Editing Descriptive set theory (section)

=== Universality properties ===

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''<sub>&delta;</sub> subspace]] of the [[Hilbert cube]], and every ''G''<sub>&delta;</sub> 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.

Because of these universality properties, and because the Baire space <math>\mathcal{N}</math> has the convenient property that it is [[homeomorphic]] to <math>\mathcal{N}^\omega</math>, many results in descriptive set theory are proved in the context of Baire space alone.