Editing Descriptive set theory (section)

== Borel sets ==

The class of '''[[Borel set]]s''' of a topological space ''X'' consists of all sets in the smallest [[sigma-algebra|&sigma;-algebra]] containing the open sets of ''X''. This means that the Borel sets of ''X'' are the smallest collection of sets such that:
* Every open subset of ''X'' is a Borel set.
* If ''A'' is a Borel set, so is <math>X \setminus A</math>. That is, the class of Borel sets are closed under complementation. 
* If ''A''<sub>''n''</sub> is a Borel set for each natural number ''n'', then the union <math>\bigcup A_n</math> is a Borel set. That is, the Borel sets are closed under countable unions.

A fundamental result shows that any two uncountable Polish spaces ''X'' and ''Y'' are [[Borel isomorphism|Borel isomorphic]]: there is a bijection from ''X'' to ''Y'' such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.

=== Borel hierarchy ===

Each Borel set of a Polish space is classified in the '''[[Borel hierarchy]]''' based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of [[countable set|countable]] [[ordinal number]]s. For each nonzero countable ordinal ''&alpha;'' there are classes <math>\mathbf{\Sigma}^0_\alpha</math>, <math>\mathbf{\Pi}^0_\alpha</math>, and <math>\mathbf{\Delta}^0_\alpha</math>. 
* Every open set is declared to be <math>\mathbf{\Sigma}^0_1</math>.
* A set is declared to be <math>\mathbf{\Pi}^0_\alpha</math> if and only if its complement is <math>\mathbf{\Sigma}^0_\alpha</math>.
* A set ''A'' is declared to be <math>\mathbf{\Sigma}^0_\delta</math>, ''&delta;'' > 1, if there is a sequence &lang; ''A''<sub>''i''</sub> &rang; of sets, each of which is <math>\mathbf{\Pi}^0_{\lambda(i)}</math> for some ''&lambda;''(''i'') < ''&delta;'', such that <math>A = \bigcup A_i</math>.
* A set is <math>\mathbf{\Delta}^0_\alpha</math> if and only if it is both <math>\mathbf{\Sigma}^0_\alpha</math> and <math>\mathbf{\Pi}^0_\alpha</math>.

A theorem shows that any set that is <math>\mathbf{\Sigma}^0_\alpha</math> or <math>\mathbf{\Pi}^0_\alpha</math> is <math>\mathbf{\Delta}^0_{\alpha + 1}</math>, and any <math>\mathbf{\Delta}^0_\beta</math> set is both <math>\mathbf{\Sigma}^0_\alpha</math> and <math>\mathbf{\Pi}^0_\alpha</math> for all ''&alpha;'' > ''&beta;''.  Thus the hierarchy has the following structure, where arrows indicate inclusion.

{{center|
<!-- replace this with a diagram -->
<math>
\begin{matrix}
& & \mathbf{\Sigma}^0_1 & & & & \mathbf{\Sigma}^0_2 & & \cdots \\
& \nearrow & & \searrow & & \nearrow \\
\mathbf{\Delta}^0_1 & & & & \mathbf{\Delta}^0_2 & & & & \cdots \\
& \searrow & & \nearrow & & \searrow \\
& & \mathbf{\Pi}^0_1 & & & & \mathbf{\Pi}^0_2 & & \cdots 
\end{matrix}\begin{matrix}
& &  \mathbf{\Sigma}^0_\alpha & & & \cdots \\
& \nearrow & & \searrow \\
\quad \mathbf{\Delta}^0_\alpha &  & & & \mathbf{\Delta}^0_{\alpha + 1} & \cdots \\
& \searrow & & \nearrow \\
& & \mathbf{\Pi}^0_\alpha & & & \cdots 
\end{matrix}
</math>
}}

=== Regularity properties of Borel sets ===

Classical descriptive set theory includes the study of regularity properties of Borel sets. For example, all Borel sets of a Polish space have the [[property of Baire]] and the [[perfect set property]].  Modern descriptive set theory includes the study of the ways in which these results generalize, or fail to generalize, to other classes of subsets of Polish spaces.