Question about elementary topology

[facenian]

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

 

[VeeEight]

Use the fact for product topologies that the closure of the product is the product of the closures.

 

[Bacle]

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.

 

[facenian]

Thank you guys, I think we can use what VeeEight says which true for the infinite casa as well

 

[VeeEight]

Yes, it is true for infinite products and is a simple proof. Hope that helps.