Let \(h_*\) be a sequence of functors \(h_n\) from pairs of topological spaces to abelian groups. The functors \(h_*\) satisfy excision if for every pair \((X, A)\) and every subspace \(Z\subseteq A\) with \(\cl_X(Z) \subseteq A^\circ\), the inclusion \((X\sm Z, A\sm Z)\injects (X, A)\) induces isomorphisms \[ h_n(X\sm Z, A\sm Z) \mapsvia{\sim} h_n(X, A) \quad\text{for all } n .\]
Excision is one of the Eilenberg–Steenrod axioms for a homology theory. Singular homology satisfies it [@Hat02]; relative homotopy groups do not.
[@Hat02].