A primitive associated to the Cantor-Bendixson derivative on the real line

We consider the class of compact countable subsets of the real numbers $\mathbb{R}$. By using an appropriate partition, up to homeomorphism, of this class we give a detailed proof of a result shown by S. Mazurkiewicz and W. Sierpinski related to the cardinality of this partition. Furthermore, for any compact subset of $\mathbb{R}$, we show the existence of a "primitive" related to its Cantor-Bendixson derivative.