  We show that the classical homology theory of Steenrod may be enriched with
descriptive set-theoretic information. We prove that the resulting definable
homology theory provides a strictly finer invariant than Steenrod homology for
compact metrizable spaces up to homotopy. In particular, we show that pro-tori
are completely classified up to homeomorphism by their definable homology. This
is in contrast with the fact that, for example, there exist uncountably many
pairwise non-homeomorphic solenoids with the same Steenrod homology groups. We
similarly develop a definable cohomology theory which strengthens \v{C}ech
cohomology and we show that it completely classifies complements of pro-tori up
to 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 solenoid
complement $X$ to the $2$-sphere, which was initiated by Borsuk and Eilenberg
in 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 showing
that each orbit of the canonical action $\mathrm{Homeo}% \left( X\right)
\curvearrowright \left[ X,S^{2}\right] $ is countable, and hence that such an
action has uncountably many orbits. This can be seen as a rigidity result, and
will 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 if
one replaces solenoids with pro-tori.
  We conclude by applying the machinery developed herein to bound the Borel
complexity of several well-studied classification problems in mathematics, such
as that of automorphisms of continuous-trace $C^{*}$-algebras up to unitary
equivalence, or that of Hermitian line bundles, up to isomorphism, over a
locally compact second countable space.
