Representations versus numberings: on the relationship of two computability notions

Abstract This paper gives an answer to Weihrauch's (Computability, Springer, Berlin, 1987) question whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in general. A condition is introduced and for countably based topological T 0 -spaces it is shown that exactly the (partial) effective maps meeting the requirement are extendable. For total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directed-complete partial orders, in which the extendability is already known. In the first case a similar result holds also for partial effective maps, but not in the second.