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Computational Dimension of Topological Spaces

  • Hideki Tsuiki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2064)

Abstract

When a topological space X can be embedded into the space Σ ⊥,n w , of n⊥-sequences of Σ, then we can define the corresponding computational notion over X because a machine with n+1 heads on each tape can input/output sequences inΣ ⊥,n w ⋅. This means that the least number n such that X can be topologically embedded into Σ ⊥, n w serves as a degree of complexity of the space. We prove that this number, which we call the computational dimension of the space, is equal to the topological dimension for separable metric spaces. First, we show that the weak inductive dimension of Σ ⊥,n w is n, and thus the computational dimension is at least as large as the weak inductive dimension for all spaces. Then, we show that the Nöbeling’s universal n-dimensional space can be embedded into Σ ⊥, n w and thus the computational dimension is at most as large as the weak inductive dimension for separable metric spaces. As a corollary, the 2-dimensional Euclidean space ℝ2 can be embedded in {0,1} ⊥,2 w but not in Σ ⊥,1 w for any character set Σ, and infinite dimensional spaces like the set of closed/open/compact subsets of IR m and the set of continuous functions from ℝl to ℝm can be embedded in Σ w but not in : ⊥,n w for any n.

Keywords

Topological Space Dimension Theory Gray Code Computational Dimension Compact Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. BH00.
    Vasco Brattka and Peter Hertling. Topological properties of real number representations. Theoretical Computer Science, 2000. to appear.Google Scholar
  2. Bla97.
    Jens Blanck. Domain representability of metric spaces. Annals of Pure and Applied Logic, 83:225–247, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Eng68.
    Ryszard Engelking. Outline of General Topology. North-Holland, Amsterdam, 1968.zbMATHGoogle Scholar
  4. Eng78.
    Ryszard Engelking. Dimension Theory. North-Holland, Amsterdam, 1978.zbMATHGoogle Scholar
  5. ES98.
    Abbas Edalat and Philipp Sünderhauf. A domain-theoretic approach to computability on the real line. Theoretical Computer Science, 210(1):73–98, 1998.CrossRefGoogle Scholar
  6. Gia99.
    Pietro Di Gianantonio. An abstract data type for real numbers. Theoretical Computer Science, 221:295–326, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  7. HW48.
    Witold Hurewicz and Henry Wallman. Dimension Theory. Princeton University Press, 1948.Google Scholar
  8. Nag65.
    Juniti Nagata. Modern Dimension Theory. North-Holland, Amsterdam, 1965.zbMATHGoogle Scholar
  9. SHT99.
    Viggo Stoltenberg-Hansen and John V. Tucker. Concrete models of computation for topological algebras. Theoretical Computer Science, 219:347–378, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Tsu01a.
    Hideki Tsuiki. Implementing real number computation in GHC. Computer Software (in Japanese), 2001. to appear.Google Scholar
  11. Tsu01b.
    Hideki Tsuiki. Real number computation through gray code embedding. Theoretical Computer Science, 2001. to appear.Google Scholar
  12. Wei00.
    Klaus Weihrauch. Computable Analysis, an Introduction. Springer-Verlag, Berlin, 2000.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Hideki Tsuiki
    • 1
  1. 1.Division of MathematicsKyoto UniversityJapan

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