Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 679–693 | Cite as

On degree-preserving homeomorphisms between trees in computable topology

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Abstract

In this paper we first give a variant of a theorem of Jockusch–Lewis– Remmel on existence of a computable, degree-preserving homeomorphism between a bounded strong \({\Pi^0_2}\) class and a bounded \({\Pi^0_1}\) class in 2ω. Namely, we show that for mathematically common and interesting topological spaces, such as computably presented \({\mathbb{R}^n}\) , we can obtain a similar result where the homeomorphism is in fact the identity mapping. Second, we apply this finding to give a new, priority-free proof of existence of a tree of shadow points computable in 0′.

Keywords

Computability theory Recursion theory Topology \({\Pi_1^0}\) Trees Recursive analysis Computable analysis Recursive topology Computable topology Avoidable points Shadow points 

Mathematical Subject Classification (2000)

03D45 03D80 03C57 54A20 

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References

  1. 1.
    Cenzer, D., Remmel, J.B.: \({\Pi_1^0}\) classes in mathematics. In: Ersov, Y., Goncharov, S., Nerode, A., Remmel, J. (eds.) Recursive Mathematics. North-Holland Studies in Logic and Foundation Mathmatics, vol. 138, pp. 623–821 (1998)Google Scholar
  2. 2.
    Jockusch, C.G., Lewis, A.A., Remmel, J.B.: \({\Pi_1^0}\) -classes and rado’s selection principle. J. Symb. Log. 56, 684–693 (1991)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Jockusch, C.G., Soare, R.I.: \({\Pi_1^0}\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jockusch, C.G., Soare, R.I.: Degrees of members of \({\Pi_1^0}\) classes. Pac. J. Math. 40, 33–56 (1972)MathSciNetGoogle Scholar
  5. 5.
    Kalantari, I., Welch, L.: Point-free topological spaces, functions and recursive points; filter foundation for recursive analysis. I. Ann. Pure Appl. Log. 93, 125–151 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kalantari, I., Welch, L.: Recursive and nonextendible functions over the reals; filter foundation for recursive analysis, II. Ann. Pure Appl. Log. 98, 87–110 (1999)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kalantari, I., Welch, L.: A blend of methods of recursion theory and topology. Ann. Pure Appl. Log. 124, 141–178 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kalantari, I., Welch, L.: A blend of methods of recursion theory and topology: a \({\Pi^0_1}\) tree of shadow points. Arch. Math. Log. 43, 991–1008 (2004)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kalantari, I., Welch, L.: Specker’s theorem, cluster points, and computable quantum functions, In: Enayat, A., Kalantari, I., Moniri, M. (eds.) Logic in Tehran. Lecture Notes in Logic, Association for Symbolic Logic vol. 26, pp. 134–159 (2006)Google Scholar
  10. 10.
    Kalantari, I., Welch, L.: On turing degrees of points in computable topology (submitted)Google Scholar
  11. 11.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsWestern Illinois UniversityMacombUSA

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