set theory – On projectively countable sets in Hilbert's cube

A subset $ A $ of a topological space $ X $ is called projectively countable if for any card continues $ f: X to mathbb R $ l & # 39; Image $ f (A) $ is countable.
It is easy to see that each countable set by projection in a cube of finite dimension $[0,1]^ n $ is countable.

Problem. Is each set countable in the Hilbert cube? $[0,1]$ omega countable?

Note. The answer is affirmative under the hypothesis of the theory of sets $ omega_1 < max { mathfrak b, mathrm {cov} ( mathcal M) } $.