Prewellordering

In set theory, a prewellordering on a set $$X$$ is a preorder $$\leq$$ on $$X$$ (a transitive and reflexive relation on $$X$$) that is strongly connected (meaning that any two points are comparable) and well-founded in the sense that the induced relation $$x < y$$ defined by $$x \leq y \text{ and } y \nleq x$$ is a well-founded relation.

Prewellordering on a set
A prewellordering on a set $$X$$ is a homogeneous binary relation $$\,\leq\,$$ on $$X$$ that satisfies the following conditions:  Reflexivity: $$x \leq x$$ for all $$x \in X.$$  Transitivity: if $$x < y$$ and $$y < z$$ then $$x < z$$ for all $$x, y, z \in X.$$ Total/Strongly connected: $$x \leq y$$ or $$y \leq x$$ for all $$x, y \in X.$$ for every non-empty subset $$S \subseteq X,$$ there exists some $$m \in S$$ such that $$m \leq s$$ for all $$s \in S.$$ 
 * This condition is equivalent to the induced strict preorder $$x < y$$ defined by $$x \leq y$$ and $$y \nleq x$$ being a well-founded relation.

A homogeneous binary relation $$\,\leq\,$$ on $$X$$ is a prewellordering if and only if there exists a surjection $$\pi : X \to Y$$ into a well-ordered set $$(Y, \lesssim)$$ such that for all $$x, y \in X,$$ $x \leq y$ if and only if $$\pi(x) \lesssim \pi(y).$$

Examples


Given a set $$A,$$ the binary relation on the set $$X := \operatorname{Finite}(A)$$ of all finite subsets of $$A$$ defined by $$S \leq T$$ if and only if $$|S| \leq |T|$$ (where $$|\cdot|$$ denotes the set's cardinality) is a prewellordering.

Properties
If $$\leq$$ is a prewellordering on $$X,$$ then the relation $$\sim$$ defined by $$x \sim y \text{ if and only if } x \leq y \land y \leq x$$ is an equivalence relation on $$X,$$ and $$\leq$$ induces a wellordering on the quotient $$X / {\sim}.$$ The order-type of this induced wellordering is an ordinal, referred to as the length of the prewellordering.

A norm on a set $$X$$ is a map from $$X$$ into the ordinals. Every norm induces a prewellordering; if $$\phi : X \to Ord$$ is a norm, the associated prewellordering is given by $$x \leq y \text{ if and only if } \phi(x) \leq \phi(y)$$ Conversely, every prewellordering is induced by a unique regular norm (a norm $$\phi : X \to Ord$$ is regular if, for any $$x \in X$$ and any $$\alpha < \phi(x),$$ there is $$y \in X$$ such that $$\phi(y) = \alpha$$).

Prewellordering property
If $$\boldsymbol{\Gamma}$$ is a pointclass of subsets of some collection $$\mathcal{F}$$ of Polish spaces, $$\mathcal{F}$$ closed under Cartesian product, and if $$\leq$$ is a prewellordering of some subset $$P$$ of some element $$X$$ of $$\mathcal{F},$$ then $$\leq$$ is said to be a $$\boldsymbol{\Gamma}$$-prewellordering of $$P$$ if the relations $$<^*$$ and $$\leq^*$$ are elements of $$\boldsymbol{\Gamma},$$ where for $$x, y \in X,$$
 * 1) $$x <^* y \text{ if and only if } x \in P \land (y \notin P \lor (x \leq y \land y \not\leq x))$$
 * 2) $$x \leq^* y \text{ if and only if } x \in P \land (y \notin P \lor x \leq y)$$

$$\boldsymbol{\Gamma}$$ is said to have the prewellordering property if every set in $$\boldsymbol{\Gamma}$$ admits a $$\boldsymbol{\Gamma}$$-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
$$\boldsymbol{\Pi}^1_1$$ and $$\boldsymbol{\Sigma}^1_2$$ both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient large cardinals, for every $$n \in \omega,$$ $$\boldsymbol{\Pi}^1_{2n+1}$$ and $$\boldsymbol{\Sigma}^1_{2n+2}$$ have the prewellordering property.

Reduction
If $$\boldsymbol{\Gamma}$$ is an adequate pointclass with the prewellordering property, then it also has the reduction property: For any space $$X \in \mathcal{F}$$ and any sets $$A, B \subseteq X,$$ $$A$$ and $$B$$ both in $$\boldsymbol{\Gamma},$$ the union $$A \cup B$$ may be partitioned into sets $$A^*, B^*,$$ both in $$\boldsymbol{\Gamma},$$ such that $$A^* \subseteq A$$ and $$B^* \subseteq B.$$

Separation
If $$\boldsymbol{\Gamma}$$ is an adequate pointclass whose dual pointclass has the prewellordering property, then $$\boldsymbol{\Gamma}$$ has the separation property: For any space $$X \in \mathcal{F}$$ and any sets $$A, B \subseteq X,$$ $$A$$ and $$B$$ disjoint sets both in $$\boldsymbol{\Gamma},$$ there is a set $$C \subseteq X$$ such that both $$C$$ and its complement $$X \setminus C$$ are in $$\boldsymbol{\Gamma},$$ with $$A \subseteq C$$ and $$B \cap C = \varnothing.$$

For example, $$\boldsymbol{\Pi}^1_1$$ has the prewellordering property, so $$\boldsymbol{\Sigma}^1_1$$ has the separation property. This means that if $$A$$ and $$B$$ are disjoint analytic subsets of some Polish space $$X,$$ then there is a Borel subset $$C$$ of $$X$$ such that $$C$$ includes $$A$$ and is disjoint from $$B.$$