Is there an order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$? Or from ${\cal P}(\omega)/(fin)$ onto $[0,1]\cap \mathbb{Q}$?
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asked Nov 3, 2017 at 13:41
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4$\begingroup$ Upper density? en.wikipedia.org/wiki/Natural_density $\endgroup$– Andreas ThomCommented Nov 3, 2017 at 14:01
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2$\begingroup$ Do you mean $\leq$-preserving or $<$-preserving? $\endgroup$– Andreas ThomCommented Nov 3, 2017 at 14:02
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$\begingroup$ The title and the body ask two completely different questions. $\endgroup$– Asaf Karagila ♦Commented Nov 3, 2017 at 15:01
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1$\begingroup$ @AsafKaragila both the title and the body are about "order-preserving surjective map $f: {\cal P}(\omega)/(fin) \to [0,1]$" (and the body asks one more similar question) - or did I miss something? Or I didn't get that yours was a humourous remark? $\endgroup$– Dominic van der ZypenCommented Nov 3, 2017 at 15:09
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1$\begingroup$ @AsafKaragila That's right, but I ask about $[0,1]$ in both body and title. So the body is an extension of the title (which is true for many MO questions). But yeah, maybe it is wrong to ask more than $1$ question in a MO question, but that would be something for meta.mathoverflow.net $\endgroup$– Dominic van der ZypenCommented Nov 3, 2017 at 15:16
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1 Answer
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Replace $\omega$ with $X:=\mathbb Q\cap [0,1]$, and map each $A\subseteq X$ to $\limsup A\in [0,1]$. $\limsup$ is weakly increasing, and invariant under finite changes. This map is onto $[0,1]$.
If you want to map onto the dyadic rationals, partition $\omega$ into countably many infinite sets $\omega=A_1\cup A_2\cup A_3\cup\cdots$, and map each set $B\subseteq \omega$ to
- $\sum_{n\in I_B} 2^{-n}$, if $I_B:=\{n: B \cap A_n \mbox{ infinite}\}$ is finite
- $1$, if $I_B$ is infinite.
If you want a map onto $\mathbb Q\cap [0,1]$, compose with some isomorphism.
