# About the closure of a commutative subgroup of a topological group.

Let $H$ a topological subgroups of a topological group $G$, and $H'$ the closure of $H$ as topological subspace. Are classic results the if $H'$ is a topological subgroup, and that it is normal if $H$ is normal this for a general topological group, and that if $G$ is topologically $T_2$ then $H'$ is commutative if $H$ is commutative. All these results are for example from "Topology I" of Bourbaki. But in literature nothing about the closure of a commutative subgroup if the topology is (more) general, and is not even a counterexample.

I ask a for a such counterexample, or for a more general condition about the commutativity of the closure of a commutative subgroup.

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I removed the algebraic-groups tag which does not seem relevant –  Yemon Choi Jun 17 '12 at 20:45
Take a noncomutative group equipped with concrete topology. –  Anton Petrunin Jun 17 '12 at 20:52
$aba^{-1}b^{-1}$ is a continuous function on a topological group cross itself, thus the set where it vanishes is closed, so if it vanishes on a set then it vanishes on the closure. –  Will Sawin Jun 18 '12 at 0:40
This requires T1. –  Will Sawin Jun 18 '12 at 0:41
Yemon Choi: I put the "algebraic-groups-tag" because I thinked about the Zariski Topology of algebraic group. –  Buschi Sergio Jun 18 '12 at 10:34

Since the question has been answered in the comments I just repeat it here as a CW answer. Take any infinite non-abelian group with the indiscrete (which I assume is what Anton means by concrete) topology and take the closure of the identity.

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