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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 Σn1{displaystyle {boldsymbol {Sigma }}_{n}^{1}} for some positive integer n{displaystyle n}. Here A{displaystyle A} is

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  Σ11{displaystyle {boldsymbol {Sigma }}_{1}^{1}} if A{displaystyle A} is analytic
  


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  Πn1{displaystyle {boldsymbol {Pi }}_{n}^{1}} if the complement of A{displaystyle A}, X∖A{displaystyle Xsetminus A}, is Σn1{displaystyle {boldsymbol {Sigma }}_{n}^{1}}
  


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  Σn+11{displaystyle {boldsymbol {Sigma }}_{n+1}^{1}} if there is a Polish space Y{displaystyle Y} and a Πn1{displaystyle {boldsymbol {Pi }}_{n}^{1}} subset C⊆X×Y{displaystyle Csubseteq Xtimes Y} such that A{displaystyle A} is the projection of C{displaystyle C}; that is, A={x∈X∣∃y∈Y(x,y)∈C}{displaystyle A={xin Xmid exists yin 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

There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters Σ{displaystyle Sigma } and Π{displaystyle Pi }) and the projective hierarchy on subsets of Baire space (denoted by boldface letters Σ{displaystyle {boldsymbol {Sigma }}} and Π{displaystyle {boldsymbol {Pi }}}). Not every Σn1{displaystyle {boldsymbol {Sigma }}_{n}^{1}} subset of Baire space is Σn1{displaystyle Sigma _{n}^{1}}. It is true, however, that if a subset X of Baire space is Σn1{displaystyle {boldsymbol {Sigma }}_{n}^{1}} then there is a set of natural numbers A such that X is Σn1,A{displaystyle Sigma _{n}^{1,A}}. A similar statement holds for Πn1{displaystyle {boldsymbol {Pi }}_{n}^{1}} 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

References

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  Kechris, A. S. (1995), Classical Descriptive Set Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94374-9.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


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  Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3