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Product space

  1. Jul 11, 2013 #1
    Is there atopological space X such that XxX (the product space) is homeomorphic to the unit circle in the plane
     
  2. jcsd
  3. Jul 11, 2013 #2

    jgens

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    No. If there were such a homeomorphism, then there would be an open set U in X such that U x U is homeomorphic to an open arc in the 1-sphere. In other words U x U is homeomorphic to the real line. It should be simple enough to see that U is path-connected and an easy argument shows that U x U minus a point is still path-connected. On the other hand the real line minus a point is not path-connected. Contradiction.
     
    Last edited: Jul 11, 2013
  4. Jul 11, 2013 #3

    micromass

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    You could also look at the fundamental groups.
     
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