In this talk we will develop a framework for enriching various classical invariants of homological algebra and algebraic topology with additional descriptive set-theoretic information. The resulting "definable invariants" can be used for much finer classification than their purely algebraic counterparts. We will illustrate how these ideas apply to the classical Cech cohomology invariants to produce a new "definable cohomology theory" which, unlike its classical counterpart, provides a complete classification to homotopy classes of mapping telescopes of d-tori, and for homotopy classes of maps from mapping telescopes of d-tori to spheres. In the process, we will prove Ulam stability for quotients of Polish abelian non-archimedean groups G by Polishable subgroups H. A special case of our Ulam stability theorem answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups.