= Solution
A <topological group> is a group with a topology for which $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are continuous; we use the Hausdorff convention. Translations and inversion are then <homeomorphisms>.
For the final generator-extraction assertion, use the <profinite Frattini subgroup>
$$
\Phi(G)=\overline{G^p[G,G]}.
$$
For a finitely generated <pro-p group>, the <Burnside basis theorem> says that $G/\Phi(G)$ is a finite-dimensional <vector space> over $\mathbb F_p$ and a subset topologically generates $G$ precisely when its image spans that <vector space>. In particular its dimension is the minimal number $d(G)$ of <topological generators>. A given $d$-element generating set shows $d(G)\leq d$. The image of any other generating set spans the same finite-dimensional <vector space>, so it contains a basis of $d(G)$ elements. Choose their preimages in that generating set and apply the <Burnside basis theorem> again. \b[Every generating set contains a generating subset of size $d(G)\leq d$.] This argument also applies to an infinite generating set.
Back to article page
