Multiple deletion lemma

Statement

Suppose $X$ is a topological space and consider closed subsets $A_{1}, A_{2}, \dots A_{n}$ 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:

$H_{q} (X, X ∖ A) \to ⨁_{i = 1}^{n} H_{q} (X, X ∖ A_{i})$

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.