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| In [[set theory]], a '''prewellordering''' is a [[binary relation]] <math>\le</math> that is [[Transitive relation|transitive]], [[total relation|total]], and [[Well-founded relation|wellfounded]] (more precisely, the relation <math>x\le y\land y\nleq x</math> is wellfounded). In other words, if <math>\leq</math> is a prewellordering on a set <math>X</math>, and if we define <math>\sim</math> by
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| :<math>x\sim y\iff x\leq y \land y\leq x</math>
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| then <math>\sim</math> is an [[equivalence relation]] on <math>X</math>, and <math>\leq</math> induces a [[wellordering]] on the [[Quotient set|quotient]] <math>X/\sim</math>. The [[order-type]] of this induced wellordering is an [[ordinal number|ordinal]], referred to as the '''length''' of the prewellordering.
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| A '''norm''' on a set <math>X</math> is a map from <math>X</math> into the ordinals. Every norm induces a prewellordering; if <math>\phi:X\to Ord</math> is a norm, the associated prewellordering is given by
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| :<math>x\leq y\iff\phi(x)\leq\phi(y)</math>
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| Conversely, every prewellordering is induced by a unique '''regular norm''' (a norm <math>\phi:X\to Ord</math> is regular if, for any <math>x\in X</math> and any <math>\alpha<\phi(x)</math>, there is <math>y\in X</math> such that <math>\phi(y)=\alpha</math>).
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| == Prewellordering property ==
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| If <math>\boldsymbol{\Gamma}</math> is a [[pointclass]] of subsets of some collection <math>\mathcal{F}</math> of [[Polish space]]s, <math>\mathcal{F}</math> closed under [[Cartesian product]], and if <math>\leq</math> is a prewellordering of some subset <math>P</math> of some element <math>X</math> of <math>\mathcal{F}</math>, then <math>\leq</math> is said to be a <math>\boldsymbol{\Gamma}</math>-'''prewellordering''' of <math>P</math> if the relations <math><^*\,</math> and <math>\leq^*</math> are elements of <math>\boldsymbol{\Gamma}</math>, where for <math>x,y\in X</math>,
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| # <math>x<^*y\iff x\in P\land[y\notin P\lor\{x\leq y\land y\not\leq x\}]</math>
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| # <math>x\leq^* y\iff x\in P\land[y\notin P\lor x\leq y]</math>
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| <math>\boldsymbol{\Gamma}</math> is said to have the '''prewellordering property''' if every set in <math>\boldsymbol{\Gamma}</math> admits a <math>\boldsymbol{\Gamma}</math>-prewellordering.
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| 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.
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| ===Examples===
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| <math>\boldsymbol{\Pi}^1_1\,</math> and <math>\boldsymbol{\Sigma}^1_2</math> both have the prewellordering property; this is provable in [[Zermelo-Fraenkel set theory|ZFC]] alone. Assuming sufficient [[large cardinal]]s, for every <math>n\in\omega</math>, <math>\boldsymbol{\Pi}^1_{2n+1}</math> and <math>\boldsymbol{\Sigma}^1_{2n+2}</math>
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| have the prewellordering property.
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| ===Consequences===
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| ====Reduction====
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| If <math>\boldsymbol{\Gamma}</math> is an [[adequate pointclass]] with the prewellordering property, then it also has the '''reduction property''': For any space <math>X\in\mathcal{F}</math> and any sets <math>A,B\subseteq X</math>, <math>A</math> and <math>B</math> both in <math>\boldsymbol{\Gamma}</math>, the union <math>A\cup B</math> may be partitioned into sets <math>A^*,B^*\,</math>, both in <math>\boldsymbol{\Gamma}</math>, such that <math>A^*\subseteq A</math> and <math>B^*\subseteq B</math>.
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| ====Separation====
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| If <math>\boldsymbol{\Gamma}</math> is an [[adequate pointclass]] whose [[dual pointclass]] has the prewellordering property, then <math>\boldsymbol{\Gamma}</math> has the '''separation property''': For any space <math>X\in\mathcal{F}</math> and any sets <math>A,B\subseteq X</math>, <math>A</math> and <math>B</math> ''disjoint'' sets both in <math>\boldsymbol{\Gamma}</math>, there is a set <math>C\subseteq X</math> such that both <math>C</math> and its [[Complement (set theory)|complement]] <math>X\setminus C</math> are in <math>\boldsymbol{\Gamma}</math>, with <math>A\subseteq C</math> and <math>B\cap C=\emptyset</math>.
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| For example, <math>\boldsymbol{\Pi}^1_1</math> has the prewellordering property, so <math>\boldsymbol{\Sigma}^1_1</math> has the separation property. This means that if <math>A</math> and <math>B</math> are disjoint [[analytic set|analytic]] subsets of some Polish space <math>X</math>, then there is a [[Borel set|Borel]] subset <math>C</math> of <math>X</math> such that <math>C</math> includes <math>A</math> and is disjoint from <math>B</math>.
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| == See also ==
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| *[[Descriptive set theory]]
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| *[[Scale property]]
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| *[[Graded poset]] – a graded poset is analogous to a prewellordering with a norm, replacing a map to the ordinals with a map to the integers
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| == References ==
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| * {{cite book | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}
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| [[Category:Mathematical relations]]
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| [[Category:Descriptive set theory]]
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| [[Category:Wellfoundedness]]
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| [[Category:Order theory]]
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