Let $X=X_1 \bigcup X_2$ and $A=X_1 \bigcap X_2$. Suppose $X_1$ and $X_2$ are closed in $X$, and $A$ is a deformation retract of an open set $U$ of $X_2$. Show that $X_1$ is a deformation retract of $U \bigcup X$; conclude that $(X_1,X_2)$ is an excisive couple.
Well, to get started, I just laid out some definitions. Eventually I will want to show that $(X_1,X_2)$ is an excisive couple, which means that the inclusion:
$S(X_1) + S(X_2) -> S(X)$ induces an isomorphism in homology.
If $A$ is a deformation retract of an open set $U$ of $X_2$ and $A=X_1 \bigcap X_2$, then, well, something fishy is going on....