Projective hierarchy

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"Projective set" redirects here. For the card game, see Projective Set (game).

In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol{\Sigma}^1_n for some positive integer n. Here A is

  • \boldsymbol{\Sigma}^1_1 if A is analytic
  • \boldsymbol{\Pi}^1_n if the complement of A, X\setminus A, is \boldsymbol{\Sigma}^1_n
  • \boldsymbol{\Sigma}^1_{n+1} if there is a Polish space Y and a \boldsymbol{\Pi}^1_n subset C\subseteq X\times Y such that A is the projection of C; that is, A=\{x\in X \mid \exists y\in Y (x,y)\in C\}

The choice of the Polish space Y in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.

Relationship to the analytical hierarchy[edit]

There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters \Sigma and \Pi) and the projective hierarchy on subsets of Baire space (denoted by boldface letters \boldsymbol{\Sigma} and \boldsymbol{\Pi}). Not every \boldsymbol{\Sigma}^1_n subset of Baire space is \Sigma^1_n. It is true, however, that if a subset X of Baire space is \boldsymbol{\Sigma}^1_n then there is a set of natural numbers A such that X is \Sigma^{1,A}_n. A similar statement holds for \boldsymbol{\Pi}^1_n sets. Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. This relationship is important in effective descriptive set theory.

A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets of Cantor space and, more generally, subsets of any effective Polish space.