Timeline for The space $\psi$
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2013 at 9:38 | vote | accept | Vahideh Bagheri | ||
| Sep 4, 2013 at 8:59 | answer | added | jonathanverner | timeline score: 4 | |
| Sep 4, 2013 at 5:48 | comment | added | Vahideh Bagheri | [quote]I can see a counterexample constructed from a Luzin gap; I don't immediately see whether there can be an F-Z $\psi$-space, however I would look at special AD families (i.e. AD families which are nowhere inseparable). – jonathanverner 19 hours ago[quote] can you please clarify? | |
| Sep 3, 2013 at 15:07 | comment | added | Joseph Van Name | For F-Z-spaces which are not normal, one should look at $P$-spaces. The $P$-spaces are precisely the completely regular spaces where every cozero-set is clopen, so clearly every $P$-space is an F-Z-space, but there are many $P$-spaces which are not normal. For instance, one can generalize the Tychonoff Plank construction to obtain a $P$-space which is not normal. | |
| Sep 3, 2013 at 14:53 | comment | added | Joseph Van Name | Another way to construct examples of normal spaces which are even compact and connected but are not FZ-spaces is to look at the order topology. For instance, if one takes the long line and adds a copy of the unit interval $[0,1]$ at the end of the long line, one obtains a ordered space $([0,1)\times\omega_{1})\cup[0,1]$ which is compact and connected since it is a complete lattice. The interval $(0,1]$ on the very right is a cozero set, but the closure $[0,1]=\overline{(0,1]}$ is not a zero set in $([0,1)\times\omega_{1})\cup[0,1]$. | |
| Sep 3, 2013 at 13:33 | history | reopened |
Joseph Van Name Yemon Choi Andrey Rekalo Daniel Moskovich Benjamin Steinberg |
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| Sep 3, 2013 at 10:15 | comment | added | jonathanverner | For a normal but not FZ-space, start with a Tychonoff plank and add a convergent sequence $\langle s_m^n:n<\omega\rangle$ to each point $[\omega_1, m]$ on the right-most vertical column. Then the set $S=\{s_m^n:n,m<\omega\}$ is a co-zero set whose closure is not $G_\delta$ hence not zero. | |
| Sep 3, 2013 at 9:40 | comment | added | jonathanverner | I can see a counterexample constructed from a Luzin gap; I don't immediately see whether there can be an F-Z $\psi$-space, however I would look at special AD families (i.e. AD families which are nowhere inseparable). | |
| Sep 3, 2013 at 8:54 | comment | added | Vahideh Bagheri | thanks dear Van Name and others. I am searching a space which is normal but not FZ-space, or is FZ-space but not normal. The space $\psi$ is non normal space. So I want to know if it is FZ-space? I dont know the form of zero sets in this space. only we fined for every $\omega_E \in Z(f)$ we have one and only one of the followings :a) $\omega_E\in int(Z(f))$ b) $Z(f)\cap E = \omega_E$ | |
| Sep 3, 2013 at 3:06 | comment | added | Joseph Van Name | I checked problem 5I and I found nothing about F-Z-spaces in that problem. Also, I edited this question to include the definition of $\psi$ and information about this space to clear any confusion. I am voting to reopen this question. | |
| Sep 3, 2013 at 3:01 | history | edited | Joseph Van Name | CC BY-SA 3.0 |
added 959 characters in body
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| Sep 3, 2013 at 2:42 | review | Reopen votes | |||
| Sep 3, 2013 at 13:35 | |||||
| Sep 3, 2013 at 1:30 | history | closed |
Ryan Budney David White Karl Schwede Steven Landsburg Daniel Moskovich |
Needs details or clarity | |
| Sep 2, 2013 at 19:02 | comment | added | Yemon Choi | If you are trying to do Problem 5I, please give some details of where you are stuck | |
| Sep 2, 2013 at 19:00 | review | Close votes | |||
| Sep 3, 2013 at 1:30 | |||||
| Sep 2, 2013 at 16:29 | history | edited | Ricardo Andrade |
removed deprecated tag 'topology'
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| Sep 2, 2013 at 16:28 | comment | added | Jon Bannon | Can you please include the definitions of each thing that appears in this question? At least, you should take the time to define \psi | |
| Sep 2, 2013 at 15:44 | history | asked | Vahideh Bagheri | CC BY-SA 3.0 |