A note on a theorem of Hausdorff

Abstract

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 \subset \ldots \subset G_{\xi } \subset \ldots , ξ > ω 1 {\xi } > {\omega }_1 , G ξ ≠ X G_{\xi } \not = X , of zero-dimensional G δ G_{\delta } -sets in X X such that for any finite Borel measure μ \mu on X X , μ ( X ∖ G ξ ) = 0 \mu (X \setminus G_{\xi }) = 0 , for some ξ {\xi } . Our approach yields also an analogous result in a more general setting of ccc σ \sigma -ideals.

Proofs of the Hausdorff theorem in the literature are based on subtle combinatorial reasonings.