Two Topological Uniqueness Theorems for Spaces of Real Numbers

TL;DR

This paper reviews two classical theorems that uniquely characterize the Cantor set and the rationals among metric spaces, highlighting their proofs, implications, and limitations.

Contribution

It provides an accessible overview of Brouwer's and Sierpinski's theorems, emphasizing their similarities, differences, and the conditions necessary for their uniqueness results.

Findings

The Cantor set is uniquely characterized as a totally disconnected, compact, perfect metric space.

The rationals are uniquely characterized as a countable, dense, metric space without isolated points.

Counterexamples show the necessity of the hypotheses in these theorems.

Abstract

A 1910 theorem of Brouwer characterizes the Cantor set as the unique totally disconnected, compact metric space without isolated points. A 1920 theorem of Sierpinski characterizes the rationals as the unique countable metric space without isolated points. The purpose of this exposition is to give an accessible overview of this celebrated pair of uniqueness results. It is illuminating to treat the problems simultaneously because of commonalities in their proofs. Some of the more counterintuitive implications of these results are explored through examples. Additionally, near-examples are provided which thwart various attempts to relax hypotheses.