Timeline for Can the real line be embedded in a space $X$ such that all the nonempty open subsets of $X$ are homeomorphic?
Current License: CC BY-SA 4.0
11 events
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| Aug 3, 2018 at 10:18 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Aug 3, 2018 at 10:18 | vote | accept | Will Brian | ||
| Aug 2, 2018 at 4:49 | answer | added | Taras Banakh | timeline score: 8 | |
| Aug 2, 2018 at 4:40 | answer | added | Taras Banakh | timeline score: 4 | |
| Jul 31, 2018 at 17:54 | comment | added | R. van Dobben de Bruyn | Could there be a subspace of the hyperreals that contains $\mathbb R$ and has similar properties to $\mathbb R \setminus \mathbb Q$? | |
| Jul 31, 2018 at 10:02 | comment | added | Will Brian | @bof: I don't think it's a dumb question, but no, $\mathbb R \times \mathbb Q$ is not a hoss because it is not homeomorphic to $\mathbb R \times \mathbb Q$ minus a point. The following (admittedly contrived) property distinguishes the two spaces (true in the first, false in the second): Suppose $\langle x_n \rangle_n$ is a sequence converging to $x$ and $\langle y_n \rangle_n$ is a sequence converging to $y$; if $x_n$ and $y_n$ are connected by an arc for every $n$, then so are $x$ and $y$. | |
| Jul 30, 2018 at 15:23 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Jul 30, 2018 at 12:40 | answer | added | bof | timeline score: 10 | |
| Jul 30, 2018 at 10:44 | comment | added | Will Brian | It isn't clear to me how to get any space satisfying the stated conditions, so I did not specify any further conditions, and would be interested to see a non-Hausdorff example. That being said, I would be even more interested to see a Hausdorff example. | |
| Jul 30, 2018 at 9:47 | comment | added | bof | The other question you referred to asked about Hausdorff spaces, but you just wrote "topological space". Do you want a Hausdorff space, or will you accept a $\text T_1$-space satisfying your stated conditions? | |
| Jul 30, 2018 at 7:59 | history | asked | Will Brian | CC BY-SA 4.0 |