Prewellordering

In set theory, a prewellordering is a binary relation that is transitive, wellfounded, and connected. In other words, if $\leq$ is a prewellordering on a set $X$, and if we define $\sim$ by

$x\sim y\iff x\leq y\land y\leq x$

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

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

$x\leq y\iff \phi (x)\leq \phi (y)$

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

Prewellordering property

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

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

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

Examples

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

Consequences

Reduction

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

Separation

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

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

References

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