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Does anyone know of examples of "pathological" topological groups that are actually used? I am particularly interested in infinite non-Hausdorff examples.

Perry Bleiberg
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    $p$-adic Lie groups and profinite groups often pop up in number theory. – anomaly Jun 12 '17 at 16:48
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    The Zariski topology is used in algebraic geometry and it almost always is non-Hausdorff. – Daniel Schepler Jun 12 '17 at 17:09
  • And in fact, the Zariski topology on a scheme (including "generic points") is almost always not even $T_1$. – Daniel Schepler Jun 12 '17 at 17:16
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    There is no reason to study topological groups that are not Hausdorff: their study immediately reduces to the study of their Hausdorffification and the kernel of the map to the Hausdorffification, namely the normal subgroup given by the closure of the identity. – Qiaochu Yuan Jun 12 '17 at 19:32

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A large and common class of topological groups that aren't Lie groups are profinite groups. These naturally appear in Galois theory as Galois groups of infinite extensions, in number theory in relation to $p$-adic Lie groups, and in algebraic geometry as étale fundamental groups.

Qiaochu Yuan
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