Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 3 / b

A topological group has the topological Hopf property when every continuous surjective endomorphism is a topological automorphism.

Let $\varphi :G\to G$ be a continuous surjective endomorphism. For each $n$, inverse image under $\varphi$ permutes the finite set of open subgroups of index at most $n$: surjectivity preserves the index, and injectivity of the inverse-image operation follows from surjectivity. HenceIf $x\in \ker \varphi$, then $x\in G_n$ for every $n$. Part 3(a)(iii) implies ${\bigcap }_n{G}_n=\{1\}$, so $x=1$. Thus $\varphi$ is bijective. A continuous bijection from compact $G$ to Hausdorff $G$ is a homeomorphism, proving the Hopf property of a topologically finitely generated profinite group.