= Solution
A <topological group> is a <group> with a <topology> for which multiplication $G\times G\to G$ and inversion $G\to G$ are continuous. A <profinite group> has either of the following equivalent descriptions: \b[a <compact> <Hausdorff> <totally disconnected> <topological group>], or \b[an <inverse limit> of finite discrete <groups>]. <Totally disconnected> means that every <connected component> is a singleton.
For clarity, an <inverse system> over a <directed set> $I$ consists of <groups> $G_i$ and maps $\phi_{ij}:G_j\to G_i$ for $i\le j$, with $\phi_{ii}=1$ and $\phi_{ik}=\phi_{ij}\phi_{jk}$. Its <inverse limit> is
$$
\varprojlim G_i=\{(g_i)\in\prod_iG_i:\phi_{ij}(g_j)=g_i\text{ whenever }i\le j\},
$$
with coordinatewise multiplication and the <subspace topology>. Equivalently, it is a <group> with compatible continuous projections such that every other compatible family of maps into the $G_i$ factors uniquely through it.
The displayed limit is a <closed subgroup> of the <compact> <Hausdorff> product of the <finite groups>, and is <totally disconnected>, since two different tuples are separated by a <clopen> coordinate condition. Conversely a <compact> <Hausdorff> <totally disconnected> <group> has a <neighbourhood basis> of <open normal subgroups>. One way to see the <group> part of this standard clopen-basis fact is to choose a <clopen> identity <neighbourhood> $U$ inside a prescribed <neighbourhood>. <Compactness> of $U$ and <continuity> give a symmetric identity <neighbourhood> $V$ with $VU\subseteq U$. The stabilizer $H=\{g:gU=U\}$ contains $V$, is open, and is contained in $U$ because $1\in U$. Its index is finite by part (ii); its <normal core> is the intersection of finitely many conjugates and is open and normal. The canonical map
$$
G\longrightarrow\varprojlim_{N\trianglelefteq_oG}G/N
$$
is <injective> because these $N$ separate points, and <surjective> because compatible <cosets> have the <finite intersection property> and $G$ is <compact>. It is a <homeomorphism> by <compactness> and the <Hausdorff> property. This explains the equivalence of the two definitions.
The <profinite Frattini subgroup> is $\Phi(G)=\bigcap M$, where $M$ ranges over all maximal proper <open subgroups>; for the trivial <group> the empty intersection is $G$. For a <pro-p group>,
$$
\boxed{\Phi(G)=\overline{G^p[G,G]}.}
$$
Here powers and <group commutators> mean generated <subgroups> and the bar supplies <closure>. Every maximal <open subgroup> in a <pro-p group> is normal of index $p$, as follows by passing to the <finite p-group> quotient by its <normal core>. Such a <subgroup> therefore contains all powers and <group commutators>. Conversely the quotient by $D=\overline{G^p[G,G]}$ is an abelian <pro-p group> of exponent $p$. If an element of this quotient is nontrivial, some finite <elementary abelian> quotient detects it; a <linear functional> to $\mathbb F_p$ then detects it as well. Its <group homomorphism kernel> pulls back to a maximal <open subgroup> of $G$. Thus the intersection of these kernels is exactly $D$.
We also use the topological generation criterion: a subset generates $G$ topologically if and only if its image generates $G/\Phi(G)$ topologically. To prove the nontrivial direction, a proper closed generated <subgroup> $K$ lies in a proper <open subgroup> $KN$ for a suitably small open normal $N$, and hence in a maximal <open subgroup> obtained from the finite quotient $G/N$. Since that maximal <subgroup> contains $\Phi(G)$, the image of $K$ cannot generate the <Frattini quotient>.
Now suppose $G$ is a nontrivial procyclic <pro-p group>. If it had distinct maximal <open subgroups> $M_1,M_2$, the <homomorphism> to $C_p\times C_p$ induced by the two quotient maps would be onto: its projections are onto, and a proper such <subgroup> would be a one-dimensional graph, forcing the two kernels to coincide. Hence $G/(M_1\cap M_2)\cong C_p^2$, contradicting procyclicity. Thus there is exactly one maximal <open subgroup> $M$, and $\Phi(G)=M$. Any $g\notin M$ generates $G/\Phi(G)\cong C_p$, so the criterion gives
$$
\boxed{G=\overline{\langle g\rangle}.}
$$
For the trivial <group> use its identity as generator. \b[The additive <group> $\mathbb Z_p$ is an infinite example], topologically generated by $1$: its finite continuous quotients are cyclic $\mathbb Z/p^n\mathbb Z$.
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