Theorem 38.2 (Excision):

Let $(X, A)$ be a pair and $U$ be a subset of $A$ such that the closure of $U$ is contained in the interior of $A$. Then, the homomorphism

$$H_n(i)\;:\; H_n(X-U,\; A-U) \longrightarrow H_n(X, A)$$

induced by inclusion $i:(X-U, A-U)\longrightarrow (X, A)$ is an isomorphism for every $n = 0, 1, 2,\dots$. In other words the set $U$ may be excised from the pair $(X, A)$ without affecting the homology groups of the pair.