What does it mean to say that a topological space is compact? Prove directly from the definition that $[0,1]$ is compact. Hence show that the unit circle $S^{1} \subset \mathbb{R}^{2}$ is compact, proving any results that you use. [You may use without proof the continuity of standard functions.]

The set $\mathbb{R}^{2}$ has a topology $\mathcal{T}$ for which the closed sets are the empty set and the finite unions of vector subspaces. Let $X$ denote the set $\mathbb{R}^{2} \backslash\{0\}$ with the subspace topology induced by $\mathcal{T}$. By considering the subspace topology on $S^{1} \subset \mathbb{R}^{2}$, or otherwise, show that $X$ is compact.