# 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 $$omega_1 < max { mathfrak b, mathrm {cov} ( mathcal M) }$$.