I was learning about topological groups from Atiyah-Macdonald's chapter on Completions, and I have the following question:
Let $G$ be an abelian topological group. Let $H$ be the intersection of all neighborhoods of $0$. Then in Lemma $1.10$ of Atiyah-Macdonald, it is shown that $H$ is in fact a subgroup of $G$. Now, the quotient group $G/H$ can be given the structure of a topological space via the canonical projection $\pi:G \rightarrow G/H$.
So, then when we want to show that $G/H$ is also a topological group, one of the verifications requires us to check that the binary operation (addition) $G/H \times G/H \rightarrow G/H$ is continuous.
My question is do we regard $G/H \times G/H$ as a topological space under the product topology, or as a topological space which makes the map $\pi \times \pi: G \times G \rightarrow G/H \times G/H$ into a quotient map? Assuming the latter, I have been able to show that addition on $G/H$, i.e., the map from $G/H \times G/H \rightarrow G/H$ is continuous.
I initially thought that the product topology on $G/H \times G/H$ and the quotient topology on $G/H \times G/H$ w.r.t. the map $\pi \times \pi$ would be the same. But, after being unable to prove this statement, I realized that exercise $22.6(b)$ in Munkres shows that this need not be the case in general.