On some problems in computable topology
Dieter Spreen · Cambridge University Press eBooks · 2007
. Computations in spaces like the real numbers are not done on the points of the space itself but on some representation. If one considers only computable points, i.e., points that can be approximated in a computable way, finite objects as the natural numbers can be used for this. In the case of the real numbers such an indexing can e.g. be obtained by taking the Gödel numbers of those total computable functions that enumerate a fast Cauchy sequence of rational numbers. Obviously, the numbering is only a partial map. It will be seen that this is not a consequence of a bad choice, but is so by necessity. The paper will discuss some consequences. All is done in a rather general topological framework.