Timeline for Separable sigma-algebra: equivalence of two definitions
Current License: CC BY-SA 2.5
14 events
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| Aug 29, 2014 at 1:29 | comment | added | Fred Dashiell | The $\sigma$-algebra $S$ above is the collection of sets $A$ such that either $A$ or its complement contains a club. This is precisely the Borel $\sigma$-algebra of $\omega_1$. See a review of these matters in "A classification of ordinals up to Borel isomorphism", Gao-Jackson-Kieftenbeld, Fund. Math. 198 (2008), 61--76. | |
| Aug 19, 2014 at 9:11 | review | Suggested edits | |||
| Aug 19, 2014 at 9:15 | |||||
| Jun 9, 2010 at 8:40 | vote | accept | Kestutis Cesnavicius | ||
| Jun 9, 2010 at 2:32 | comment | added | Joel David Hamkins | I have now added a proof of the forward implication. | |
| Jun 9, 2010 at 2:29 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jun 9, 2010 at 1:31 | comment | added | Joel David Hamkins | I updated my argument. The example shows that from separability in the semi-metric space, one cannot necessarily find the $\sigma$-algebra in the completion of a countably generated $\sigma$-algebra. | |
| Jun 9, 2010 at 1:01 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jun 9, 2010 at 0:55 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jun 8, 2010 at 23:36 | comment | added | Joel David Hamkins | Nate, of course you are right, since the closure process generally proceeds in an $\omega_1$ hierarchy, so this part of my argument is wrong. But I've realized that the equivalence is not correct even when one considers the completion, and I'll update my answer shortly to explain. | |
| Jun 8, 2010 at 21:41 | comment | added | Nate Eldredge | Indeed, the Borel $\sigma$-field on $\mathbb{R}$ is generated by the countable family of intervals with rational endpoints, but certainly not every Borel set is a countable union of such sets and their complements! (Any closed bounded set is already a counterexample.) | |
| Jun 8, 2010 at 21:01 | comment | added | Kestutis Cesnavicius | Why is every $A\in S$ obtained by a union of the form you're claiming? | |
| Jun 8, 2010 at 19:42 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jun 8, 2010 at 19:25 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
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| Jun 8, 2010 at 19:19 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |