Generation of a fundamental group by two open sets

If $X=U_1\cup U_2$, the sets $U_1,U_2$, and their intersection are path-connected, and the base point lies in the intersection, then ${\pi }_1(X)$ is generated by the images of ${\pi }_1(U_1)$ and ${\pi }_1(U_2)$. Subdivide each loop into arcs lying in cover members and join every transition point to the base point within the intersection; the inserted paths telescope.