Solution
= Solution
The sphere is the inverse image of $\{1\}$ under the continuous map $u\mapsto\|u\|^2$, so it is closed in $\mathbb R^3$. It is bounded. The <Heine-Borel theorem> therefore makes it compact. Every <metric space> is <Hausdorff>: distinct points have disjoint open balls of radii smaller than half their distance. The Euclidean subspace metric supplies that property here. Hence \b[$S^2$ is compact and Hausdorff].