# One point compactification

• Mar 25th 2010, 10:44 PM
math.dj
One point compactification
1) Show that one point compactification of N(set of natural numbers) is homeomorphic wit {1/n : n in N} union {0}.

2)Show that one point compactification of R(set of real numbers) is homeomoerphic with S1(unit circle)..

any help is appreciated..
• Mar 26th 2010, 10:13 PM
Drexel28
Quote:

Originally Posted by math.dj
1) Show that one point compactification of N(set of natural numbers) is homeomorphic wit {1/n : n in N} union {0}.

Denote the one-point compactification of $\displaystyle \mathbb{N}$ by $\displaystyle \mathbb{N}_{\infty}$ and $\displaystyle \mathcal{N}=\left\{\frac{1}{n}:n\in\mathbb{N}\righ t\}\cup\{0\}$ and do what's natural. Define $\displaystyle \varphi:\mathbb{N}_{\infty}\to \mathcal{N}$ by $\displaystyle n\mapsto\begin{cases}\frac{1}{n} & \mbox{if} \quad n\ne\infty \\ 0 & \mbox{if} \quad n=\infty\end{cases}$.

That would be my initial guess, but until you tell us what topology is put on these two sets there can be no discussion of continuity.

The second one is done similarly.