Timeline for Topological spaces in which countable intersections of dense open sets have dense interior
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 6 at 21:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 6 at 18:51 | comment | added | Gro-Tsen | @NotMike Is the precise statement you are referring to proposition 5.5.2 of chapter 17 in the Handbook of Boolean Algebras (screenshot here)? I ask because this one has “open dense” in the conclusion, which is different from “dense interior” in general (but maybe it's equivalent in the context of Boolean algebras / Stone spaces, I'm not too familiar with it). If you have the time, I would be grateful if you could write an answer with whatever equivalents of “super Baire” you can provide for Stone spaces. | |
| Aug 6 at 0:38 | comment | added | Not Mike | @Gro-Tsen Yes. They are dual notions. | |
| Aug 5 at 16:00 | comment | added | Gro-Tsen | @NotMike Does this mean that the Stone space of an $\omega_1$-distributive Boolean algebra will be “super Baire” in the sense defined in the present question? | |
| Aug 5 at 15:25 | comment | added | Gro-Tsen | @YCor The “property P” you mention appears to be known as being an “almost P-space” by some people. I added an update to the question summarizing what I can say about some of the properties mentioned so far. | |
| Aug 5 at 15:24 | history | edited | Gro-Tsen | CC BY-SA 4.0 |
long update summarizing the (known to me) implications between some similar topological properties
|
| Aug 4 at 22:10 | comment | added | Ali Taghavi | Me too I do not know how or why it disappeared | |
| Aug 4 at 21:53 | comment | added | Gro-Tsen | @AliTaghavi I do remember seeing a comment of yours regarding a translation in terms of essential ideals (and I read it). I don't know how or why it disappeared. | |
| Aug 4 at 20:49 | comment | added | Ali Taghavi | Yesrerday immediate after the first version of your question I commented on your question but unfortunately I can not find my comments. any way my comment was identical to my existing answer | |
| Aug 4 at 20:46 | answer | added | Ali Taghavi | timeline score: 0 | |
| Aug 4 at 15:08 | comment | added | David Gao | Another example would be that any hyperstonean space $X$ has this “super Baire” property, as nowhere dense sets in $X$ are exactly those sets which are null w.r.t. any normal measure on $X$. | |
| Aug 4 at 12:00 | comment | added | Will Brian | @NotMike: Good observation. The property of $\omega^*$ mentioned in YCor's comment is, in general, equivalent to the algebra of regular open sets being countably closed, which is a little stronger than being $\omega_1$-distributive. | |
| Aug 4 at 11:37 | comment | added | Not Mike | In the context of forcing or general properties of Boolean algebras, the exact phrasing you have is equivalent to being $\omega_1$-distributive. Perhaps something along the lines of $\omega_1$-Baire would be a good choice. | |
| Aug 4 at 10:54 | history | edited | LSpice | CC BY-SA 4.0 |
Link to comment; more information about answer
|
| Aug 4 at 10:18 | comment | added | YCor | Note: there is a (closely?) related notion, say "Property P": every nonempty $G_\delta$ subset has nonempty interior. It occurs in the following theorem of ZFC+CH due to Parovicenko: if $X$ is a Stone space with Property P, no isolated point, and with continuum many clopen subsets, then it is homeomorphic to $\beta\omega-\omega$ (the Stone-Cech remainder of a countable discrete set). | |
| Aug 4 at 9:34 | history | edited | Gro-Tsen | CC BY-SA 4.0 |
add another remark with an example given to my by Noé de Rancourt on Twitter
|
| Aug 4 at 9:18 | comment | added | Gro-Tsen | @NotMike Indeed, I should have mentioned the P-space property. I added a remark to the question on this subject. | |
| Aug 4 at 9:17 | history | edited | Gro-Tsen | CC BY-SA 4.0 |
add some remarks, mentioning the P-space property
|
| Aug 4 at 7:01 | comment | added | Not Mike | It's been awhile, but I believe what you have described is adjacent to (or possibly equivalent to) the notion of P-space. | |
| Aug 3 at 21:07 | history | asked | Gro-Tsen | CC BY-SA 4.0 |
