Prewellordering

In set theory, a prewellordering is a binary relation ${\displaystyle \leq }$ that is transitive, connex, and wellfounded (more precisely, the relation ${\displaystyle x\leq y\land y\nleq x}$ is wellfounded). In other words, if ${\displaystyle \leq }$ is a prewellordering on a set ${\displaystyle X}$, and if we define ${\displaystyle \sim }$ by

${\displaystyle x\sim y\iff x\leq y\land y\leq x}$

then ${\displaystyle \sim }$ is an equivalence relation on ${\displaystyle X}$, and ${\displaystyle \leq }$ induces a wellordering on the quotient ${\displaystyle X/\sim }$. The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set ${\displaystyle X}$ is a map from ${\displaystyle X}$ into the ordinals. Every norm induces a prewellordering; if ${\displaystyle \phi :X\to Ord}$ is a norm, the associated prewellordering is given by

${\displaystyle x\leq y\iff \phi (x)\leq \phi (y)}$

Conversely, every prewellordering is induced by a unique regular norm (a norm ${\displaystyle \phi :X\to Ord}$ is regular if, for any ${\displaystyle x\in X}$ and any ${\displaystyle \alpha <\phi (x)}$, there is ${\displaystyle y\in X}$ such that ${\displaystyle \phi (y)=\alpha }$).

Prewellordering property

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

1. ${\displaystyle x<^{*}y\iff x\in P\land [y\notin P\lor \{x\leq y\land y\not \leq x\}]}$
2. ${\displaystyle x\leq ^{*}y\iff x\in P\land [y\notin P\lor x\leq y]}$

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

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

Reduction

If ${\displaystyle {\boldsymbol {\Gamma }}}$ 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 ${\displaystyle {\boldsymbol {\Gamma }}}$, the union ${\displaystyle A\cup B}$ may be partitioned into sets ${\displaystyle A^{*},B^{*}}$, both in ${\displaystyle {\boldsymbol {\Gamma }}}$, such that ${\displaystyle A^{*}\subseteq A}$ and ${\displaystyle B^{*}\subseteq B}$.

Separation

If ${\displaystyle {\boldsymbol {\Gamma }}}$ is an adequate pointclass whose dual pointclass has the prewellordering property, then ${\displaystyle {\boldsymbol {\Gamma }}}$ 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 ${\displaystyle {\boldsymbol {\Gamma }}}$, there is a set ${\displaystyle C\subseteq X}$ such that both ${\displaystyle C}$ and its complement ${\displaystyle X\setminus C}$ are in ${\displaystyle {\boldsymbol {\Gamma }}}$, with ${\displaystyle A\subseteq C}$ and ${\displaystyle B\cap C=\emptyset }$.

For example, ${\displaystyle {\boldsymbol {\Pi }}_{1}^{1}}$ has the prewellordering property, so ${\displaystyle {\boldsymbol {\Sigma }}_{1}^{1}}$ 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 ${\displaystyle B}$.

See also

• Descriptive set theory
• Scale property
• Graded poset – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the integers

References

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.