Timeline for Characterizing "bounded" distributivity in terms of dense open sets
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 10, 2019 at 4:26 | comment | added | Joseph Van Name | Now people are continuing to upvote incorrect comments after I have refuted them. | |
| Apr 10, 2019 at 0:05 | comment | added | Joseph Van Name | This duality extends to a duality between $P_{<\kappa}$-proximity spaces and $<\kappa$-algebras of sets for all uncountable regular $\kappa$. | |
| Apr 10, 2019 at 0:05 | comment | added | Joseph Van Name | If anyone wants a correspondence between $\sigma$-algebras and more topological objects, one should look at proximity spaces since I have generalized the notion of a $P$-space to proximity spaces, and I have proven that the category of all $\sigma$-algebras is isomorphic to the category of all $P$-proximity spaces in the only paper on $P$-proximity spaces (the paper is a very easy read). In that case, the sets in the $\sigma$-algebra are precisely the sets $R$ with $R\overline{\delta}R^{c}$. | |
| Apr 10, 2019 at 0:04 | comment | added | Joseph Van Name | Therefore, if $(X,\mathcal{M})$ is the Borel $\sigma$-algebra on a Polish space, then since $\mathcal{M}$ is atomic, every $\sigma$-complete ultrafilter on $\mathcal{M}$ is principal, but $\mathcal{M}$ is not complete, we conclude that $\mathcal{M}$ cannot be isomorphic to the algebra of clopen sets of any $P$-space. | |
| Apr 10, 2019 at 0:04 | comment | added | Joseph Van Name | You can also observe that for completely regular spaces, the points on the Hewitt realcompactification $\upsilon X$ of $X$ is in a one-to-one correspondence with the $\sigma$-complete ultrafilters on the Baire $\sigma$-algebra (by the Baire $\sigma$-algebra we mean the $\sigma$-algebra generated by the clopen sets) where the principal $\sigma$-complete ultrafilters correspond to $X$. In particular, $\upsilon X=X$ precisely when $X$ is realcompact. Since every Polish space is realcompact, we can conclude that $\sigma$-complete ultrafilter on $(X,\mathcal{M})$ is principal. | |
| Apr 10, 2019 at 0:03 | comment | added | Joseph Van Name | Let $(X,\mathcal{M})$ be the Borel $\sigma$-algebra on a Polish space $X$. Then one can show directly that all the $\sigma$-complete ultrafilters on $\mathcal{M}$ are the principal ultrafilters. | |
| Apr 9, 2019 at 23:59 | comment | added | Joseph Van Name | If $B$ is any $<\kappa$-complete Boolean algebra which is not both complete and atomic and where every $<\kappa$-complete ultrafilter on $B$ is principal, then $B$ is not isomorphic to the algebra of all clopen sets of any $P_{<\kappa}$-space. If $X$ is a $P_{<\kappa}$-space and $M$ is the $<\kappa$-complete algebra of clopen sets and every $<\kappa$-complete ultrafilter on $M$ is principal, then for each $x\in X$, the set $\{R\in M|x\in R\}$ is a $<\kappa$-complete ultrafilter on $M$ and hence principal. Therefore, $X$ is the discrete space, and hence $B\simeq M=P(X)$. | |
| Apr 9, 2019 at 23:59 | comment | added | Joseph Van Name | @NotMike. You are incorrect about your claim about representability. So $<\kappa$-representability is typically defined as being isomorphic as a Boolean algebra to a $<\kappa$-field of sets. This is not equivalent to being $\mathrm{CO}(X)$ for a $P_{<\kappa}$-space (here I am assuming that $\mathrm{CO}(X)$ denotes the algebra of clopen sets). | |
| Apr 9, 2019 at 14:14 | comment | added | Joseph Van Name | @NotMike. If you have a simplification of these results, then please post another answer. | |
| Apr 9, 2019 at 14:09 | comment | added | Not Mike | (See Proposition 14.4, p. 214; The Handbook of Boolean Algebras, Vol. 1. ) | |
| Apr 9, 2019 at 13:55 | comment | added | Not Mike | I downvoted this answer; it's a confused and overly complicated exposition of the elementary result that $\kappa$-representable $\kappa$-complete Boolean Algebras are $(\mu, \lambda)$-distributive for every $\mu, \lambda < \kappa$ (a Boolean Algebra is $\kappa$-representable, iff, it can be realized as, $CO(X)$ for some regular $P_{<\kappa}$-space $X$, iff, it can be realized as a $\kappa$-complete field of sets.) | |
| Apr 6, 2019 at 18:35 | history | answered | Joseph Van Name | CC BY-SA 4.0 |