# A Borel perfectly everywhere dominating family of functions

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.

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

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 $$\omega^\omega$$.