A remark on Cantor derivative

TL;DR

This paper explores the algebraic structure of topological spaces under the Cantor derivative, showing that, when considering an equivalence relation based on finite correspondences, the class forms an integral semi-ring with the Cantor derivative acting as a derivation.

Contribution

It introduces an equivalence relation on topological spaces that reveals the semi-ring structure and characterizes the Cantor derivative as a derivation within this framework.

Findings

Topological spaces form an integral semi-ring under a specific equivalence relation.

The Cantor derivative functions as a derivation in this semi-ring.

The framework provides new algebraic insights into the structure of topological spaces.

Abstract

It is shown that, modulo an equivalence relation induced by finite correspondences preserving Cantor rank, the class of topological spaces is an integral semi-ring on which the Cantor derivative is precisely a derivation.