Topology: Clopen basis of a space

[talolard]

Homework Statement

So, I'm going through a proof and it is shamelessly asserted that the collection of clopen sets of {0,1}^{\mathbb{N}} is a countable basis. Can anyone reasure me of this, point me in the direction of proving it.
Thanks
Tal

 

[ystael]

I assume you mean that \{0, 1\} is to be given the discrete topology and then \{0, 1\}^\mathbb{N} gets the product topology based on that.

Remember that the basic open sets of the product topology on a product \textstyle\prod_\lambda X_\lambda are the the sets \textstyle\prod_\lambda U_\lambda where each U_\lambda is open in X_\lambda and only finitely many of the U_\lambda differ from X_\lambda.

How many finite subsets does \mathbb{N} have?

 

[talolard]

Got it.
Thanks.