Topological group with box topology

Question

I read the below theory in discussions topology groups:

theory: let $\alpha \in I $, $G_{\alpha}$ be a topological group,then direct multiplication Group $G =\prod G_{\alpha}$ is topological group with product Topology.

my question is:

is theory true with box topology ?

Answer 1

The (set-)direct product $\prod G_\alpha$, equipped with box topoloyg, is a topological group, i.e., multiplication and inverse are continuous maps (because they are so componentwise). However, this does no longer have the universal property we expect for a product in the category of topological groups.