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One point compactification of the positive integers

  1. Apr 28, 2009 #1
    How do we show the one point compactification of the positive integers is homeomorphic to the set K={0} U {1/n : n is a positive integer}?

    Say Y is the one point compactification of the positive integers. I know Y must contain Z+ and Y\Z+ is a single point. Also Y is a compact Hausdorff space.

    But I am not sure how to show this.
     
  2. jcsd
  3. Apr 28, 2009 #2
    Did you try defining f : Y to K by the most natural formula and showing directly that f is a homeomorphism?
     
  4. Apr 28, 2009 #3
    yes, I am just trying to figure out how to show that f is continuous. [f defined by f(n)=1/n and f(p)=0 where p is the single point for the one point compactification]
     
  5. Apr 28, 2009 #4

    Vid

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    Are you using the discrete topology? If so, it's immediately continuous.
     
  6. Apr 28, 2009 #5
    I would agree, except possibly at the point at infinity. Here you must use definition of open nbhd of the point at infinity, right?
     
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