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Product of hausdorff spaces

  1. Sep 26, 2010 #1
    Hello, everyone.

    Theorem) If each space Xa(a∈A) is a Hausdorff space, then X=∏Xa is a Hausdorff space in both the box and product topologies.

    I understand if a box topology, the theorem holds.
    but if a product toplogy, I do not understand clearly.

    I think if there are distinct points c,d in X, then Uc, Ud (arbitrary open sets in X contain c, d respectively) are equals Xa except for finitely many values of a, so Uc and Ud are not disjoint.
    If I have a mistake, please point out it....
     
  2. jcsd
  3. Sep 27, 2010 #2
    Start with this: if c and d are different points of X then there is an index [tex]a\in A[/tex] for which the projections of c and d differ. Exploit this value of the index.
     
  4. Sep 27, 2010 #3
    Thanks a lot, arkajad. I understand it.
    if only one coordinate is different, they are disjoint.
     
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