Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 21F / a / Solution

Regard the 3-sphere asIt is the union of the two closed solid toriExplicit homeomorphisms from the two copies in the question areandOn their boundaries both maps send $(u,v)\in S^1\times S^1$ to $(u,v)/\sqrt{2}$. They therefore descend through the quotient to a continuous bijection $S\to S^3$. Since $S$ is compact and $S^3$ is Hausdorff, this bijection is a homeomorphism.