Consider the sphere $S^{2}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\}$, a subset of $\mathbb{R}^{3}$, as a subspace of $\mathbb{R}^{3}$ with the Euclidean metric.

(i) Show that $S^{2}$ is compact and Hausdorff as a topological space.

(ii) Let $X=S^{2} / \sim$ be the quotient set with respect to the equivalence relation identifying the antipodes, i.e.

$$(x, y, z) \sim\left(x^{\prime}, y^{\prime}, z^{\prime}\right) \Longleftrightarrow\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(x, y, z) \text { or }(-x,-y,-z)$$

Show that $X$ is compact and Hausdorff with respect to the quotient topology.