Topological properties of real number representations

  • Authors:

  • Vasco Brattka;Peter Hertling

  • Affiliations:

  • Theoretische Informatik I, Fern Universität Hagen, D-58084 Hagen, Germany;Theoretische Informatik I, Fern Universität Hagen, D-58084 Hagen, Germany

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2002

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Abstract

We prove three results about representations of real numbers (or elements of other topological spaces) by infinite strings. Such representations are useful for the description of real number computations performed by digital computers or by Turing machines. First, we show that the so-called admissible representations, a topologically natural class of representations introduced by Kreitz and Weihrauch, are essentially the continuous extensions (with a well-behaved domain) of continuous and open representations. Second, we show that there is no admissible representation of the real numbers such that every real number has only finitely many names. Third, we show that a rather interesting property of admissible real number representations holds true also for a certain non-admissible representation, namely for the naive Cauchy representation: the property that continuity is equivalent to relative continuity with respect to the representation.