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

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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

Related facts

Similar facts

Opposite facts