  A compact set has computable type if any homeomorphic copy of the set which
is semicomputable is actually computable. Miller proved that finite-dimensional
spheres have computable type, Iljazovi\'c and other authors established the
property for many other sets, such as manifolds. In this article we propose a
theoretical study of the notion of computable type, in order to improve our
general understanding of this notion and to provide tools to prove or disprove
this property.
  We first show that the definitions of computable type that were distinguished
in the literature, involving metric spaces and Hausdorff spaces respectively,
are actually equivalent. We argue that the stronger, relativized version of
computable type, is better behaved and prone to topological analysis. We obtain
characterizations of strong computable type, related to the descriptive
complexity of topological invariants, as well as purely topological criteria.
We study two families of topological invariants of low descriptive complexity,
expressing the extensibility and the null-homotopy of continuous functions. We
apply the theory to revisit previous results and obtain new ones.
