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Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?

This is a refinement of the question I asked in: A Borel perfectly everywhere surjective function on the Cantor set. I think the argument which yields a "no" answer there would also give that there is no function $2^\omega\to\omega^\omega$ with comeager, or possibly even nonmeager, image on every perfect set. But while every nonmeager family of functions in $\omega^\omega$ is unbounded, the converse (even for dominating families) is not true.

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  • $\begingroup$ Why do you say "closed perfect set"? What is your definition of perfect set that does not include being closed? $\endgroup$
    – bof
    Commented Jun 30, 2021 at 7:20
  • $\begingroup$ @bof Just there for emphasis, I suppose, don't want it read as only "has no isolated points". $\endgroup$
    – Iian Smythe
    Commented Jun 30, 2021 at 13:17

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The answer is no. By Lusin's Theorem there exists $A \subseteq 2^\omega$ closed and of positive measure such that $f \restriction A$ is continuous. Since $A$ cannot be countable it must contain a perfect set $P$. Now $f[P] \subseteq \omega^\omega$ is compact, hence bounded in $\omega^\omega$.

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