Topology and subsets - the story of a theorem

TL;DR

This paper explores the topology of the space of at most three-element subsets of a circle, illustrating various algebraic topology methods through a historical and conceptual perspective.

Contribution

It introduces a specific topological space related to subsets of a circle and demonstrates diverse algebraic topology techniques using this example.

Findings

Identification of the space of at most 3-element subsets of the circle

Application of algebraic topology methods to this space

Historical context from Borsuk and Bott's work

Abstract

In this exposition the space of at most 3-element subsets of the circle, first identified by Borsuk and Bott, is used as a motivation to introduce the readers to a variety of methods in (algebraic) topology.