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The question is from the definition of to topological group. I can find an example such that the inversion map is continuous but the group operation is not continuous, but I cannot find an example such that the group operation is continuous but the inversion map is not continuous.

I guess that such a space (if exists) is not T3, but I just know some easy examples of Hausdorff but not T3 spaces.

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    $\begingroup$ The Sorgenfrey line. This space is even Hausdorff and (monotonically) normal and paracompact. $\endgroup$
    – Tyrone
    Commented Mar 12, 2021 at 14:58
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    $\begingroup$ If the space is completely metrizable, then inversion is automatically continuous. In fact, you can assume less: multiplication is separately continuous. The proof uses Baire category theorem, but the nonseparable case is tricky. $\endgroup$
    – Gerald Edgar
    Commented Mar 12, 2021 at 16:25
  • $\begingroup$ @Tyrone ... Would this work taking the rationals with the Sorgenfrey topology? $\endgroup$
    – Gerald Edgar
    Commented Mar 12, 2021 at 16:28
  • $\begingroup$ @GeraldEdgar I believe it suffices to assume that the underlying space is Cech complete CRH. $\mathbb{Q}_\ell$ isn't Cech complete. It's second-countable and regular so metrisable, but doesn't have any isolated points so isn't complete. In particular it can't even be Baire. Or maybe I misunderstood your question? $\endgroup$
    – Tyrone
    Commented Mar 12, 2021 at 18:15
  • $\begingroup$ I meant: is the Sorgenfrey rationals an example where $+$ is continuous, but $x \mapsto -x$ is not. $\endgroup$
    – Gerald Edgar
    Commented Mar 12, 2021 at 21:41

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