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!

Give example & prove there is a set with countable infinite set of accumulation point

  1. Jun 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Give an example and prove there is a set with a countable infinite set of accumulations points.

    2. Relevant equations
    An example would be s = {k + 1/n l k element integers, n element natural numbers}

    integers are countable infinitie a bijection exists with natural numbers

    Def: Let S be a set of real numbers. A, element reals, is an accumulation point iff every neighborhood of A contains infinitely many elements of S.

    Def: Let x element reals. Then a set Q, subset Reals, is called a neighborhood of x iff there exists epsilon > 0 such that (x -e, x + e) is a subset of Q.


    3. The attempt at a solution

    I've spent hours and don't know how to start to prove this. Would appreciate any help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 15, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Give example & prove there is a set with countable infinite set of accumulation p

    Do you know a set that's countable and dense in R? What about that set?
     
  4. Jun 15, 2011 #3
    Re: Give example & prove there is a set with countable infinite set of accumulation p

    Rationals are countable and dense in R. Still not sure where I am to take this.
     
  5. Jun 15, 2011 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Give example & prove there is a set with countable infinite set of accumulation p

    Aren't the rationals a set consisting of all accumulation points?
     
  6. Jun 15, 2011 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Give example & prove there is a set with countable infinite set of accumulation p

    Are you thinking about rationals in Q? If you are thinking about rationals in R, then all points of R are accumulation points. That's not countable. What's wrong with {k+1/n}?
     
  7. Jun 16, 2011 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Give example & prove there is a set with countable infinite set of accumulation p

    Oh my, it appears I've been reading the questio entirely wrong :blushing:

    Yep {k+1/n | k,n naturals} are fine!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Give example & prove there is a set with countable infinite set of accumulation point
  1. Countable infinite sets (Replies: 29)

Loading...