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Product space

  1. Aug 14, 2009 #1
    Hi,
    I'm starting to read an introduction to topology book.
    In the chapter about the topology of the product space (product of topological spaces), it says: "It should be clearly understood that an unrestricted product of open sets in the coordinate spaces need not be open in the product topology".

    Can anybody please give me an example of this?
    Thank you
     
  2. jcsd
  3. Aug 14, 2009 #2

    Hurkyl

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    Wikipedia mentions that the Cantor set is homeomorphic to a product of countably many copies of the discrete space {0,1}.

    The Cantor set is not discrete. But note that every subset of {0,1}N is a union of set-theoretic products of open subsets of {0,1}.
     
  4. Aug 14, 2009 #3
    Thanks for the reply Hurkyl.

    This is what I was thinking: We get into the quoted situation when the amount of spaces in the product is infinite. If we take the product of infiinitely many open sets smaller than the coordinate spaces, then we have infinitely many projections smaller than the coordinate spaces. But then this resulting set wouldn't be in the product topology because only a finite amount of projections of a set in the product topology can be different than the coordinate spaces.
    So for example if our product space is the product of the real line infinitely many times, then taking the product of (0,1) infinitely many times does not result in an open set in the product topology.
    Is this right?
     
  5. Aug 15, 2009 #4
    Our space C is the infinite product of {0,1}. In the space {0,1}, the singleton {0} is open. But the product of {0} in every factor (which gives us a singleton in C) is not open in C.
     
  6. Aug 15, 2009 #5
    Thanks g_edgar,
    The singleton in C is not open because of what I explained in my previous reply, right?
     
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