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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.

edited Jun 17 at 20:45

Yemon Choi
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asked Jun 17 at 20:40

Buschi Sergio
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I removed the algebraic-groups tag which does not seem relevant – Yemon Choi Jun 17 at 20:45

Take a noncomutative group equipped with concrete topology. – Anton Petrunin Jun 17 at 20:52

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$aba^{-1}b^{-1}$ 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 at 0:40

This requires T1. – Will Sawin Jun 18 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 at 10:34

Will Sawin · Answer 1 · 2012-06-18 03:22:54Z UTC

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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.

edited Jun 18 at 3:22

Will Sawin

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Benjamin Steinberg 67%

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

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