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Question about elementary topology

  1. Jul 8, 2010 #1
    Hello, I've got a simple question
    is the product of closed sets closed in the product topology?
    I think the answer is yes but need to sure
     
    Last edited: Jul 8, 2010
  2. jcsd
  3. Jul 8, 2010 #2
    Use the fact for product topologies that the closure of the product is the product of the closures.
     
  4. Jul 8, 2010 #3
    It is true for finite product, but I am not so sure it is true for infinite products.

    For a product of finitely many spaces, the base is given by the product of
    all open sets , i.e., given spaces X_1,..,X_n , and U_i open in X_i , then
    U_1 xU_2 x...xU_n is open in the product and a basis element.

    For infinite products, you need a different basis.
     
  5. Jul 8, 2010 #4
    Thank you guys, I think we can use what VeeEight says which true for the infinte casa as well
     
  6. Jul 9, 2010 #5
    Yes, it is true for infinite products and is a simple proof. Hope that helps.
     
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