Polish space

A Polish space is a topological space that is separable and completely metrizable. It is the natural habitat of descriptive set theory: every uncountable Polish space contains a homeomorphic copy of the Cantor set, and the Borel hierarchy, analytic sets, and projective sets are all defined relative to Polish spaces. The real numbers, the Baire space ω^ω, the Cantor space 2^ω, and every separable Banach space are Polish spaces.

The significance of Polish spaces in mathematics is that they are rich enough to support all of classical analysis and topology, yet tame enough that definable subsets behave well under determinacy assumptions. The Axiom of Determinacy implies that every subset of a Polish space is Lebesgue measurable and has the Baire property — regularity properties that fail in ZFC for arbitrarily constructed sets.