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Can someone give an example of a topological group $G$ that is not Hausdorff but that contains a fundamental system of neighbourhoods of $1\in G$ consisting of quasi-compact subgroups?

Thanks in advance.

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Every topological group is completely regular, so a $T_0$ topological group is Tikhonov and therefore Hausdorff. Thus, no example of what you want can be $T_0$. Any group with the indiscrete topology is a non-Hausdorff topological group in which each point has a (rather trivial!) local base of compact nbhds.

Added: For a slightly more interesting example, let $D$ be the discrete group of order $2$, and let $G$ be, say, the integers with the indiscrete topology. Then $G\times D$ has the desired properties and has $16$ open sets rather than just the $2$ of the indiscrete topology.

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  • $\begingroup$ I agree: T0 and T1 are equivalent for topological groups. Thanks for your answer! Could you also come up with a 'non-indiscrete example'? $\endgroup$ – AYK Jun 15 '15 at 18:42
  • $\begingroup$ @AYK: You’re welcome! I’ve added a slightly less trivial example: it’s not indiscrete, but it still has only finitely many open sets. $\endgroup$ – Brian M. Scott Jun 15 '15 at 18:51

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