Topological closure of subgroup belonging to a subvariety of the variety of groups is in the same subvariety

In terms of varieties
Suppose $$G$$ is a topological group and $$H$$ is a subgroup that belongs to a subvariety $$\mathcal{V}$$ of the variety of groups. Then, the closure of $$H$$ in $$G$$ is a closed subgroup of $$G$$ that also belongs to $$\mathcal{V}$$.

In terms of group property
Suppose $$G$$ is a topological group and $$H$$ is a subgroup that satisfies a varietal group property (or quasivarietal group property) $$p$$ of the variety of groups. Then, the closure of $$H$$ in $$G$$ is a closed subgroup of $$G$$ that also satisfies $$p$$.

Similar facts

 * Topological closure of subgroup is subgroup

Opposite facts

 * Topological closure of cyclic subgroup need not be cyclic (in fact, such a closure is termed a topologically cyclic group)
 * Topological closure of finitely generated subgroup need not be finitely generated (in fact, such a closure is termed a [[topologically finitely generated group)