Definable $\mathrm{K}$-homology of separable C*-algebras

In this paper we show that the $\mathrm{K}$-homology groups of a separable C*-algebra can be enriched with additional descriptive set-theoretic information, and regarded as definable groups. Using a definable version of the Universal Coefficient Theorem, we prove that the corresponding definable $\mathrm{K}$-homology is a finer invariant than the purely algebraic one, even when restricted to the class of UHF C*-algebras, or to the class of unital commutative C*-algebras whose spectrum is a $1$-dimensional connected subspace of $\mathbb{R}^{3}$.