Excision isomorphism

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:

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