I have to prove that the unit circle $S^1$ is not homeomorphic to the sphere $S^2$ using connectedness. Intuitively I know this is true, but I'm not sure how to prove this.. Can someone help me?

  • 2
    $\begingroup$ 1) Is "$S^1$ minus two different points" connected ? 2) Is "$S^2$ minus two different points" connected ? 3) If $S^1$ were homeomorphic to $S^2$ the answers to questions 1) and 2) would be identical. $\endgroup$ – Clément Guérin Jan 13 '16 at 12:57
  • $\begingroup$ Thank you very much! I thought I could only answer the question with removing one point, but ofcourse 2 points removing is also allowed :) $\endgroup$ – jbuser430 Jan 13 '16 at 12:59

Assume that $f:S^1\to S^2$ is a homeomorphism, and $a,b\in S^1$, with $a\ne b$.

Then $$f: S^1\setminus\{a,b\}\to S^2 \setminus\{\,f(a),f(b)\},$$ would also be a homeomorphism.

However, $S^1\setminus\{a,b\}$ is not connected, while $S^2 \setminus\{\,f(a),f(b)\}$ is connected.


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