= Solution
Use $F:[0,1]\to S^1$, $F(t)=e^{2\pi it}$. This is <continuous>, identifies the two endpoints, and is otherwise injective. Its fibres are exactly the <equivalence classes> for <collapsing a subset to a point> with $A=\{0,1\}$. Hence it factors uniquely through a <bijection>
$$
\overline F:X_A\longrightarrow S^1,\qquad\overline F([t])=e^{2\pi it}.
$$
By the <universal property of the quotient topology>, $\overline F$ is <continuous>. The interval is <compact>, so its quotient $X_A$ is compact. The <unit circle> is <Hausdorff>. The <compact-to-Hausdorff continuous bijection theorem> now shows that $\overline F$ is a <homeomorphism>:
$$
\boxed{[0,1]/\{0\sim1\}\cong S^1.}
$$
Indeed the theorem makes the inverse continuous because images of closed subsets of a compact space are compact and hence closed in a Hausdorff target.
Back to article page
