Recall that Goursat's Lemma has the following useful consequence. Let $G_1, G_2$ be finite groups with no common simple non-abelian quotients, and suppose $\gcd(|G_1^{\operatorname{ab}}|, |G_2^{\operatorname{ab}}|) = 1$, where superscript $\operatorname{ab}$ denotes abelianization. If $H \subset G_1 \times G_2$ is a subgroup with the property that the natural projections $H \to G_1$ and $H \to G_2$ are surjective, then $H = G_1 \times G_2$. A statement/proof of this version of Goursat's Lemma can be found in Lemma A.4 of Zywina's article ``Elliptic Curves with Maximal Galois Action on their Torsion Points'' (see http://www.math.cornell.edu/~zywina/papers/MaximalGalois.pdf).

I would like to obtain a similar version Goursat's Lemma in the following more general situation. Let $I$ be an at-most-countable index set, and let $\{G_i\}_{i \in I}$ be a collection of topological groups. Let the product group $G = \prod_{i \in I} G_i$ be equipped with the usual product topology (a base of opens is given by sets of the form $U_{i_1} \times \cdots \times U_{i_n} \times \prod_{i \neq i_1, \dots, i_n} G_i$, where $U_{i_j} \subset G_{i_j}$ is open for each $1 \leq j \leq n$). Perhaps a statement of Goursat's Lemma in this situation would be something like the following:

Suppose no two of the $G_i$'s have any common simple non-abelian quotients, and suppose further that no two of the $G_i$'s have any common abelian quotients (this is the analogue of saying that the abelianizations have coprime order). If $H \subset G$ is a closed subgroup with the property that the natural projections $H \to G_i$ are surjective for every $i \in I$, then $H = G$.

Is the above blocked statement true, or are further assumptions required to make it true? For example, would the statement hold if the groups $G_i$ were profinite?

Here is what I have found so far:

- Simply inducting on the size of $I$ with Goursat's Lemma for finite products isn't going to get the desired result.
- Even if I consider the projection $H'$ of $H$ onto a product of $G_{i_1}, \dots, G_{i_n}$ and argue that that $H' = \prod_{j = 1}^n G_{i_j}$ with Goursat's Lemma for finite products, I still do not immediately know that $H \supset H'$.