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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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  • $\begingroup$ I removed the algebraic-groups tag which does not seem relevant $\endgroup$
    – Yemon Choi
    Jun 17, 2012 at 20:45
  • $\begingroup$ Take a noncomutative group equipped with concrete topology. $\endgroup$
    – Anton Petrunin
    Jun 17, 2012 at 20:52
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    $\begingroup$ $aba^{-1}b^{-1}$ 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. $\endgroup$
    – Will Sawin
    Jun 18, 2012 at 0:40
  • $\begingroup$ This requires T1. $\endgroup$
    – Will Sawin
    Jun 18, 2012 at 0:41
  • $\begingroup$ Yemon Choi: I put the "algebraic-groups-tag" because I thinked about the Zariski Topology of algebraic group. $\endgroup$
    – Buschi Sergio
    Jun 18, 2012 at 10:34

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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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