Let $G$ be an abelian first-countable topological group. Define $CS(G) := \{(x_n) \in G^{\mathbb{N}} \ | \ (x_n) \subset G \text{ is Cauchy-sequence}\}$ and $NS(G):=\{(x_n) \in CS(G) \ | \ (x_n) \subset G \text{ is null-sequence}\}$. Then the completion of $G$ is defined (algebraically) as $\hat G := CS(G)/NS(G)$.
One possibility to topologize $CS(G)$ and thus $\hat G$ (as quotient group) is to give $G^{\mathbb{N}}$ the product topoology and $CS(G)$ the subspace topology.
Is this the right topology?
I don't think so because if $G$ had a neighborhood basis consisting of subgroups $G_0 \supseteq G_1 \supseteq G_2 \supseteq ...$, the canonical map $\phi_n: CS(G) \to G/G_n, \ (x_n) \to \bar x_N$ with $x_k-x_l \in G_n$ for $k,l \geq N$ would not be continuous, because $G/G_n$ is discrete and $\phi_n$ is not locally constant.
Would the "right" topology instead then be the initial topology with respect to the family $(\phi_n)$?