Timeline for When can we divide continuous functions?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2022 at 8:53 | history | became hot network question | |||
| Aug 1, 2022 at 7:07 | vote | accept | erz | ||
| Aug 1, 2022 at 4:50 | answer | added | Joseph Van Name | timeline score: 16 | |
| Jul 31, 2022 at 23:53 | comment | added | David Handelman | I think the condition on $X $ is that all countable suprema of projections exist in $C(X)$; this is the same as the closure of a countable union of clopen sets is clopen. Monotone $\sigma-$complete, I think is the term. | |
| Jul 31, 2022 at 23:40 | answer | added | Nik Weaver | timeline score: 9 | |
| Jul 31, 2022 at 23:38 | comment | added | LSpice | @erz, as long as your space is infinite, it has an injective sequence, which has a limit point by compactness. Maybe it's not a limit point of a subsequence, but doesn't @NikWeaver's argument still apply? (I'm not sure where connectedness, which @NikWeaver doesn't seem to use, comes into it.) | |
| Jul 31, 2022 at 23:33 | comment | added | erz | @NikWeaver thank you for your comment, but the whole point is that spaces like this are very non-sequential and also very disconnected. | |
| Jul 31, 2022 at 23:32 | comment | added | Nik Weaver | @LSpice yeah, good point. | |
| Jul 31, 2022 at 22:01 | comment | added | LSpice | @NikWeaver, that seems like an answer; but, if you post it as such, then you might consider avoiding $h$, which is used with another meaning in the body of the question. | |
| Jul 31, 2022 at 20:50 | comment | added | Nik Weaver | If $x_n \to x$ (all distinct), define $g(x_n) = 1/n$, $g(x) =0$ and set $h(x_n) = 1/n$ or $0$ depending on whether $n$ is even or odd, and $h(x) =0$. Extend $g$ and $h$ to positive continuous functions on $X$ by Tietze, then set $f = g \wedge h$. | |
| Jul 31, 2022 at 19:27 | comment | added | Nik Weaver | Seems to me that if $X$ contains a nonconstant convergent sequence it can't have this property. | |
| Jul 31, 2022 at 19:03 | history | asked | erz | CC BY-SA 4.0 |