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Rudin Chapter 2 Problem 25

  1. Jul 5, 2012 #1
    Rudin's problem asks: Prove that every compact metric space K has a countable base.

    My concern is how valid this statement really is. Wouldn't a finite compact metric space be unable to have a countable base?
  2. jcsd
  3. Jul 5, 2012 #2
    Rudin probably defines countable as either finite or in bijection with [itex]\mathbb{N}[/itex]. So finite things are countable to him.

    You should look up his definition to be sure.
  4. Jul 5, 2012 #3
    Rudin distinguishes finite sets and countable sets in his book. Would this affect the validity of the statement?
  5. Jul 5, 2012 #4
    Yes, it does. You want the base to be either finite or countable.
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