Timeline for Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2022 at 13:54 | history | edited | Joseph Van Name | CC BY-SA 4.0 |
added 1155 characters in body
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| Feb 22, 2022 at 10:03 | vote | accept | Overflowian | ||
| Feb 22, 2022 at 10:02 | comment | added | Overflowian | Thank you for your clear answer. Three comments: 1) the existence of the function $h$ is crucial and is due to the fact that $X$ is normal (so we can apply Tietze) 2) the set O is open and also non-empty for the same reason. 3) in the last paragraph there's a typo: $x\in \partial K$ where you probably wanted to write $k\in \partial K$. | |
| Feb 22, 2022 at 8:18 | comment | added | Pierre PC | Please, this is your answer! The value of me adding my own would be very slim, I tend to write things akin to “by a standard partition of unity argument, $\mathcal C^\infty(M,\mathcal R)$ is dense in $\mathcal C^0(M,\mathcal R)$ with respect to the strong Whitney topology” which might be a bit terse. | |
| Feb 22, 2022 at 2:31 | comment | added | Joseph Van Name | @PierrePC. That is a good observation. Did you want to make a new answer explaining the partition of unity argument or should I just edit my answer and explain the partition of unity argument myself? | |
| Feb 22, 2022 at 0:35 | comment | added | Pierre PC | Nice answer. I don't think we use the full generality of the approximation theorem here since $N=\mathbb R$, it is really just a simple partition of unity argument. | |
| Feb 22, 2022 at 0:29 | history | answered | Joseph Van Name | CC BY-SA 4.0 |