Timeline for Is there a Hausdorff space that is also a group such that the group operation is continuous but the inversion map is not continuous?
Current License: CC BY-SA 4.0
11 events
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| Apr 7, 2021 at 10:03 | comment | added | Colin Reid | If you're looking for references on this, the term for a group equipped with a topology with continuous multiplication (but not necessarily continuous inverse) is a "paratopological group". See for instance the answers to this question: mathoverflow.net/questions/130419/… | |
| Mar 15, 2021 at 11:44 | comment | added | Tyrone | @GeraldEdgar to answer your question, yes $\mathbb{Q}_\ell$ is a countable, metrisable example. My previous comment explains why there is no contradiction to the fact contained in your initial comment (I thought this is what you were asking before). | |
| Mar 12, 2021 at 21:41 | comment | added | Gerald Edgar | I meant: is the Sorgenfrey rationals an example where $+$ is continuous, but $x \mapsto -x$ is not. | |
| Mar 12, 2021 at 18:15 | comment | added | Tyrone | @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? | |
| Mar 12, 2021 at 16:28 | comment | added | Gerald Edgar | @Tyrone ... Would this work taking the rationals with the Sorgenfrey topology? | |
| Mar 12, 2021 at 16:25 | comment | added | Gerald Edgar | 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. | |
| Mar 12, 2021 at 15:41 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
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| Mar 12, 2021 at 15:18 | history | edited | YCor |
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| Mar 12, 2021 at 14:58 | comment | added | Tyrone | The Sorgenfrey line. This space is even Hausdorff and (monotonically) normal and paracompact. | |
| Mar 12, 2021 at 14:42 | review | First posts | |||
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| Mar 12, 2021 at 14:39 | history | asked | Zyis | CC BY-SA 4.0 |