Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The Cartesian product theorem for dimension 0

  1. May 28, 2013 #1
    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. jcsd
  3. May 28, 2013 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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?
     
  4. May 28, 2013 #3
    The proof seems to hold for uncountable product.The proof is attached.
     

    Attached Files:

    • 001.jpg
      001.jpg
      File size:
      13.9 KB
      Views:
      57
    Last edited: May 28, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: The Cartesian product theorem for dimension 0
  1. Inner Product (Replies: 5)

  2. Product space (Replies: 2)

Loading...