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.



Infinite products of topological groups
You can define the product of an arbitrary family $(G\_i)\_{i \in I}$ of topological groups $G\_i$ by equipping the group-theoretic product $G = \prod\_{i \in I} G\_i$ with the product topology; the product topology is indeed compatible with the group structure (confer Bourbaki, General topology, III.2.9, but it's pretty obvious actually).


Perhaps your problem is the product topology? Note that a basis for the product topology are the sets $(U\_i)\_{i \in I}$ where $U\_i \subseteq G\_i$ is open and $U\_i = G\_i$ for all but finitely many $i \in I$. (confer [wiki](http://en.wikipedia.org/wiki/Product_topology) for the product topology).


