the excision theorem
Relative homology H_n(X, A) is supposed to measure 'X beyond A'. Intuitively, what happens deep inside A, far from where A meets the rest of X, should not affect that measurement — you ought to be able to cut away an interior chunk of A without changing the relative homology. Excision is the theorem that makes this intuition exact, and it is one of the two facts (with the long exact sequence) that makes homology computable at all.
Precisely: let Z be a subset whose closure lies inside the interior of A (so Z is 'safely buried' in A). Then the inclusion of pairs (X minus Z, A minus Z) into (X, A) induces an isomorphism on relative homology, H_n(X minus Z, A minus Z) isomorphic to H_n(X, A) for every n. In words: you may excise — cut out — the buried set Z from both X and A, and the relative homology does not notice. There is an equivalent, often more convenient form: if X is covered by the interiors of two subspaces A and B, then H_n(X, A) is isomorphic to H_n(B, A intersect B).
Why it is indispensable: excision is precisely what lets you localize a homology computation to a neighborhood of the interesting part. Combined with the long exact sequence of a pair, it powers the computation of the homology of spheres, the proof that cellular homology agrees with singular homology, the Mayer-Vietoris sequence, and the suspension isomorphism. Without excision, relative homology would be a definition with no leverage; with it, relative groups become as concrete as local data.
The honest catch is the closure condition. You may only excise a set whose closure sits strictly inside the interior of A — you cannot cut right up to the boundary where A touches the rest of X. Excision fails if Z reaches the 'seam'. The theorem is also genuinely about good subspace pairs; the cleanest statements assume the cover by interiors, and for arbitrary pairs one passes to 'good pairs' or uses CW structure.
To compute H_n(S^k), write S^k as two hemispheres. Let A be the closed lower hemisphere (contractible) and U a small open cap around the south pole inside it; excising U leaves a pair whose relative homology is that of the upper hemisphere relative to the equator. Iterating, excision plus the long exact sequence yields H_n(S^k) = Z for n = 0 and n = k, zero otherwise.
Excision localizes the sphere computation to one hemisphere relative to the equator.
You may only excise a set whose closure lies in the interior of A — excision fails for sets that touch the seam where A meets the rest of X. The hypothesis is not a technicality; it is where the theorem can break.