  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. 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. 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.
