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From the first two, we can say every topological vector space is completely regular, can't we?

Does a TVS need to be hausdorff in order to be completely regular?

Thanks!

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    In the first and second ones, *completely regular* does not include $T_0$; in the third it does (or might as well, since Hausdorff implies $T_0$).2012-02-28

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