  W. Hurewicz proved that analytic Menger sets of reals are $\sigma$-compact
and that co-analytic completely Baire sets of reals are completely metrizable.
It is natural to try to generalize these theorems to projective sets. This has
previously been accomplished by $V = L$ for projective counterexamples, and the
Axiom of Projective Determinacy for positive results. For the first problem,
the first author, S. Todorcevic, and S. Tokg\"oz have produced a finer analysis
with much weaker axioms. We produce a similar analysis for the second problem,
showing the two problems are essentially equivalent. We also construct in ZFC a
separable metrizable space with $\omega$-th power completely Baire, yet lacking
a dense completely metrizable subspace. This answers a question of Eagle and
Tall in Abstract Model Theory.
