Timeline for Uncountable Pre-Image
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2013 at 3:20 | comment | added | Nik Weaver | Oh, one more comment. I wish I'd thought of saying this earlier. The proof that the complement of any countable subset of ${\bf R}^2$ is connected goes back to Cantor. I remember reading it in Dauben's biography. | |
| Apr 23, 2013 at 5:00 | comment | added | Nik Weaver | @John: The complement of $f^{-1}(x)$ is disconnected because it is the disjoint union of two nonempty open sets, $U$ and $V$. | |
| Apr 23, 2013 at 3:18 | comment | added | John Bluto | Many thanks! One follow up question: How does your definition of U and V imply that the complement of $f^{-1}(x)$ is disconnected? | |
| Apr 23, 2013 at 1:04 | vote | accept | John Bluto | ||
| Apr 23, 2013 at 1:01 | comment | added | Nik Weaver | Also, my argument actually shows that $f^{-1}(x)$ has cardinality $2^{\aleph_0}$, for those who care about such niceties. | |
| Apr 23, 2013 at 1:00 | comment | added | Nik Weaver | Actually, I only used the fact that the range of $f$ is open, not that $f$ is an open map. | |
| Apr 23, 2013 at 0:53 | history | answered | Nik Weaver | CC BY-SA 3.0 |