interesting examples of one point compactifications? Just wondering if anyone had any interesting examples of one point compactifications.  I am studying for a test and I would like to get some good practice proving homeomorphisms of one point compactifications
 A: What is the one-point compactification of the open interval $(0,1)$? What about the half-open interval $[0,1)$?
What is the one-point compactification of $(0,1)\cup(2,3)$?
What happens if you try to compute the one-point compactification of a set that is already compact?  What do you get when you apply the construction to a closed interval, or a circle?  Is the result compact?
Consider an open ball in $\mathbb R^2$ with one point deleted: $ \{(x,y)\in\mathbb R^2 \mid 0 < x^2+y^2 < 1\}$.  What is the one-point compactification of this set?
Let $S = \{(x,y)\in\mathbb R^2 \mid xy = 1\}$.  What is the one-point compactification of $S$?
Suppose $S$ and $T$ are not homeomorphic.  Can they have homeomophic compactifications, or is that impossible?
A: A key example is the one-point compactification of the complex plane, which is homeomorphic to a sphere, the so-called Riemann sphere.
Along similar lines, the one-point compactification of $\mathbb R$ is a circle.
Another particularly useful one is the one-point compactification of $\mathbb N$. Exercise: a function $f : \mathbb N^+ \to \mathbb R$ is continuous iff $\lim_{n \to \infty} f(n) = f(\infty)$. This allows you to understand convergent sequences in terms of continuous functions, which have nice properties. (In particular, the fact that continuous functions map convergent sequences to convergent sequences becomes the fact that the composition of continuous functions is continuous).
A: One of the simplest but most important one-point compactifications is the one-point compactification of $\Bbb N$, where $\Bbb N$ has the discrete topology. Call that one-point compactification $\Bbb N^*$. If $X$ is any topological space, and $f:\Bbb N\to X$ is any function, then $f$ is continuous; such a function is nothing more nor less than a sequence in $X$. A function $\hat f:\Bbb N^*\to X$ is called an extension of $f$ if $\hat f\upharpoonright\Bbb N=f$. Under what circumstances does $f$ have a continuous extension $\hat f$?
