Prewellordering

In set theory, a prewellordering is a binary relation that is transitive, connex, and wellfounded (more precisely, the relation is wellfounded). In other words, if is a prewellordering on a set , and if we define by

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

A norm on a set is a map from into the ordinals. Every norm induces a prewellordering; if is a norm, the associated prewellordering is given by

Conversely, every prewellordering is induced by a unique regular norm (a norm is regular if, for any and any , there is such that ).

Prewellordering propertyEdit

If   is a pointclass of subsets of some collection   of Polish spaces,   closed under Cartesian product, and if   is a prewellordering of some subset   of some element   of  , then   is said to be a  -prewellordering of   if the relations   and   are elements of  , where for  ,

  1.  
  2.  

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

ExamplesEdit

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

ConsequencesEdit

ReductionEdit

If   is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space   and any sets  ,   and   both in  , the union   may be partitioned into sets  , both in  , such that   and  .

SeparationEdit

If   is an adequate pointclass whose dual pointclass has the prewellordering property, then   has the separation property: For any space   and any sets  ,   and   disjoint sets both in  , there is a set   such that both   and its complement   are in  , with   and  .

For example,   has the prewellordering property, so   has the separation property. This means that if   and   are disjoint analytic subsets of some Polish space  , then there is a Borel subset   of   such that   includes   and is disjoint from  .

See alsoEdit

ReferencesEdit

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