On Borsuk's non-retract theorem

TL;DR

This paper revisits Borsuk's non-retract theorem, demonstrating that while the sphere is not a continuous retract of the ball, it can be a piecewise continuous retract, offering a nuanced perspective on classical topology.

Contribution

It introduces the concept of piecewise continuous retracts, extending the classical theorem by showing the sphere's retractability in a piecewise manner.

Findings

Sphere is not a continuous retract of the ball.

Sphere is a piecewise continuous retract of the ball.

Provides a new perspective on classical topological theorems.

Abstract

The classical Borsuk's non-retract theorem asserts that a unit sphere in $\mathbb{R}^n$ is not a continuous retract of the unit closed ball. We will show that such a unit sphere is a piecewise continuous retract of the unit closed ball.