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