Prewellordering

All definitions tacitly require the homogeneous relation

A term's definition may require additional properties that are not listed in this table.

(a transitive and reflexive relation on

) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation

is a homogeneous binary relation

that satisfies the following conditions:[1] A homogeneous binary relation

is a prewellordering if and only if there exists a surjection

the binary relation on the set

of all finite subsets of

denotes the set's cardinality) is a prewellordering.

is an equivalence relation on

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

Every norm induces a prewellordering; if

Conversely, every prewellordering is induced by a unique regular norm (a norm

α < ϕ ( x ) ,

ϕ ( y ) = α

is a pointclass of subsets of some collection

closed under Cartesian product, and if

is said to have the prewellordering property if every set in

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.

both have the prewellordering property; this is provable in ZFC alone.

Assuming sufficient large cardinals, for every

is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space

is an adequate pointclass whose dual pointclass has the prewellordering property, then

has the separation property: For any space

has the separation property.

are disjoint analytic subsets of some Polish space

then there is a Borel subset