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.