Fundamental groups of spheres:

An important example of this is the case $U = S^n - \{{\bf e}_n\}$ and $V = \{{\bf e}_n\}$. When $n\geq 2$, the spaces $U$ and $V$ are homeomorphic to $\mathbb{R}^n$ via the stereo-graphic projection and since $U\cap V$ is path connected we conclude that $U \cup V = S^n$ is simply connected.