Prewellordering

In set theory, a prewellordering is a binary relation that is transitive, wellfounded, and total. In other words, if $leq$ is a prewellordering on a set $X$, and if we define $sim$ by:$xsim\; yiff\; xleq\; y\; land\; yleq\; x$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 $phi:X\; o\; Ord$ is a norm, the associated prewellordering is given by:$xleq\; yiffphi(x)leqphi(y)$Conversely, every prewellordering is induced by a unique regular norm (a norm $phi:X\; o\; Ord$ is regular if, for any $xin\; X$ and any

Prewellordering property

If 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 -prewellordering of $P$ if the relations and $leq^*$ are elements of , where for $x,yin\; X$,
#
# $xleq^*\; yiff\; xin\; Pland\; [y\; otin\; Plor\; xleq\; y]$

is said to have the prewellordering property if every set in admits a -prewellordering.

Examples

and both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every $ninomega$, and have the prewellordering property.

Consequences

Reduction

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

eparation

If is an adequate pointclass whose dual pointclass has the prewellordering property, then has the separation property: For any space $Xinmathcal\{F\}$ and any sets $A,Bsubseteq\; X$, $A$ and $B$ "disjoint" sets both in , there is a set $Csubseteq\; X$ such that both $C$ and its complement $Xsetminus\; C$ are in , with $Asubseteq\; C$ and $Bcap\; C=emptyset$.

For example, has the prewellordering property, so 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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