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

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.

The open interval does not have the fixed-point property. The continuous self-maphas its only possible fixed point at the excluded endpoint $(0,0)$.

The closed interval does have the fixed-point property. If $f(x,0)=(F(x),0)$, then $F(0)-0\ge 0$ and $F(1)-1\le 0$. The intermediate value theorem supplies $x\in [0,1]$ with $F(x)-x=0$.

The annulus does not have the fixed-point property. The antipodal mapis a continuous self-map of the annulus and has no fixed point there.