I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the empty set.
http://www.cs.man.ac.uk/~hsimmons/DOCUMENTS/PAPERSandNOTES/CB-examples.pdf
On page 3, the author says: A closed set X (of the reals) is scattered if $X^{\alpha}=\emptyset$, where $\alpha$ is the Cantor-Bendixson rank of $X$.
From what I could find, scattered means that any proper subset contains isolated points. However, for a countable subset of the reals, the Cantor-Bendixson-derivatives eventually vanish. As well, if I understand the definition of scattered correctly, a converging sequence together with its limit would be a closed, non-scattered set (the set with only the limit point in it will not have isolated points), but the CB-derivatives still eventually vanish. It seems to me that a closed set of the reals has vanishing CB-derivatives if and only if it is countable.
Am I misreading the definition of scattered?