Timeline for Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2021 at 0:17 | vote | accept | Louis Deaett | ||
| Jun 25, 2021 at 18:01 | answer | added | Julian Rosen | timeline score: 13 | |
| Jun 25, 2021 at 13:47 | comment | added | LSpice | @YCor, since that (together with @IosifPinelis's reference) seems to be a complete answer, maybe it could be posted as such? | |
| Jun 25, 2021 at 13:23 | comment | added | Iosif Pinelis | @YCor : Thank you. I have indeed found this answer: mathoverflow.net/a/281159/36721 . | |
| Jun 25, 2021 at 13:02 | comment | added | YCor | @IosifPinelis this inverse image is dense, and $\mathrm{Homeo}(\mathbf{R}^n)$ acts transitively on dense countable subsets of $\mathbf{R}^n$ (I think the latter fact appears somewhere on MO). | |
| Jun 25, 2021 at 12:56 | comment | added | Iosif Pinelis | @YCor : How is such a self-homeomorphism constructed? | |
| Jun 25, 2021 at 12:15 | comment | added | YCor | Yes for $n=1$. No for $n\ge 2$. For instance, for $n=2$ take the complex exponential plus an irrational ($f:z\mapsto\exp(z)+z_0$), and precompose with a self-homeomorphism of $\mathbf{R}^2$ mapping $\mathbf{Q}^2$ onto $f^{-1}(\mathbf{Q}^2)$. | |
| Jun 25, 2021 at 11:11 | history | asked | Louis Deaett | CC BY-SA 4.0 |