Multiple deletion lemma

Statement
Suppose $$X$$ is a topological space and consider closed subsets $$A_1, A_2, \ldots A_n$$ of $$X$$ such that there exist open subsets $$U_i \supset X_i$$, such that the $$U_i$$ are pairwise disjoint. Let $$A$$ be the union of the $$A_i$$s. Then the following natural map is an isomorphism:

$$H_q (X, X \setminus A) \to \bigoplus_{i=1}^n H_q(X, X \setminus A_i)$$

Particular cases

 * When $$X$$ is a normal space, and all the $$A_i$$s are pairwise disjoint, then the existence of $$U_i$$s is guaranteed.
 * When $$X$$ is a Hausdorff space, and all the $$A_i$$s are distinct points, then again the existence of $$U_i$$s is guaranteed, so we have the isomorphism.