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Homework Help: Hausdorff Spaces

  1. Apr 3, 2010 #1
    The Klein bottle is the identification space [itex]K=I^2 / \sim[/itex] of the unit square [itex]I^2[/itex] with

    [itex](x,0) \sim (x,1) , (0,y) \sim (1,1-y) \quad ( 0 \leqslant x,y \leqslant 1)[/itex]

    show that K is Hausdorff.

    I have a proposition in my ntoes that says:
    If [itex]\tilde{\Delta}[/itex] is a closed subset of [itex]X \times X[/itex] and the projection [itex]p : X \rightarrow X / \sim[/itex] sends open subsets [itex]W \subseteq X[/itex] to open subsets [itex]p(W) \subseteq X / \sim[/itex] then [itex]X / \sim[/itex] is Hausdorff.

    so i need to show that [itex]\tilde{\Delta} = \{ (x,y,x',y') \in I^2 \times I^2 | (x,y) \sim (x',y') \}[/itex] is closed or rather that [itex](I^2 \times I^2) \backslash \tilde{\Delta}[/itex] is open.

    now i'm stuck though. any ideas on how to to show this is open?

    thanks a lot.
     
  2. jcsd
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