The Cartesian product theorem for dimension 0

1. May 28, 2013

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

2. May 28, 2013

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?

3. May 28, 2013

hedipaldi

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

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Last edited: May 28, 2013