Example 38.2:

In general, given a pair of open sets $\{U, V\}$ of a topological space $X$, let $W = (U\cup V) - V$. Then, the closure of $W$ in $U\cap V$ is contained in $U$ and since

$$(X - W,\; U - W) = (V, U\cap V),$$

the excision theorem gives

$$H_n(U\cup V,\; U) \cong H_n(V,\; U\cap V), \quad n = 0, 1, 2, \dots$$