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?

  • $\begingroup$ do continuous surjections preserve simple-connectivity? $\endgroup$ Sep 11, 2019 at 12:36
  • $\begingroup$ yes but both are connected in my case. $\endgroup$ Sep 11, 2019 at 12:37
  • $\begingroup$ $S^1$ is not simply connected. $\endgroup$
    – Berci
    Sep 11, 2019 at 12:48
  • $\begingroup$ but simple Connectedness doesn't preserved under continuity. $\endgroup$ Sep 11, 2019 at 12:58
  • 1
    $\begingroup$ we can wrap real line around unit circle continuously. $\endgroup$ Sep 11, 2019 at 12:59

1 Answer 1


Since you mention in the comments that we can wrap real line around unit circle continuously, try $(x,y) \mapsto x \mapsto (\cos x, \sin x)$.


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