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

into a well-ordered set

the binary relation on the set

denotes the set's cardinality) is a prewellordering.

induces a wellordering on the quotient

The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

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 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

have the prewellordering property.

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

may be partitioned into sets

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

has the separation property: For any space

disjoint sets both in

has the prewellordering property, so

has the separation property.

are disjoint analytic subsets of some Polish space

Hasse diagram of the prewellordering on the non-negative integers, shown up to 29. Cycles are indicated in red and denotes the floor function .
Hasse diagram of the prewellordering on the non-negative integers, shown up to 18. The associated equivalence relation is it identifies the numbers in each light red square.