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)