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A notable feature of the TTE approach to computability is the representationof the argument values and the corresponding function values by means ofinfinitistic names. Two ways to eliminate the using of such names in certaincases are indicated in the paper. The first one is intended for the case oftopological spaces with selected indexed denumerable bases. Suppose a partialfunction is given from one such space into another one whose selected base hasa recursively enumerable index set, and suppose that the intersection of baseopen sets in the first space is computable in the sense of Weihrauch-Grubba.Then the ordinary TTE computability of the function is characterized by theexistence of an appropriate recursively enumerable relation between indices ofbase sets containing the argument value and indices of base sets containing thecorresponding function value.This result can be regarded as an improvement of aresult of Korovina and Kudinov. The second way is applicable to metric spaceswith selected indexed denumerable dense subsets. If a partial function is givenfrom one such space into another one, then, under a semi-computabilityassumption concerning these spaces, the ordinary TTE computability of thefunction is characterized by the existence of an appropriate recursivelyenumerable set of quadruples. Any of them consists of an index of element fromthe selected dense subset in the first space, a natural number encoding arational bound for the distance between this element and the argument value, anindex of element from the selected dense subset in the second space and anatural number encoding a rational bound for the distance between this elementand the function value. One of the examples in the paper indicates that thecomputability of real functions can be characterized in a simple way by usingthe first way of elimination of the infinitistic names.

Approximation systems for functions in topological and in metric spaces