Timeline for Borel sets preserved under open maps?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 19 at 0:44 | comment | added | 183orbco3 | There is even a continuous $f$, using a continuous open map $h:\mathbb{R}^3\rightarrow\mathbb{R}^4$ | |
| Sep 13, 2012 at 16:11 | vote | accept | Jing Zhang | ||
| Sep 11, 2012 at 20:39 | comment | added | Clinton Conley | Now if $B \subseteq \mathbb{R}^2$ is any Borel set with non-Borel projection, then $g[B]$ is again Borel (as $g$ is injective), but $f[g[B]]$ is the projection of $B$, which was non-Borel. | |
| Sep 11, 2012 at 20:39 | comment | added | Clinton Conley | Here's a cheap trick that gives an open $f \colon \mathbb{R} \to \mathbb{R}$. Let $E$ be Vitali equivalence on $\mathbb{R}$, and let $g \colon \mathbb{R}^2 \to \mathbb{R}$ be any Borel function sending distinct points to $E$-unrelated points. Then define $f \colon \mathbb{R} \to \mathbb{R}$ by $f(x) = y$ if $\exists z\ (x \mathrel{E} g(y,z))$, and say $f(x) = 0$ if no such $(y,z)$ exists. This function is open (since the image of any open set is $\mathbb{R}$ by density of $E$-classes). [cont.] | |
| Sep 11, 2012 at 20:05 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 14 characters in body
|
| Sep 11, 2012 at 20:00 | comment | added | Joel David Hamkins | I have realized how to do it with $\mathbb{R}^n\to\mathbb{R}^n$ itself, and edited the answer. | |
| Sep 11, 2012 at 19:59 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Making domain and codomain match
|
| Sep 11, 2012 at 18:56 | comment | added | Joel David Hamkins | If one uses Baire space or Cantor space, which are homeomorphic to their squares, rather than $\mathbb{R}$, then one can easily fold in another dimension to transfer this example to have the same domain and codomain. | |
| Sep 11, 2012 at 18:38 | comment | added | Joel David Hamkins | But I admit, this doesn't quite get a counterexample with an open map $\mathbb{R}^n\to\mathbb{R}^n$ as requested... | |
| Sep 11, 2012 at 18:20 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |