Let \(X\) be a topological space, \(A\subseteq X\) a subset, and \(n\geq 0\). The local homology of \(X\) at \(A\) in degree \(n\) is the relative homology group \[ H_n(X\mid A) \coloneqq H_n(X, X\sm A; \ZZ) .\] For a point \(x\in X\), write \(H_n(X\mid x)\coloneqq H_n(X\mid\ts{x})\).
[@Hat02].