Prewellordering

Prewellordering property

If ${\displaystyle \mathbf{\Gamma }}$ is a pointclass of subsets of some collection ${\displaystyle \mathcal{F}}$ of Polish spaces, ${\displaystyle \mathcal{F}}$ closed under Cartesian product, and if ${\displaystyle \le }$ is a prewellordering of some subset ${\displaystyle P}$ of some element ${\displaystyle X}$ of ${\displaystyle \mathcal{F},}$ then ${\displaystyle \le }$ is said to be a ${\displaystyle \mathbf{\Gamma }}$-prewellordering of ${\displaystyle P}$ if the relations ${\displaystyle {<}^{\ast }}$ and ${\displaystyle {\le }^{\ast }}$ are elements of ${\displaystyle \mathbf{\Gamma },}$ where for ${\displaystyle x,y\in X,}$

1. ${\displaystyle x{<}^{\ast }y\text{ if and only if }x\in P\wedge (y\notin P\vee (x\le y\wedge y\nleqq x))}$
2. ${\displaystyle x{\le }^{\ast }y\text{ if and only if }x\in P\wedge (y\notin P\vee x\le y)}$

${\displaystyle \mathbf{\Gamma }}$ is said to have the prewellordering property if every set in ${\displaystyle \mathbf{\Gamma }}$ admits a ${\displaystyle \mathbf{\Gamma }}$-prewellordering.

The prewellordering property is related to the stronger scale property; in practice, many pointclasses having the prewellordering property also have the scale property, which allows drawing stronger conclusions.

Examples

${\displaystyle {\mathbf{\Pi }}_1^1}$ and ${\displaystyle {\mathbf{\Sigma }}_2^1}$ both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every ${\displaystyle n\in \omega ,}$ ${\displaystyle {\mathbf{\Pi }}_{2n+1}^1}$ and ${\displaystyle {\mathbf{\Sigma }}_{2n+2}^1}$ have the prewellordering property.

Consequences

Reduction

If ${\displaystyle \mathbf{\Gamma }}$ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space ${\displaystyle X\in \mathcal{F}}$ and any sets ${\displaystyle A,B\subseteq X,}$ ${\displaystyle A}$ and ${\displaystyle B}$ both in ${\displaystyle \mathbf{\Gamma },}$ the union ${\displaystyle A\cup B}$ may be partitioned into sets ${\displaystyle A^{\ast },B^{\ast },}$ both in ${\displaystyle \mathbf{\Gamma },}$ such that ${\displaystyle A^{\ast }\subseteq A}$ and ${\displaystyle B^{\ast }\subseteq B.}$

Separation

If ${\displaystyle \mathbf{\Gamma }}$ is an adequate pointclass whose dual pointclass has the prewellordering property, then ${\displaystyle \mathbf{\Gamma }}$ has the separation property: For any space ${\displaystyle X\in \mathcal{F}}$ and any sets ${\displaystyle A,B\subseteq X,}$ ${\displaystyle A}$ and ${\displaystyle B}$ disjoint sets both in ${\displaystyle \mathbf{\Gamma },}$ there is a set ${\displaystyle C\subseteq X}$ such that both ${\displaystyle C}$ and its complement ${\displaystyle X\setminus C}$ are in ${\displaystyle \mathbf{\Gamma },}$ with ${\displaystyle A\subseteq C}$ and ${\displaystyle B\cap C=\varnothing .}$

For example, ${\displaystyle {\mathbf{\Pi }}_1^1}$ has the prewellordering property, so ${\displaystyle {\mathbf{\Sigma }}_1^1}$ has the separation property. This means that if ${\displaystyle A}$ and ${\displaystyle B}$ are disjoint analytic subsets of some Polish space ${\displaystyle X,}$ then there is a Borel subset ${\displaystyle C}$ of ${\displaystyle X}$ such that ${\displaystyle C}$ includes ${\displaystyle A}$ and is disjoint from ${\displaystyle B.}$