Excision isomorphism

This fact is related to: excisive triad

Statement

Let $(X;X_1,X_2)$ be an excisive triad, viz $X$ is a topological space and $X_1$ and $X_2$ are subspaces such that the union of their interiors is $X$. Then the following map induced by inclusion of pairs, is an isomorphism:

$H_n(X_1,X_1\cap X_2)\to H_n(X,X_2)$

This is called the excision isomorphism.

Alternative formulation

Let $X$ be a topological space, $U$ be an open subset, and $A$ a closed subset inside $U$. Then the following inclusion-induced map is an isomorphism:

$H_n(X\setminus A,U\setminus A)\to H_n(X,U)$