Descriptive set theory: Difference between revisions

Revision as of 09:51, 5 May 2014

Revision as of 18:45, 8 September 2014

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Descriptive set theory begins with the study of Polish spaces and their [[Borel set]]s.

A '''[[Polish space]]''' is a [[second countable]] [[topological space]] that is [[metrizable]] with a [[complete metric]]. Equivalently, it is a complete [[separable metric space]] whose metric has been "forgotten". Examples include the [[real line]] <math>\mathbb{R}</math>, the [[Baire space (set theory)|Baire space]] <math>\mathcal{N}</math>, the [[Cantor space]] <math>\mathcal{C}</math>, and the [[Hilbert cube]] <math>I^{\mathbb{N}}</math>.

=== Universality properties ===

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* 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.

== Borel sets ==