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We show that the classical homology theory of Steenrod may be enriched withdescriptive set-theoretic information. We prove that the resulting definablehomology theory provides a strictly finer invariant than Steenrod homology forcompact metrizable spaces up to homotopy. In particular, we show that pro-toriare completely classified up to homeomorphism by their definable homology. Thisis in contrast with the fact that, for example, there exist uncountably manypairwise non-homeomorphic solenoids with the same Steenrod homology groups. Wesimilarly develop a definable cohomology theory which strengthens \v{C}echcohomology and we show that it completely classifies complements of pro-tori upto homeomorphism.  We also apply definable cohomology theory to the study of the space $\left[X,S^{2}\right] $ of homotopy classes of continuous functions from a solenoidcomplement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenbergin 1936. It was proved by Eilenberg and Steenrod in 1940 that the space $\left[X,S^{2}\right] $ is uncountable. We will strengthen this result, by showingthat each orbit of the canonical action $\mathrm{Homeo}% \left( X\right)\curvearrowright \left[ X,S^{2}\right] $ is countable, and hence that such anaction has uncountably many orbits. This can be seen as a rigidity result, andwill be deduced from a rigidity result for definable automorphisms of the\v{C}ech cohomology of $X$. We will also show that these results still hold ifone replaces solenoids with pro-tori.  We conclude by applying the machinery developed herein to bound the Borelcomplexity of several well-studied classification problems in mathematics, suchas that of automorphisms of continuous-trace $C^{*}$-algebras up to unitaryequivalence, or that of Hermitian line bundles, up to isomorphism, over alocally compact second countable space.

Definable (co)homology, pro-torus rigidity, and (co)homological classification