Timeline for When $C (X) $ is zero dimensional
Current License: CC BY-SA 3.0
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| Apr 10, 2017 at 16:11 | history | edited | Laurent Moret-Bailly | CC BY-SA 3.0 |
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| Apr 10, 2017 at 14:42 | comment | added | Gro-Tsen | @NikWeaver Ah, thanks, I had misread the statement "if $X$ is a P-space then $\beta X$ is an F-space" in Gilman & Jerison's book (I took the "F" for a "P"). In fact, $\beta X$ is never a P-space if $X$ is infinite because compact P-spaces are indeed discrete. | |
| Apr 10, 2017 at 14:40 | comment | added | Gro-Tsen | @LaurentMoret-Bailly A P-space is precisely what the original question is asking about, i.e., one in which every zero-set is open. My claim that $\beta\mathbb{N}$ is a nondiscrete P-space was wrong, but such spaces do exist: Gillman & Jerison (exercise 4N) give the example of the union of an uncountable discrete set and a point $s$ whose neighborhoods are the co-countable sets containing $s$. | |
| Apr 10, 2017 at 14:11 | comment | added | Laurent Moret-Bailly | @Anonymous: what is a P-space? | |
| Apr 10, 2017 at 13:54 | comment | added | Nik Weaver | @Gro-Tsen: define $f: \mathbb{N} \to [0,1]$ by $f(n) = 1/n$ and extend continuously to $\beta\mathbb{N}$. This function is not locally constant, as it is constantly zero on $\beta\mathbb{N}\setminus\mathbb{N}$, which is not open. | |
| Apr 10, 2017 at 13:16 | comment | added | Anonymous | The condition that every continuous real-valued function on a space X is locally constant does not imply that the space is discrete, even for Tychonoff spaces. The condition on X that is equivalent to C(X) having Krull dimension 0 is that X be a P-space, and there are certainly non-discrete P-spaces. | |
| Apr 10, 2017 at 13:00 | history | answered | Laurent Moret-Bailly | CC BY-SA 3.0 |