I have proved that if $G$ is a Hausdorff Topological Group and $H \subset G$ is a subgroup, then $\bar H$ is a subgroup, and that if $H$ is abelian, so is $\bar H$.
Is it possible to drop the Hausdorff Hypothesis?
2026-03-30 01:52:07.1774835527
Closure of an abelian subgroup of a topological group is abelian
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No. Any nonabelian group is a topological group in the indiscrete topology, and in such a group the closure of, say, the trivial subgroup is the whole group and hence is not abelian.