if there is a Polish space and a subset such that is the projection of onto ; that is,
The choice of the Polish space 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 hierarchyEdit
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters and ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters and ). Not every subset of Baire space is . It is true, however, that if a subset X of Baire space is then there is a set of natural numbersA such that X is . A similar statement holds for 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.
Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN978-0-262-68052-3
October 21, 2023
projective, hierarchy, projective, redirects, here, card, game, projective, game, mathematical, field, descriptive, theory, subset, displaystyle, polish, space, displaystyle, projective, displaystyle, boldsymbol, sigma, some, positive, integer, displaystyle, h. Projective set redirects here For the card game see Projective Set game In the mathematical field of descriptive set theory a subset A displaystyle A of a Polish space X displaystyle X is projective if it is S n 1 displaystyle boldsymbol Sigma n 1 for some positive integer n displaystyle n Here A displaystyle A is S 1 1 displaystyle boldsymbol Sigma 1 1 if A displaystyle A is analytic P n 1 displaystyle boldsymbol Pi n 1 if the complement of A displaystyle A X A displaystyle X setminus A is S n 1 displaystyle boldsymbol Sigma n 1 S n 1 1 displaystyle boldsymbol Sigma n 1 1 if there is a Polish space Y displaystyle Y and a P n 1 displaystyle boldsymbol Pi n 1 subset C X Y displaystyle C subseteq X times Y such that A displaystyle A is the projection of C displaystyle C onto X displaystyle X that is A x X y Y x y C displaystyle A x in X mid exists y in Y x y in C The choice of the Polish space Y displaystyle 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 EditThere is a close relationship between the relativized analytical hierarchy on subsets of Baire space denoted by lightface letters S displaystyle Sigma nbsp and P displaystyle Pi nbsp and the projective hierarchy on subsets of Baire space denoted by boldface letters S displaystyle boldsymbol Sigma nbsp and P displaystyle boldsymbol Pi nbsp Not every S n 1 displaystyle boldsymbol Sigma n 1 nbsp subset of Baire space is S n 1 displaystyle Sigma n 1 nbsp It is true however that if a subset X of Baire space is S n 1 displaystyle boldsymbol Sigma n 1 nbsp then there is a set of natural numbers A such that X is S n 1 A displaystyle Sigma n 1 A nbsp A similar statement holds for P n 1 displaystyle boldsymbol Pi n 1 nbsp 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 Table EditThis box viewtalkedit Lightface BoldfaceS00 P00 D00 sometimes the same as D01 S00 P00 D00 if defined D01 recursive D01 clopenS01 recursively enumerable P01 co recursively enumerable S01 G open P01 F closedD02 D02S02 P02 S02 Fs P02 GdD03 D03S03 P03 S03 Gds P03 Fsd S0 lt w P0 lt w D0 lt w S10 P10 D10 arithmetical S0 lt w P0 lt w D0 lt w S10 P10 D10 boldface arithmetical D0a a recursive D0a a countable S0a P0a S0a P0a S0wCK1 P0wCK1 D0wCK1 D11 hyperarithmetical S0w1 P0w1 D0w1 D11 B BorelS11 lightface analytic P11 lightface coanalytic S11 A analytic P11 CA coanalyticD12 D12S12 P12 S12 PCA P12 CPCAD13 D13S13 P13 S13 PCPCA P13 CPCPCA S1 lt w P1 lt w D1 lt w S20 P20 D20 analytical S1 lt w P1 lt w D1 lt w S20 P20 D20 P projective References EditKechris A S 1995 Classical Descriptive Set Theory Berlin New York Springer Verlag ISBN 978 0 387 94374 9 Rogers Hartley 1987 1967 The Theory of Recursive Functions and Effective Computability First MIT press paperback edition ISBN 978 0 262 68052 3 Retrieved from https en wikipedia org w index php title Projective hierarchy amp oldid 1132189250, wikipedia, wiki, book, books, library,