It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞.
Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (−π, π) on the x-axis; then bend the ends of this interval upwards (in positive y-direction) and move them towards each other, until you get a circle with one point (the topmost one) missing.
A bit more formally: we represent a point on the unit circle by its angle, in radians, going from −π to π for simplicity.
What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below.
In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
In this case, there is a unique (up to homeomorphism) "most general" Hausdorff compactification, the Stone–Čech compactification of X, denoted by βX; formally, this exhibits the category of Compact Hausdorff spaces and continuous maps as a reflective subcategory of the category of Tychonoff spaces and continuous maps.
"Most general" or formally "reflective" means that the space βX is characterized by the universal property that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way.
More explicitly, βX is a compact Hausdorff space containing X such that the induced topology on X by βX is the same as the given topology on X, and for any continuous map f : X → K, where K is a compact Hausdorff space, there is a unique continuous map g : βX → K for which g restricted to X is identically f. The Stone–Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval [0, 1].
[3] [4] Walter Benz and Isaak Yaglom have shown how stereographic projection onto a single-sheet hyperboloid can be used to provide a compactification for split complex numbers.
In fact, the hyperboloid is part of a quadric in real projective four-space.
Complex projective space CPn is also a compactification of Cn; the Alexandroff one-point compactification of the plane C is (homeomorphic to) the complex projective line CP1, which in turn can be identified with a sphere, the Riemann sphere.
Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems.
In the study of discrete subgroups of Lie groups, the quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.