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

From Groupprops

## Contents

## Statement

### In terms of varieties

Suppose is a topological group and is a subgroup that belongs to a subvariety of the variety of groups. Then, the closure of in is a closed subgroup of that also belongs to .

### In terms of group property

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

## Particular cases

## Related facts

### Similar facts

### 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)