On Computable Metrization

• Authors: Tanja Grubba;Klaus Weihrauch
• Venue: Electronic Notes in Theoretical Computer Science (ENTCS)
• Year: 2007

Every second-countable regular topological space X is metrizable. For a given ''computable'' topological space satisfying an axiom of computable regularity M. Schroder [M. Schroder, Effective metrization of regular spaces, in: K.-I. Ko, A. Nerode, M. B. Pour-El, K. Weihrauch and J. Wiedermann, editors, Computability and Complexity in Analysis, Informatik Berichte 235 (1998), pp. 63-80, cCA Workshop, Brno, Czech Republic, August, 1998.] has constructed a computable metric. In this article we study whether this metric space (X,d) can be considered computationally as a subspace of some computable metric space [K. Weihrauch, Computable Analysis, Springer, Berlin, 2000]. While Schroder's construction is ''pointless'', i.e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points is needed. By partial completion we extend (X,d) to a metric space (X@?,d@?) with computable metric and canonical representation. We construct a computable sequence (x"i)"i"@?"N of points which is dense in (X@?,d@?). The isometric embedding of X into X@? is computable. Its inverse is computable if some further computability axiom holds true. The space (X@?,d@?) can be embedded computationally into the computable metric space generated by the sequence (x"i)"i"@?"N of points. The inverse of this embedding is continuous.