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

## Statement

### 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 $p$.

## Particular cases

varietal group property corresponding statement
abelian group topological closure of abelian subgroup is abelian
nilpotent group of nilpotency class $c$ topological closure of nilpotent group is nilpotent of same class
solvable group of derived length $\ell$ topological closure of solvable group is solvable of same derived length