Topological group not implies normal

Statement

The underlying topological space of a T0 topological group need not be normal. This is noteworthy because any topological group has a uniform structure, and hence the underlying topological space of a T0 topological group is a uniform space, thus is completely regular.

Examples

The standard example is $\mathbb {R} ^{J}$, where $J$ is an uncountable indexing set, given the product topology and the external direct product structure.