A topological group is pro-p exactly when it is profinite and has an open normal basis with finite p-group quotients

If $G$ is pro-$p$, then [F1] gives a presentation with each $P_i$ a finite $p$-group. By [L1] this already makes $G$ profinite, and [L2] supplies an open normal basis whose quotients are the coordinate images ${\pi }_i(G)\le P_i$, hence finite $p$-groups.

Conversely, suppose $G$ is profinite and let $\mathcal{N}$ be an identity-neighbourhood basis of open normal subgroups with each $G\mathrm{/}N$ a finite $p$-group. Replace it by the basis of finite intersections of its members. The new basis has the same properties, because a quotient by $N_1\cap \cdots \cap N_r$ embeds in the product of the finite $p$-groups $G\mathrm{/}{N}_j$, and it is directed by reverse inclusion. The quotient maps define a continuous homomorphism . Because the basis separates points, $\eta$ is injective. A basic cylinder in the target prescribes finitely many compatible cosets, and their common refinement in $\mathcal{N}$ has a representative in $G$, so $\eta$ is surjective. Its coordinate maps are exactly the quotient maps by open normal subgroups from the refined basis, so $\eta$ is a homeomorphism. The target is an inverse limit of finite $p$-groups, hence [F1] makes $G$ pro-$p$.

Steps 1.1 and 1.2 prove both implications. The trivial group is included: it is the inverse limit of the constant system on $1$, and its only finite quotient is the trivial $p$-group.