What is the one point compactification of the reals?

Question

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a circular number line. Does it apply to complex numbers too? And do numbers such as aleph-null or ordinal infinities mess up this theorem?

Answer 1

Problem: Can we write one to one, onto and continious from the circle $S_1$ to $\mathbb R$ ?

The answer is no and it can be shown that if we extract one point from $S_1$, it is possible.

Now the idea is that instead of extractinting one point from $S_1$, let's add one point to $\mathbb R$ so that we can write the map from $S_1$ to $\mathbb R \cup {a} $. As $S_1$ is compact then $\mathbb R\cup {a}$ is compact.

One point compactification is the generelezation of this idea. In general, if you add one point to $\mathbb R^n$ in a appropriate way you will get $S_n$.

Answer 2

The one point compactification of the reals is the circumference. Indeed, stereographic projection from the north pole gives a homeo from the circumference without that pole onto the real line, and the inverse is the embedding of the reals into a compact space consisting of just one single additional point. In full generality, for non-compact locally compact Haussdorf space $X$, the one point (or Alexandroff) compactification is just a compact Haussdorf space $X^*$ obtained adding a point to the given one, and all such $X^*$'s are homeomorphic. For complex numbers, which topologically are the real plane, the one point compactification is the sphere, and stereographic projection works the same.