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.