2008-27.md

course	course_year	question_number	tags	title	year
Metric and Topological Spaces	IB	27	IB 2008 Metric and Topological Spaces	2.I.4F	2008

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

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