T0 topological group may be internal direct product of dense subgroups

Statement
It is possible to have a T0 topological group $$G$$ and dense subgroups $$H,K$$ of $$G$$ such that $$G$$ is an internal direct product of $$H$$ and $$K$$.

Proof
Let $$\alpha$$ be any irrational number. Consider the additive subgroup $$G$$ of $$\R$$ given by:

$$G := \{ a + b\alpha \mid a,b \in \mathbb{Q} \}$$

Consider the subgroups of $$G$$ given by:

$$H = \mathbb{Q}$$

and:

$$K = \mathbb{Q}\alpha$$

Then, both $$H$$ and $$K$$ are dense in $$G$$ and $$G$$ is an internal direct product of these subgroups.