1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Rudin 2.14

  1. Jun 29, 2015 #1
    1. The problem statement, all variables and given/known data

    Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.)

    2. Relevant equations

    Every separable metric space has a countable base.

    3. The attempt at a solution

    Let F be closed. Using the above fact, I've shown that the isolated points of F are at most countable, likewise their closure. I'm trying to construct a perfect set by removing non-limit points of F' points from F', the set of limit points of F, but it's not quite falling into place yet. Is this a good direction to go?
  2. jcsd
  3. Jun 29, 2015 #2


    User Avatar
    Science Advisor
    Gold Member

    Careful, that is a countable _local_ base, not a countable global base, i.e., every metric space is 1st-countable, but not necessarily 2nd-countable.
    EDIT: Sorry, I did not read carefully: the hypothesis of separable implies 2nd countable.
  4. Jun 29, 2015 #3
    It's a countable global base since the space is separable.
  5. Jun 29, 2015 #4


    User Avatar
    Science Advisor
    Gold Member

    Yes, thanks, I just corrected it as you were writing; I did not read carefully-enough.
  6. Jun 29, 2015 #5
    Hint: consider the points ##x\in F## for which every neighborhood of ##x## meets ##F## in uncountably many points.
  7. Jun 30, 2015 #6
    Got it, thanks!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads for Rudin
Rudin POMA: chapter 4 problem 14