It is shown to be consistent that every function of first Baire class can be decomposed into continuous functions yet the least cardinal of a dominating family in is . The model used in the one obtained by adding Miller reals to a model of the Continuum Hypothesis.
We consider a triple ⟨E₀,E₁,E₂⟩ of equivalence relations on ℝ² and investigate the possibility of decomposing the plane into three sets ℝ² = S₀ ∪ S₁ ∪ S₂ in such a way that each intersects each -class in finitely many points. Many results in the literature, starting with a famous theorem of Sierpiński, show that for certain triples the existence of such a decomposition is equivalent to the continuum hypothesis. We give a characterization in ZFC of the triples for which the decomposition exists....