The real line is an example of a locally compact Hausdorff space. In a Hausdorff space, two distinct points have disjoint neighbourhoods. As the old joke says, “any two points can be *housed off* from each other”. We will define local compactness below. The one-point compactification is a way of embedding a locally compact Hausdorff space in a compact space. In particular, it is a way to “make the real line compact”.

**Closed and Bounded Sets and Compactness**

In undergraduate mathematics we learn that a set of real numbers is compact iff it is closed and bounded. We then study the *Heine-Borel theorem*, which basically says that any open cover of a closed, bounded set — that is, any collection of open intervals that covers — can be reduced to a finite cover:

It is this characteristic of compactness that applies to topological spaces in general: in a nutshell, a set in a topological space is compact *if and only if every open cover of the set has a finite sub cover.*

Compact spaces have attractive properties, and the proof of theorems is often simpler for these spaces. However, many of the spaces we study are not compact: neither the real line , nor -dimensional Euclidean space space are compact. But these spaces are locally compact: *every point has a compact neighbourhood.*

When we study a non-compact topological space , it is often convenient to embed in a space that is compact. Then we can use standard compactness arguments for the extended space, and many proofs are simplified in this setting. For example, the real line can be extended by adding two points, and . We may denote the extended line as . Many results are simpler for than for .

**Adding a Point**

The simplest way to extend a locally compact space to a compact space is to adjoin a single point, usually denoted . The real line can be extended by adding a point at infinity, which converts the topology to that of a circle. We imagine the line to be bent around so that the extremes meeting at the point (see Figure above). This gives us a space topologically equivalent to a circle.

The same idea in two dimensions gives us the 2-sphere or Riemann sphere, which plays a crucial role in complex analysis (see Figure below). The Riemann sphere is constructed by adjoining a point at infinity to the complex plane and defining neighbourhoods of infinity to be the complements of compact subsets of the plane.

The same idea works in higher dimensions. The one-point compactification of Euclidean -space is the -sphere:

An -sphere is the set of points in that are at a distance (the radius) from a fixed point, the centre. Taking a unit radius and the centre at the origin, we can write the equations

An especially interesting case arises when . One way to construct the 3-sphere is as the one-point compactification of Euclidean three-space: just add to and define neighbourhoods of to be sets that are complements of compact sets in . There are several other constructions; I hope to describe them in a later post.

**Conclusion**

The 3-sphere was considered by Einstein as a possible model of a compact universe [see earlier post here]. Remarkably, it was envisaged many centuries earlier by Dante who, in *The Divine Comedy*, used a 3-sphere to model a universe that unified the physical and spiritual domains in a compact universe; we will write about that next week.