1. Jul 8, 2010

### 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

Last edited: Jul 8, 2010
2. Jul 8, 2010

### VeeEight

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

3. Jul 8, 2010

### 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.

4. Jul 8, 2010

### facenian

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

5. Jul 9, 2010

### VeeEight

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