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.