Corollary 26.5

Suppose that $U$, $V$ are open path-connected subsets of a topological space such that $U\cap V$ and $U$ are simply connected then with a base point $x_0\in U\cap V$,

$$\pi_1(U\cup V, x_0) = \pi_1(V, x_0).$$

We turn to an important example to illustrate the use of this corollary. Regard $\mathbb{R}^3$ as a subset of $S^3$ via the stereo-graphic projection and $K$ be a compact subset of $\mathbb{R}^3$ such that the complement $\mathbb{R}^3 - K$ is connected. We then have the following result.