Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 19I / a / Solution

A compact topological group is a group $G$ equipped with a compact Hausdorff topology for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous.

The special unitary group isIt is closed in the finite-dimensional real vector space $M_n(\mathbb{C})$ because the equations $U^{\ast }U=I$ and $\det U=1$ are closed conditions. It is bounded because every column of a unitary matrix has norm one. The Heine-Borel theorem therefore makes it compact. Matrix multiplication is continuous, and inversion is continuous because $U^{-1}=U^{\ast }$. Hence $SU(n)$ is a compact topological group.