= Solution
A <topological group> is a <group> equipped with a <topology> for which multiplication $G\times G\to G$ and inversion $G\to G$ are <continuous maps>. The product uses the <product topology>. Translation and inversion are therefore <homeomorphisms>. Hausdorffness is sometimes included in the definition; the two open-subgroup arguments below do not require it.
For the <inverse system>, take a <directed set> $I$, <topological spaces> $X_i$, and <continuous maps> $f_{ij}:X_j\to X_i$ for $i\leq j$, with $f_{ii}=\mathrm{id}$ and $f_{ik}=f_{ij}f_{jk}$. An <inverse limit> is a space $X$ with compatible <continuous maps> $\pi_i:X\to X_i$ such that every compatible family $q_i:Y\to X_i$ factors uniquely as $q_i=\pi_iq$ through a <continuous map> $q:Y\to X$. Existence follows from the concrete construction
$$
X=\left\{(x_i)\in\prod_{i\in I}X_i:f_{ij}(x_j)=x_i\text{ whenever }i\leq j\right\},
$$
with the subspace <topology> of the <product topology>. The only possible map $q$ is $q(y)=(q_i(y))_i$. Compatibility puts its image in $X$, and coordinatewise continuity makes it continuous. This proves the <universal property of an inverse limit>, even when $X$ is empty. Given two <inverse limits>, their universal properties supply maps each way; the composites have the same coordinates as the identity, so uniqueness makes them identities. \b[The inverse limit exists and is unique up to the unique homeomorphism respecting its projections.]
A <profinite group> can be defined as an <inverse limit> of finite discrete <groups>, or as a <compact Hausdorff space> that is a <totally disconnected space>, equipped with continuous <group> operations. For the first implication, use the <Tychonoff theorem>. The compatibility equalities define a closed <subgroup> of the compact Hausdorff product. Two distinct points are distinguished by a finite discrete coordinate, so no connected subset contains both. Consequently the <inverse limit> is a compact Hausdorff totally disconnected <topological group>.
For the converse, use the topological result that a compact Hausdorff totally disconnected space has a basis of <clopen sets>. Choose a <clopen set> $U$ containing $1$ inside any specified identity neighbourhood. For each $u\in U$, continuity gives an identity neighbourhood $V_u$ and an open neighbourhood $W_u$ of $u$ with $V_uW_u\subseteq U$. Finitely many $W_u$ cover the <compact set> $U$. Intersect their $V_u$ and its inverse to obtain a symmetric identity neighbourhood $V$ with $VU\subseteq U$. Symmetry also gives $V^{-1}U\subseteq U$, and hence $vU=U$ for every $v\in V$. The set
$$
H=\{g\in G:gU=U\}
$$
is a <subgroup> containing $V$, so it is open. It is contained in $U$, since $1\in U$ and $h=h\cdot1\in hU=U$. By part (ii), $H$ has finite index. Its <normal core of a subgroup> is a finite intersection of conjugates of $H$, hence an open <normal subgroup> $N$ contained in $U$. This proves the <open normal subgroup basis of a profinite group>.
The natural <group homomorphism>
$$
G\longrightarrow\varprojlim_{N\trianglelefteq_oG}G/N
$$
is injective because the intersection of these $N$ is $\{1\}$. For a compatible family of cosets, any finitely many have nonempty intersection: their intersection subgroup is another indexing subgroup, and its prescribed coset lies in each. The <finite intersection property> and compactness give a point belonging to all the cosets, proving surjectivity. Finally a continuous bijection from a <compact space> to a <Hausdorff space> is a <homeomorphism>. \b[The two definitions of profinite group are equivalent.]
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