Does there exists continuous serjective map from $ \mathbb{R}^2 $ onto unit circle $ \mathbb{S}^1 $?. I feel that no such map exist. I tried normalized map but i have trouble with origin. Can anyone give some hint?
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$\begingroup$ do continuous surjections preserve simple-connectivity? $\endgroup$– mathworker21Sep 11, 2019 at 12:36
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$\begingroup$ yes but both are connected in my case. $\endgroup$– CHOUDHARY bhim senSep 11, 2019 at 12:37
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$\begingroup$ $S^1$ is not simply connected. $\endgroup$– BerciSep 11, 2019 at 12:48
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$\begingroup$ but simple Connectedness doesn't preserved under continuity. $\endgroup$– CHOUDHARY bhim senSep 11, 2019 at 12:58
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1$\begingroup$ we can wrap real line around unit circle continuously. $\endgroup$– CHOUDHARY bhim senSep 11, 2019 at 12:59
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1 Answer
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Since you mention in the comments that we can wrap real line around unit circle continuously, try $(x,y) \mapsto x \mapsto (\cos x, \sin x)$.
