I try to show that a compact subset $A\subset \mathbb{C}$ is at most countable if and only if there exists a countable ordinal number $\alpha$ (i.e $\alpha <\omega_{1},$ where $\omega_{1}$ is the first uncountable ordinal) such that $\mbox{acc}^{\alpha}\,A=\emptyset;$ where $\mbox{acc}^{\alpha}\,A$ is the Cantor-Bendixson derivative set defined by: \ $$ \begin{cases} \mbox{acc}^{0}\,A=A\\ \mbox{acc}^{\alpha}\,A=\mbox{acc}^{\alpha-1}\,(\mbox{acc}\,A) \text{ if $\alpha$ is a successor ordinal.}\\ \mbox{acc}^{\lambda}\,A=\bigcap_{\alpha<\lambda}\mbox{acc}^{\alpha}\,A \text{ if $\lambda$ is a limit ordinal.} \end{cases} $$ Where $\mbox{acc}\,A$ is the set of all accumulation points of $A.$ Is this equivalence hold?
Please direct me to a book or to an article that discusses the Cantor-Bendixson derivative set in detail.