One point compactification of the positive integers

  • Thread starter math8
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  • #1
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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.
 

Answers and Replies

  • #2
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Did you try defining f : Y to K by the most natural formula and showing directly that f is a homeomorphism?
 
  • #3
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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]
 
  • #4
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Are you using the discrete topology? If so, it's immediately continuous.
 
  • #5
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Are you using the discrete topology? If so, it's immediately continuous.
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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