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

TL;DR

This paper investigates the structure of compact countable subsets of the real line, providing a detailed classification and introducing a new concept called a 'primitive' associated with their Cantor-Bendixson derivatives.

Contribution

It offers a detailed partition-based classification of these subsets and introduces the concept of a primitive linked to the Cantor-Bendixson derivative.

Findings

Partition of compact countable subsets up to homeomorphism

Cardinality results related to the partition

Existence of a primitive associated with the Cantor-Bendixson derivative

Abstract

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.