I try to show $\mathbb{S}^1\cong[0,1)$, by the map $f(x) = (\cos2\pi x,\sin2\pi x)$, for $x\in[0,1)$. It's clear that $f$ is continuous and bijective. But I don't know how to show the inverse map $f^{-1}$ is continuous also. Any ideas?
2026-03-27 08:58:55.1774601935
continuity of isomorphism of unit circle
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