Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 12G / a / i / Solution

The Brouwer fixed-point theorem says that every continuous self-map of the closed unit disc has a fixed point.

A union of two disjoint closed discs does not have the fixed-point property. Define a map that is constant from the first component to a point in the second and constant from the second component to a point in the first. It is continuous because the components are disjoint and has no fixed point.