Timeline for A group where the Weil topology induced by the Haar measure does not coincide with the original topology
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15 events
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| Sep 6, 2020 at 18:02 | vote | accept | Saúl Pilatowsky-Cameo | ||
| Sep 6, 2020 at 8:30 | answer | added | Robert Furber | timeline score: 6 | |
| Sep 5, 2020 at 18:36 | comment | added | Saúl Pilatowsky-Cameo | Sure, go ahead! | |
| Sep 5, 2020 at 9:28 | comment | added | Robert Furber | Do you want me to post my comment as an answer? Then you can accept it and the question will be considered "answered". Otherwise the question recirculates to the front page in search of answers for ever and ever. | |
| Sep 4, 2020 at 15:25 | comment | added | Saúl Pilatowsky-Cameo | You assumed wrong! Thank you so much, I completely missed this. | |
| Sep 4, 2020 at 8:55 | comment | added | Robert Furber | If $G$ is a locally compact group, then the topology on $G$ agrees with the Weil topology. One way follows from local compactness and continuity of $(g,h) \mapsto gh^{-1}$, and the other is Weil's extension of Steinhaus's theorem to locally compact groups (that if $E$ has strictly positive Haar measure, $EE^{-1}$ is a neighbourhood of the identity). I assumed that you knew this, and thought you were looking for a (necessarily non-locally compact) topological group that had a Haar measure such that the topology differed from the Weil topology. | |
| Sep 3, 2020 at 5:39 | comment | added | Saúl Pilatowsky-Cameo | This seems like a good path, although the distinct topology on $H$ needs to be also locally compact. Halmos gives an example of a thick subgroup in $G=\mathbb{R}^2$ in [1] p. 276 exercise (6), using some basic linear algebra techniques. But I would not know either how to give this subgroup a distinct locally compact topology so the Baire sets coincide. | |
| Sep 3, 2020 at 2:01 | comment | added | Robert Furber | It seems that what needs to be done is to take a locally compact group $G$, and find a subgroup $H$ of full outer measure, and a distinct topology on $H$ that both a group topology and has the same Baire sets as the subspace topology on $H$. Unfortunately I don't know how to do this. | |
| Sep 2, 2020 at 22:57 | history | edited | Saúl Pilatowsky-Cameo |
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| Sep 2, 2020 at 22:33 | history | edited | Saúl Pilatowsky-Cameo | CC BY-SA 4.0 |
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| Sep 2, 2020 at 20:27 | history | edited | Saúl Pilatowsky-Cameo | CC BY-SA 4.0 |
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| Sep 2, 2020 at 20:02 | history | edited | Saúl Pilatowsky-Cameo | CC BY-SA 4.0 |
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| Sep 2, 2020 at 19:50 | history | edited | Saúl Pilatowsky-Cameo | CC BY-SA 4.0 |
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| Sep 2, 2020 at 19:50 | review | First posts | |||
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| Sep 2, 2020 at 19:45 | history | asked | Saúl Pilatowsky-Cameo | CC BY-SA 4.0 |