One point compactification of the positive integers

In summary, to show that the one point compactification of the positive integers is homeomorphic to the set K={0} U {1/n : n is a positive integer}, you can define a function f : Y to K, where Y is the one point compactification and K is the given set, by f(n)=1/n and f(p)=0, where p is the single point for the one point compactification. This function will be continuous, using the discrete topology, except possibly at the point at infinity where the definition of open neighborhood must be used.
  • #1
math8
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0
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.
 
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  • #2
Did you try defining f : Y to K by the most natural formula and showing directly that f is a homeomorphism?
 
  • #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]
 
  • #4
Are you using the discrete topology? If so, it's immediately continuous.
 
  • #5
Vid said:
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?
 

1. What is the one point compactification of the positive integers?

The one point compactification of the positive integers is a way of adding an extra point, called the "point at infinity," to the set of positive integers in order to make it a compact space. This is done by considering the positive integers as a subset of the real numbers and then adding the point at infinity as the limit point of the positive integers.

2. Why is the one point compactification of the positive integers useful?

The one point compactification of the positive integers is useful in topology and analysis because it allows us to treat the set of positive integers as a compact space. This can be helpful in proving certain theorems and in making calculations easier.

3. How is the one point compactification of the positive integers different from the real numbers?

The one point compactification of the positive integers is different from the real numbers in that it includes an extra point, the point at infinity, which is not a part of the real numbers. This point acts as the "boundary" of the positive integers, making it a compact space.

4. Can the one point compactification of the positive integers be performed on other sets?

Yes, the one point compactification can be performed on any locally compact Hausdorff space. This means that the space must be Hausdorff and each point must have a compact neighborhood. The one point compactification of a set is unique up to homeomorphism.

5. What are some applications of the one point compactification of the positive integers?

The one point compactification has various applications in mathematics, including in algebraic geometry, topology, and analysis. It is also used in some areas of physics, such as string theory and cosmology, where compactification is important in understanding the behavior of physical systems.

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