The Cartesian product theorem for dimension 0

[hedipaldi]

The cartesian product ∏X = Xi of a countable family {Xi} of regular spaces is zero-dimensional
i f and only i f all spaces Xi , are zero-dimensional.
I wonder if the countability assumption is just to ensure the regularity of the product space ,or it is crucial for the clopen basis.
Thank's

 

[micromass]

Products of an arbitrary number of regular spaces are always regular, so the problem isn't there. Did you check the proof?? Where did they use countable?
What book is this anyway?

 

[hedipaldi]

The proof seems to hold for uncountable product.The proof is attached.