  We provide a detailed study of two properties of spaces and pairs of spaces,
the surjection property and the epsilon-surjection property, that were recently
introduced to characterize the notion of computable type arising from
computability theory. For a class of spaces including the finite simplicial
complexes, we develop techniques to prove or disprove these properties using
homotopy and homology theories, and give applications of these results. In
particular, we answer an open question on the computable type property, showing
that it is not preserved by taking products. We also observe that computable
type is decidable for finite simplicial complexes.
