One point compactification of the positive integers

[math8]

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.

 

[Billy Bob]

Did you try defining f : Y to K by the most natural formula and showing directly that f is a homeomorphism?

 

[math8]

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]

 

[Vid]

Are you using the discrete topology? If so, it's immediately continuous.

 

[Billy Bob]

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?