Projective subsets of separable metric spaces

Annals of Pure and Applied Logic 50 (1):53-69 (1990)
  Copy   BIBTEX

Abstract

In this paper we will consider two possible definitions of projective subsets of a separable metric space X. A set A subset of or equal to X is Σ11 iff there exists a complete separable metric space Y and Borel set B subset of or equal to X × Y such that A = {x ε X : there existsy ε Y ε B}. Except for the fact that X may not be completely metrizable, this is the classical definition of analytic set and hence has many equivalent definitions, for example, A is Σ11 iff A is relatively analytic in X, i.e., A is the restriction to X of an analytic set in the completion of X. Another definition of projective we denote by ΣX1 or abstract projective subset of X. A set of A subset of or equal to X is ΣX1 iff there exists an n ε ω and a Borel set B subset of or equal to X × Xn such that A = {x ε X:there existsy ε Xn ε B}. These sets ca n be far more pathological. While the family of sets Σ11 is closed under countable intersections and countable unions, there is a consistent example of a separable metric space X where ΣX1 is not closed under countable intersections or countable unions. This takes place in the Cohen real model. Assuming CH, there exists a separable metric space X such that every Σ11 set is Borel in X but there exists a Σ11 set which is not Borel in X2. The space X2 has Borel subsets of arbitrarily large rank while X has bounded Borel rank. This space is a Luzin set and the technique used here is Steel forcing with tagged trees. We give examples of spaces X illustrating the relationship between Σ11 and ΣX1 and give some consistent examples partially answering an abstract projective hierarchy problem of Ulam

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,593

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Embeddings of countable closed sets and reverse mathematics.Jeffry L. Hirst - 1993 - Archive for Mathematical Logic 32 (6):443-449.
Compact Metric Spaces and Weak Forms of the Axiom of Choice.E. Tachtsis & K. Keremedis - 2001 - Mathematical Logic Quarterly 47 (1):117-128.
Projective spinor geometry and prespace.F. A. M. Frescura - 1988 - Foundations of Physics 18 (8):777-808.
Metric Boolean algebras and constructive measure theory.Thierry Coquand & Erik Palmgren - 2002 - Archive for Mathematical Logic 41 (7):687-704.
Uniform domain representations of "Lp" -spaces.Petter K. Køber - 2007 - Mathematical Logic Quarterly 53 (2):180-205.
On Metric Types That Are Definable in an O-Minimal Structure.Guillaume Valette - 2008 - Journal of Symbolic Logic 73 (2):439 - 447.
Located sets and reverse mathematics.Mariagnese Giusto & Stephen G. Simpson - 2000 - Journal of Symbolic Logic 65 (3):1451-1480.
Relativity of the metric.William M. Honig - 1977 - Foundations of Physics 7 (7-8):549-572.

Analytics

Added to PP
2014-01-16

Downloads
17 (#742,076)

6 months
1 (#1,040,386)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Add more citations

References found in this work

Steel forcing and barwise compactness.Sy D. Friedman - 1982 - Annals of Mathematical Logic 22 (1):31-46.
Forcing with tagged trees.John R. Steel - 1978 - Annals of Mathematical Logic 15 (1):55.
Analytic determinacy and 0#. [REVIEW]Leo Harrington - 1978 - Journal of Symbolic Logic 43 (4):685 - 693.
Steel forcing and Barwise compactness.S. D. Friedman - 1982 - Annals of Mathematical Logic 22 (1):31.

View all 7 references / Add more references