I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
2026-04-03 08:21:20.1775204480
Unit circle can't be covered by one chart
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If so, then the image is homeomorphic to $\Bbb R^n$, but $\Bbb R^n$ is simply connected, has trivial reduced homology/fundamental group, et cetera.
If you already know it is a $1$-manifold, then you also know that removing a single point from $\Bbb R$ disconnects it, but doing so for a circle will produce something still connected, so one chart restricted to a punctured $\Bbb R$ would be a continuous function which sends a connected set, $S^1\setminus P$, continuously onto a disconnected one, $\Bbb R\setminus P'$, which is impossible.