- Prewellordering
In
set theory , a prewellordering is abinary relation that is transitive, wellfounded, and total. In other words, if leq is a prewellordering on a set X, and if we define sim by:xsim yiff xleq y land yleq xthen sim is anequivalence relation on X, and leq induces awellordering on the quotient X/sim. The order-type of this induced wellordering is anordinal , referred to as the length of the prewellordering.A norm on a set X is a map from X into the
ordinal s. Every norm induces a prewellordering; if phi:X o Ord is a norm, the associated prewellordering is given by:xleq yiffphi(x)leqphi(y)Conversely, every prewellordering is induced by a unique regular norm (a norm phi:X o Ord is regular if, for any xin X and any alpha, there is yin X such that phi(y)=alpha). Prewellordering property
If oldsymbol{Gamma} is a
pointclass of subsets of some collection mathcal{F} ofPolish space s, mathcal{F} closed underCartesian 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 oldsymbol{Gamma}-prewellordering of P if the relations and leq^* are elements of oldsymbol{Gamma}, where for x,yin X,
# x<^*yiff xin Pland [y otin Plor{xleq yland y otleq x}]
# xleq^* yiff xin Pland [y otin Plor xleq y]oldsymbol{Gamma} is said to have the prewellordering property if every set in oldsymbol{Gamma} admits a oldsymbol{Gamma}-prewellordering.
Examples
oldsymbol{Pi}^1_1, and oldsymbol{Sigma}^1_2 both have the prewellordering property; this is provable in ZFC alone. Assuming sufficient
large cardinal s, for every ninomega, oldsymbol{Pi}^1_{2n+1} and oldsymbol{Sigma}^1_{2n+2}have the prewellordering property.Consequences
Reduction
If oldsymbol{Gamma} is an
adequate pointclass with the prewellordering property, then it also has the reduction property: For any space Xinmathcal{F} and any sets A,Bsubseteq X, A and B both in oldsymbol{Gamma}, the union Acup B may be partitioned into sets A^*,B^*,, both in oldsymbol{Gamma}, such that A^*subseteq A and B^*subseteq B.eparation
If oldsymbol{Gamma} is an
adequate pointclass whosedual pointclass has the prewellordering property, then oldsymbol{Gamma} has the separation property: For any space Xinmathcal{F} and any sets A,Bsubseteq X, A and B "disjoint" sets both in oldsymbol{Gamma}, there is a set Csubseteq X such that both C and its complement Xsetminus C are in oldsymbol{Gamma}, with Asubseteq C and Bcap C=emptyset.For example, oldsymbol{Pi}^1_1 has the prewellordering property, so oldsymbol{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.
References
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