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Let $\newcommand{\o}{\omega}\o$ be the set of non-negative integers, and for any set $X$, let $\newcommand{\oo}{[\o]^{<\o}}X^{<\o}$ denote the collection of all finite subsets of $X$.

What is an example of a function $f:\oo\to\{0,1\}$ with the following property?

Whenever $X\subseteq \o$ is infinite, then the restriction $f|_{X^{<\o}}:X^{<\o}\to \{0,1\}$ is non-constant.

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    $\begingroup$ A boring example would be the function that sends the empty set to zero and all other sets to one. $\endgroup$
    – HenrikRüping
    Commented Nov 12 at 12:50

1 Answer 1

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For $P \subset \omega$ finite, let $f(P) = \#P \bmod{2}$, i.e., $f(P) = 0$ if $P$ has even cardinal, and $1$ otherwise.

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  • $\begingroup$ Or $f(P)=g(\#P)$ where $g:\omega\to\{0,1\}$ is any nonconstant function. Or $f(P)=1$ if $\#P=\min P$ and $f(P)=0$ otherwise. $\endgroup$
    – bof
    Commented Nov 13 at 0:24
  • $\begingroup$ @bof Yes, there are tons of examples. $\endgroup$
    – Najib Idrissi
    Commented Nov 13 at 9:54

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