.

Thus my question is: is there something special about $\mathbb{R}$, or have I misunderstood a definition?

Briefly, the example is the subset of $[0,1]^2$ given by the union of subsets of horizontal lines of height $1/n$, with each containing $n$ equidistributed points.

• I'm inclined to say that you've misunderstood a definition, as what Brian said should be true of all topological spaces under appropriate definitions. But as you haven't indicated what definitions you are using (or the example you came up with) it's difficult to say for certain. Oct 13, 2015 at 14:52
• What's your example? Have you double-checked the definition of "scattered"? Oct 13, 2015 at 14:57
• Well, my definitions are as in the question linked. In particular I take the definition of the C-B as given in the linked article on that question, which I now notice requires the set to be closed. So I think my confusion stems from the fact that a subspace may not contain all of its limits points, and so when taking limit points it needs to be clarified whether to include the ones not in the set but present in the ambient space. I'll add the example now. Oct 13, 2015 at 15:00
• Regarding your example, what are the isolated points (as per the definition of "scattered") for the subset $[0,1] \times \{0\}?$ Oct 13, 2015 at 15:08
• That set is not a subset, since all the points must have positive height. Perhaps the example is not clear? Oct 13, 2015 at 16:19

• Thanks for the prompt reply. From the examples on the "Derived set" wikipedia page, one clearly does not treat the set as subspaces, but uses the topology of the parent set (having in general no partial order between $A$ and $A'$, whereas for you $A'\subset A$). Why would one restrict the definition of the C-B derivative to using the subspace topology? (I can ask this as a separate question, if you find this interesting enough) Oct 21, 2021 at 11:24