Timeline for Is the "hereditarily indecomposable" property separably determined?
Current License: CC BY-SA 4.0
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| Nov 27, 2021 at 16:44 | history | edited | Onur Oktay |
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| Nov 22, 2021 at 16:23 | vote | accept | Onur Oktay | ||
| Nov 22, 2021 at 16:23 | |||||
| Nov 21, 2021 at 18:06 | comment | added | Onur Oktay | @KevinBeanland Thanks for your comment. For the record, Milman's characterization is given in Proposition 1.1, and the embedding is proved in Proposition 1.3 in the Memoirs of Argyros-Tolias [dx.doi.org/10.1090/memo/0806] I can see at this moment that these beautiful results would be overkill to answer a perhaps simple question. | |
| Nov 21, 2021 at 18:00 | answer | added | Onur Oktay | timeline score: 4 | |
| Nov 21, 2021 at 16:05 | comment | added | Kevin Beanland | I don't know, but I think the answer should be yes. All HI spaces embed into $\ell_\infty$ and so they can't be too large. The proof of this fact is short and uses a characterization of HI spaces due to Milman. It is the Memoirs of Argyros-Tolias that I don't have with me. Maybe one can use that characterization to prove that the HI property is separable determined? In the other direction, there is a space of Argyros, Todocevic and Lopez-Adad $X_{\omega_1}$ that is reflexive non-separable and contains no UBS so it is HI saturated. That may be a candidate for a counterexample. | |
| Nov 20, 2021 at 6:36 | history | asked | Onur Oktay | CC BY-SA 4.0 |