Excision theorem

In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space ${\displaystyle X}$ and subspaces ${\displaystyle A}$ and ${\displaystyle U}$ such that ${\displaystyle U}$ is also a subspace of ${\displaystyle A}$, the theorem says that under certain circumstances, we can cut out (excise) ${\displaystyle U}$ from both spaces such that the relative homologies of the pairs ${\displaystyle (X\setminus U,A\setminus U)}$ into ${\displaystyle (X,A)}$ are isomorphic.

This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.