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.