Analytic and Coanalytic Sets

Analytic sets ( $\boldsymbol{\Sigma}^1_1$ ) are the first level beyond the Borel hierarchy, arising as projections of Borel sets. Their theory is remarkably rich and mostly settable in ZFC.

Analytic Sets

Definition9.3Analytic set

A subset $A$ of a Polish space $X$ is analytic (or $\boldsymbol{\Sigma}^1_1$ ) if it is the continuous image of a Borel subset of a Polish space. Equivalently:

$A$ is the projection of a closed subset of $X \times \omega^\omega$ .
$A = \{x \in X : \exists y \in \omega^\omega,\, (x,y) \in F\}$ for some closed (or Borel) $F \subseteq X \times \omega^\omega$ .
$A$ is the image of $\omega^\omega$ under a continuous function (if $A \neq \emptyset$ ).

ExampleExamples of analytic sets

Every Borel set is analytic.
The set of real numbers that are limits of convergent subsequences of a given sequence is analytic.
In $C([0,1])$ : the set of functions that attain their maximum at a unique point is coanalytic ( $\boldsymbol{\Pi}^1_1$ ).
The set of well-founded trees on $\omega$ is coanalytic but not analytic (a key example).

The Suslin Theorem

Theorem9.2Suslin's theorem

A set is Borel if and only if it is both analytic ( $\boldsymbol{\Sigma}^1_1$ ) and coanalytic ( $\boldsymbol{\Pi}^1_1$ ):

$\mathbf{Borel} = \boldsymbol{\Sigma}^1_1 \cap \boldsymbol{\Pi}^1_1.$

RemarkBeyond Borel: proper analytic sets

Not every analytic set is Borel. The universal analytic set $U \subseteq 2^\omega \times 2^\omega$ (defined by $U = \{(x, y) : y \in A_x\}$ where $A_x$ ranges over all analytic sets parametrized by $x$ ) is analytic but not Borel (by a diagonalization argument).

The set $\mathrm{WO} = \{x \in 2^{\omega \times \omega} : x \text{ codes a well-ordering of } \omega\}$ is a canonical example of a coanalytic set that is not Borel.

Regularity Properties

Theorem9.3Regularity of analytic sets

Every analytic set is:

Lebesgue measurable.
Has the Baire property.
Has the perfect set property: every uncountable analytic set contains a perfect subset.

These properties are provable in ZFC. For coanalytic sets, (1) and (2) hold in ZFC, but (3) (the perfect set property for $\boldsymbol{\Pi}^1_1$ ) is independent of ZFC.

RemarkThe perfect set property and large cardinals

The statement "every $\boldsymbol{\Pi}^1_1$ set has the perfect set property" is equivalent to $\omega_1^L < \omega_1$ (the first uncountable ordinal of $L$ is countable in $V$ ), which follows from the existence of $0^\sharp$ . This is the first connection between regularity properties and large cardinals.