Exercise 59.1

Proof. Let $U=X\setminus \{a\},V=X\setminus \{b\}$ such that $a,b$ are in different copies of $S^2$; $U,V$ are open since $X$ is Hausdorff. Then, since $S^2\setminus \{a\},S^2\setminus \{b\}$ are homeomorphic to ${ℝ}^2$ by the proof in Theorem $59.3$, we see that there exists a deformation retraction from $X\setminus \{a\}$ onto $S^2$ by taking the copy of $S^2$ containing $a$ and retracting it into $\{x_0\}$ the intersection of the two copies of $S^2$; likewise, there exists a deformation retraction from $X\setminus \{b\}$ onto $S^2$. Thus, both $U,V$ are simply connected by Theorem $59.3$. $U\cap V$ is path connected since it is homeomorphic to two copies of ${ℝ}^2$ adjoined at $x_0$, and so $X$ is simply connected by Corollary $59.2$. □