Product of closed sub-groups in topological groups

Question

We know that if $G$ is a topological group and $C \subseteq G$ is compact and $A \subseteq G$ is closed, then $AC$ , $CA$ are closed.

Is it right to say:

The product of two closed subgroups in topological groups is closed?

Answer 1

This is false: sum of two closed subspaces of a Banach space need not be closed.