  A uniform approach to computing with infinite objects like real numbers,
tuples of these, compacts sets, and uniformly continuous maps is presented. In
work of Berger it was shown how to extract certified algorithms working with
the signed digit representation from constructive proofs. Berger and the
present author generalised this approach to complete metric spaces and showed
how to deal with compact sets. Here, we unify this work and lay the foundations
for doing a similar thing for the much more comprehensive class of compact
Hausdorff spaces occurring in applications. The approach is of the same
computational power as Weihrauch's Type-Two Theory of Effectivity.
