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:

$H_{n}(X\setminus A,U\setminus A)\to H_{n}(X,U)$