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

The quotient maps define a continuous homomorphismIts kernel is ${\bigcap }_{U\in \mathcal{U}}U=\{1\}$, because $\mathcal{U}$ is a neighborhood basis and $G$ is Hausdorff. Thus $\Phi$ is injective. The standard compactness argument for inverse limits makes it surjective: a compatible family of cosets has the finite intersection property, and the corresponding closed cosets in compact $G$ have nonempty intersection. Finally, a continuous bijection from compact $G$ to the Hausdorff inverse limit is a homeomorphism. Henceas topological groups.

Suppose $G$ is topologically generated by $d$ elements. An open subgroup $H$ of index $n$ gives a continuous transitive coset actionA continuous homomorphism $G\to S_n$ is determined by the images of the $d$ topological generators, so there are at most $|S_n{|}^d$ such homomorphisms. Each has only finitely many point stabilizers. Therefore $G$ has only finitely many open subgroups of index $n$.

By part ii, $G_n$ is the intersection of finitely many open subgroups, so it is open. Conjugation permutes the subgroups of each index, hence $G_n$ is also a normal subgroup. The groups $G_n$ form a descending family. If $U$ is any open normal subgroup and $[G:U]=m$, then $G_m\subseteq U$. Thus $(G_n)$ is cofinal among the open normal neighborhoods of the identity. Part i now gives