Stating carefully any results on compactness which you use, show that if $X$ is a compact space, $Y$ is a Hausdorff space and $f: X \rightarrow Y$ is bijective and continuous, then $f$ is a homeomorphism.

Hence or otherwise show that the unit circle $S=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is homeomorphic to the quotient space $[0,1] / \sim$, where $\sim$ is the equivalence relation defined by

$$x \sim y \Leftrightarrow \text { either } x=y \text { or }\{x, y\}=\{0,1\} .$$