Editing Descriptive set theory (section)

=== 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>
}}