One point compactification of the positive integers

Click For Summary

Homework Help Overview

The discussion revolves around the one point compactification of the positive integers and its homeomorphism to the set K={0} U {1/n : n is a positive integer}. Participants explore the properties of compact Hausdorff spaces in this context.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss defining a function f from the one point compactification to the set K and the need to demonstrate that this function is continuous. There is also consideration of the topology being used and its implications for continuity.

Discussion Status

The conversation is ongoing, with participants exploring the continuity of the function and the implications of using the discrete topology. There is recognition of the need to address the point at infinity in the compactification.

Contextual Notes

Participants are considering the definitions of open neighborhoods, particularly in relation to the point at infinity, and how these affect the continuity of the proposed function.

math8
Messages
143
Reaction score
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.
 
Physics news on Phys.org
Did you try defining f : Y to K by the most natural formula and showing directly that f is a homeomorphism?
 
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]
 
Are you using the discrete topology? If so, it's immediately continuous.
 
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?
 

Similar threads

Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
If ## a ## is a positive integer and ## \sqrt[n]{a} ## is rational?
  • · Replies 20 ·
Replies
20
Views
3K
One-point compactification problem
  • · Replies 13 ·
Replies
13
Views
6K
Proving Induction for All Integers
  • · Replies 3 ·
Replies
3
Views
2K
Compact Hausdorff Spaces: Star-Isomorphic Unital C*-Algebras
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Is the one-point compactification of X-S homeomorphic to X/S?
  • · Replies 1 ·
Replies
1
Views
3K
Topology of the Set {1/n} for Positive Integers Z
  • · Replies 23 ·
Replies
23
Views
4K
Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?
  • · Replies 9 ·
Replies
9
Views
3K
No Divisors Between an Integer and Twice it
  • · Replies 3 ·
Replies
3
Views
1K
How should I find the equilibrium points and the general equation?
  • · Replies 3 ·
Replies
3
Views
2K