Timeline for topological properties of $G_{\delta}$ sets in a compact Hausdorff space
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2019 at 17:04 | comment | added | user350168 | Thanks Ramiro! Yes, that reference was very useful! | |
| Aug 30, 2019 at 13:41 | comment | added | Ramiro de la Vega | If it helps at all (e.g. to find literature), $G_\delta$ subsets of compact Hausdorff spaces are sometimes called Čech-complete spaces. Also, these spaces are characterized by having a complete sequence of open covers (i.e a sequence $\{\mathcal{U}_n\}_{n \in \omega}$ of open covers of the space such that any filter on the space that intersects them all has an accumulation point). | |
| Aug 26, 2019 at 15:08 | comment | added | Emil Jeřábek | Maybe I should also state the obvious: all subspaces of $S(A)$ are Hausdorff and zero-dimensional (hence $T_{3\frac12}$). | |
| Aug 26, 2019 at 9:26 | comment | added | Emil Jeřábek | Also, the description in the comment doesn’t work out. Assuming you mean countable intersections and unions (otherwise it doesn’t make sense at all), if the complement of $\mathcal F$ is a countable union of countable intersections of open sets, that only makes $\mathcal F$ an $F_{\sigma\delta}$ set (i.e, $\Pi_3$ in the Borel hierarchy), not $G_\delta$. | |
| Aug 26, 2019 at 8:40 | comment | added | Emil Jeřábek | I don’t know what topological properties you are interested in, but as a trivial observation, since $S(A)$ is compact Hausdorff, its $G_\delta$ subspaces are Baire spaces. If moreover $S(A)$ is second-countable (i.e., when $A$ and the underlying first-order language are countable), then its $G_\delta$ subspaces are completely metrizable (i.e., Polish spaces). | |
| Aug 26, 2019 at 2:13 | comment | added | user350168 | Ok, my apologies for the ambiguity. I meant first order types, i.e. complete consistent sets of formulas over some set of parameters $A$ in some model $M$ of a first order theory $T$. | |
| Aug 25, 2019 at 21:18 | comment | added | Andrej Bauer | Thanks, @EmilJeřábek. I retracted my close vote. | |
| Aug 25, 2019 at 21:17 | history | edited | Andrej Bauer | CC BY-SA 4.0 |
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| Aug 25, 2019 at 11:17 | comment | added | Emil Jeřábek | @AndrejBauer en.wikipedia.org/wiki/Type_(model_theory) | |
| Aug 25, 2019 at 9:50 | comment | added | Andrej Bauer | "Types" as in type theory, or "types" as in homotopy theory, or "types" as in a programming language? The word "type" has many meanings, I think you should explain your terminology or provide a reference. What is a "type over $A$"? | |
| Aug 25, 2019 at 4:00 | comment | added | user350168 | The problem is that I don't know precisely what I am looking for. More precisely, I know that $S^{*}(A) \backslash \mathcal{F}$ can be expressed as a union of sets $C_{\phi}$ where $C_{\phi}$ is the intersection of open sets. I am trying to characterize the set of types $\mathcal{F}$ in a different way, so it will be useful to know if there is some level of weak compactness that one could use. Or which topological properties does the $G_{\delta}$ set ($\mathcal{F}$) preserves from the old space. | |
| Aug 25, 2019 at 3:07 | comment | added | Nik Weaver | Good, thank you. It could also help in parts 1 and 3 if you gave some idea of what you're looking for, or an example of the kind of answer you would want ... otherwise these questions are too open-ended. | |
| Aug 25, 2019 at 2:21 | comment | added | user350168 | thanks, I just edited the question | |
| Aug 25, 2019 at 2:20 | history | edited | user350168 | CC BY-SA 4.0 |
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| Aug 24, 2019 at 22:46 | comment | added | Nik Weaver | The notation $S(A)$ could mean anything. If you don't explain your notation no one will be able to answer. | |
| Aug 24, 2019 at 20:30 | review | Close votes | |||
| Aug 25, 2019 at 0:12 | |||||
| Aug 24, 2019 at 19:13 | history | asked | user350168 | CC BY-SA 4.0 |