Timeline for Spaces without maximal homogeneous subspaces
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2018 at 6:50 | history | edited | Pietro Majer | CC BY-SA 4.0 |
corrected
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| Jun 14, 2018 at 20:47 | comment | added | Pietro Majer | Thank you Taras! yes, now I see it -- for a moment I thought it could be made even simpler, but it can't | |
| Jun 14, 2018 at 18:27 | comment | added | Taras Banakh | In 1) "regular" can be removed, the whole condition 3) can be removed; instead of 4) one should require that each subset of $X_{n+1}$, which is homeomorphic to a subset of $X_n$ is not dense in $X_{n+1}$. Then the same argument works (under the condition of regularity of $X$). | |
| Jun 14, 2018 at 17:44 | comment | added | Taras Banakh | I am afraid that this is an oversimplification. It is not clear to me why the topological sum of $\mathbb R^n$ should not contain maximal homogeneous subspaces. For example, something dense and zero-dimensional? The trick with the sum of Hilbert space of the incresing density is that each homogeneous subspace is nowhere dense in large piece. So there is a place to enlarge this homogeneous space by another its piece. But in separable space there can happen that no place for such enlargement is available. | |
| Jun 14, 2018 at 15:35 | comment | added | Pietro Majer | (So, if I didn't oversimplified Taras' construction, $X_k:=\mathbb{R}^k$ works as well). | |
| Jun 14, 2018 at 15:31 | history | answered | Pietro Majer | CC BY-SA 4.0 |