Question about elementary topology

  • Thread starter facenian
  • Start date
  • #1
394
15
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
 
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Answers and Replies

  • #2
614
0
Use the fact for product topologies that the closure of the product is the product of the closures.
 
  • #3
662
1
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.
 
  • #4
394
15
Thank you guys, I think we can use what VeeEight says which true for the infinte casa as well
 
  • #5
614
0
Yes, it is true for infinite products and is a simple proof. Hope that helps.
 

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