Timeline for $T_2$-spaces where all non-empty open sets are homeomorphic
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 16, 2018 at 22:40 | comment | added | Henno Brandsma | And a zero-dimensional space $X$ is called strongly homogeneous when every non-empty clopen subset of $X$ is homeomorphic to $X$. | |
| May 16, 2018 at 22:39 | comment | added | Henno Brandsma | There are related notions of $B$-homogenous: a space is called that if there is a base $\mathcal{B}$ for the open sets such for every $B_1, B_2 \in \mathcal{B}$ there is a homeomorphism $h$ of $X$ such that $h[B_1] = B_2$. Open problem: is every topological vector space $B$-homogenous (yes for locally convex). | |
| May 16, 2018 at 22:26 | comment | added | Henno Brandsma | You might be interested in the classic Toronto space problem: is there an uncountable Hausdorff space $X$ such that any $Y \subseteq X$ with $|Y| = |X|$ is homeomorphic to $X$? It's still open, even for size $\aleph_1$, AFAIK. Note that your problem is different as it asks for open sets etc. | |
| May 15, 2018 at 17:31 | vote | accept | Dominic van der Zypen | ||
| May 15, 2018 at 13:40 | vote | accept | Dominic van der Zypen | ||
| May 15, 2018 at 14:33 | |||||
| May 15, 2018 at 12:40 | answer | added | N. de Rancourt | timeline score: 20 | |
| May 15, 2018 at 12:04 | comment | added | Joel David Hamkins | I wonder whether $2^{<\kappa}$ for uncountable ordinals $\kappa$ would be an uncountable example analogous to $\mathbb{Q}$, but I worry that cofinality differences in the order could show up in the homeomorphism types of different open sets. | |
| May 15, 2018 at 11:47 | comment | added | Dominic van der Zypen | I thought I had :-) but now it looks like $\mathbb{Q}$ is the only example I have. It would also be interesting to see another countable example, not homeomorphic to $\mathbb{Q}$. | |
| May 15, 2018 at 11:44 | comment | added | Joel David Hamkins | Do you have any uncountable examples of this phenomenon? | |
| May 15, 2018 at 11:32 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
my example of $2^\lambda$ is false
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| May 15, 2018 at 11:27 | comment | added | Dominic van der Zypen | You are right - thanks Joel! I'll add this as a note in the question | |
| May 15, 2018 at 10:45 | comment | added | Joel David Hamkins | If you remove a point from $2^\omega$, it is no longer compact, and so not homeomorphic to the whole space $2^\omega$. Doesn't this refute your claim about $\{0,1\}^\lambda$? | |
| May 15, 2018 at 8:14 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |