Transitive binary relations  

 
indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation be transitive: for all if and then 
In set theory, a prewellordering on a set is a preorder on (a transitive and reflexive relation on ) that is strongly connected (meaning that any two points are comparable) and wellfounded in the sense that the induced relation defined by is a wellfounded relation.
A prewellordering on a set is a homogeneous binary relation on that satisfies the following conditions:^{[1]}
A homogeneous binary relation on is a prewellordering if and only if there exists a surjection into a wellordered set such that for all if and only if ^{[1]}
Given a set the binary relation on the set of all finite subsets of defined by if and only if (where denotes the set's cardinality) is a prewellordering.^{[1]}
If is a prewellordering on then the relation defined by is an equivalence relation on and induces a wellordering on the quotient The ordertype 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 ).
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.
and both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every and have the prewellordering property.
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
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