A note on a theorem of Hausdorff

We give a topological proof (using directly some known results about infinite-dimensional spaces) of the following strengthening of a remarkable Hausdorff theorem: each separable metrizable space admitting a continuous map onto [ 0 , 1 ] [0,1] is the union of a sequence G 1 ⊂ … ⊂ G ξ ⊂ … G_1 ⊂ … ⊂ Gξ ⊂ … , ξ > ω 1 {ξ } > {ω }_1 , G ξ ≠ X Gξ = X , of zero-dimensional G δ Gδ -sets in X X such that for any finite Borel measure μ μ on X X , μ ( X ∖ G ξ ) = 0 μ (X Gξ) = 0 , for some ξ {ξ } . Our approach yields also an analogous result in a more general setting of ccc σ σ -ideals. Proofs of the Hausdorff theorem in the literature are based on subtle combinatorial reasonings.