While studying for a topological groups course, I wondered if we could define the product of uncountably many topological groups such that the product is still a topological group. That is: let $G_i$ be a topological group with product law $p_i$ for each $i \in I$ (with $I$ uncountable). We can give $G = \prod_{i \in I} G_i$ the (Tychonoff) product topology and define the product law of $G$ by:

$\pi_i \circ p = p_i$ for all $i \in I$.

However, when trying to prove that this mapping is continuous end up needing $I$ to be at most countable or that the topologies of $G_i$ be discrete.

Is there any way to get around this?

Thanks.

notendowed with the trivial topology, the product is not first-countable. In particular, metrizability is lost in uncountable products. $\endgroup$