Timeline for A rare property of Hausdorff spaces
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 9, 2016 at 21:14 | comment | added | Forever Mozart | not so rare, huh? | |
| Aug 9, 2016 at 17:38 | comment | added | Gerald Edgar | Locally constant in $\{0,1\}^A$? No. Think of the projection onto $\{0,1\}^B$, where $B \subset A$ is countable. (In fact, the general continuous real-valued function factors through such a projection.) But we do have each nonempty level set $f^{-1}(x)$ has the same power as the whole space. | |
| Aug 9, 2016 at 17:18 | comment | added | Joel David Hamkins | Nice example, Gerald. It is similar to my example, but do you get the locally-constant property? It seems not, since the projection functions to the unit interval are continuous, but not locally constant. But if you used $2^A$ instead of $[0,1]^A$, then at least the projection functions would be locally constant. Is every continuous function locally constant in this case? | |
| Aug 9, 2016 at 13:42 | history | answered | Gerald Edgar | CC BY-SA 3.0 |