We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are (...) not necessarily direct or expected. (shrink)

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{}$-computable Polish space; this answers a question of Nies. (...) We also prove analogous results for compact Polish groups and locally path-connected spaces. (shrink)

Computability theory is used to evaluate the complexity of classifying various kinds of Lebesgue spaces and associated isometric isomorphism problems.