Approximation systems for functions in topological and in metric spaces

A notable feature of the TTE approach to computability is the representation of the argument values and the corresponding function values by means of infinitistic names. Although convenient for the development of the theory, using such objects as data deviates from the computational practice and has a drawback from a logico-philosophical point of view. Two ways to eliminate the using of such names in certain cases are indicated in the paper. The first one is intended for the case of topological spaces with selected indexed denumerable bases. Suppose a partial function is given from one such space into another one whose selected base has a recursively enumerable index set, and suppose that the intersection of base open sets in the first space is computable in the sense of Weihrauch-Grubba. Then the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable relation between indices of base sets containing the argument value and indices of base sets containing the corresponding function value. This result can be regarded as an improvement of a result of Korovina and Kudinov. The second way is applicable to metric spaces with selected indexed denumerable dense subsets. If a partial function is given from one such space into another one, then, under a semi-computability assumption concerning these spaces, the ordinary TTE computability of the function is characterized by the existence of an appropriate recursively enumerable set of quadruples. Any of them consists of an index of element from the selected dense subset in the first space, a natural number encoding a rational bound for the distance between this element and the argument value, an index of element from the selected dense subset in the second space and a natural number encoding a rational bound for the distance between this element and the function value (in general, this set of quadruples is different from the one which straightforwardly arises from the relation used in the first way of elimination). One of the examples in the paper indicates that the computability of real functions can be characterized in a simple way by using the first way of elimination of the infinitistic names. 2012 ACM CCS: [Theory of computation]: Models of computation—Computability; [Mathematics of computing]: Mathematical analysis—Numerical analysis.