Descriptive set theory: Difference between revisions

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The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.

* Every Polish space is [[homeomorphic]] to a [[Gδ space|''G''_δ [[subspace topology|subspace]] of the [[Hilbert cube]], and every ''G''_δ subspace of the Hilbert cube is Polish.

* Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.