A topological group has the topological Hopf property when every continuous surjective endomorphism is a topological automorphism.
Every topologically finitely generated group that is profinite has the topological Hopf property. There are only finitely many open subgroups of each given index, and inverse image under a surjective endomorphism permutes them. The endomorphism kernel consequently lies in every open normal subgroup and is trivial.

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