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 property

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 ,

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.

Examples

and both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every , 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 and any sets , and both in , the union may be partitioned into sets , both in , such that and .

Separation

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 also

References

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