Timeline for ultrafilters' succession

4 events

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Jan 7, 2013 at 14:22	comment	added	Andreas Blass		@Joseph Van Name: I think your proof is essentially the same as mine. The existence of a countable dense subset in $\mathbb N^I$ is the same fact as the existence of an $I$-indexed family of independent functions $\mathbb N\to\mathbb N$.
Jan 7, 2013 at 2:56	comment	added	Joseph Van Name		If $I$ is a set of cardinality continuum, then I was thinking simply to take a countable dense subset $A\subseteq\mathbb{N}^{I}\subseteq(\beta\mathbb{N})^{I}$. Then the inclusion map $A\hookrightarrow(\beta\mathbb{N})^{I}$ extends to a countinuous surjection $\iota:\beta A\rightarrow(\beta\mathbb{N})^{I}$. In particular, if $x_{i}\in\beta\mathbb{N}$ for $i\in I$, then there is some $x\in\beta A$ with $\iota(x)=(x_{i})_{i\in I}$. Therefore $\pi_{i}\iota(x)=x_{i}$ for $i\in I$ where each $\pi_{i}$ is the projection, so $x_{i}\leq_{RK}x$ for $i\in I$.
Dec 23, 2011 at 22:17	comment	added	Andreas Blass		My (ancient) Ph.D. thesis, a scanned pdf of which is available on my web site, contains the stronger result that, in the Rudin-Keisler ordering of ultrafilters on $\omega$, any continuum many (or fewer) ultrafilters have a (strict) upper bound. The idea is similar to the answer above but using, instead of the functions $h_n$, a family of continuum many independent functions $\omega\to\omega$.
Dec 23, 2011 at 22:14	history	answered	Andreas Blass	CC BY-SA 3.0