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Showing two sets are not homeomorphic in subspace topology.

  1. Oct 15, 2011 #1
    1. The problem statement, all variables and given/known data
    The true problem is too complicated to present here, but hopefully somebody can give me a hand with this simplified version. Consider the set [itex] H = \{ (x,y) \in \mathbb R^2 : y \geq 0 \} [/itex]. Denote by [itex] \partial H = \{ (x,0) \}[/itex]. Let U and V be open sets (relative to H) such that [itex] U \cap \partial H = \emptyset [/itex] and [itex] V \cap \partial H \neq \emptyset [/itex]. I want to show that U and V cannot be diffeormorphic. It should be possible to show that these sets are not homeomorphic so we content ourselves with that.

    3. The attempt at a solution

    It seems to me that this shouldn't be too hard, but I don't want to make any silly mistakes. First of all, these are both open sets in H viewed with the subspace topology inherited from [itex] \mathbb R^2 [/itex]. However, the fact that V intersects the "boundary" of H would imply that there is some topological property which we shouldn't be able to transfer between the two of them. Should I look for an intrinsic property in H? Or is it possible to view U and V as subsets of [itex] \mathbb R^2 [/itex] and get a contradiction there?
     
  2. jcsd
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