# Is a continuos and injective mapping from circle to a jordan curve is surjective ? [closed]

Is a continuos and injective mapping from circle to a jordan curve is surjective ?

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That's a cool question for an exam in point set topology! –  Alain Valette Jan 5 at 14:37

## closed as too localized by Bill Johnson, Captain Oates, Fernando Muro, Alain Valette, Michael RenardyJan 5 at 12:33

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Yes. Let $f:S^{1}\rightarrow S^{1}$ be continuous and injective. Then by compactness of $S^{1}$, the image $f[S^{1}]$ is homeomorphic to $S^{1}$. In particular, $f[S^{1}]$ is a compact connected subset of $S^{1}$, so $f[S^{1}]$ is either an interval or $S^{1}$ and $f[S^{1}]$ is not an interval since $f[S^{1}]$ is homeomorphic to $S^{1}$. Therefore $f[S^{1}]=S^{1}$.

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