Adding a Point to Make a Space Compact

Stereographic projection between ${\mathbb{R}}$ and ${S^1}$ . There is no point on the real line corresponding to the “North Pole” in ${S^1}$ . Can we add another point to ${\mathbb{R}}$ ?

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 $F$ — that is, any collection of open intervals that covers $F$ — can be reduced to a finite cover:

$\displaystyle X = \bigcup_{i\in I} G_i \quad\mbox{means there exists a finite cover\ }\quad X = \bigcup_{k=1}^N G_k \,.$

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 ${\mathbb{R}}$ , nor ${n}$ -dimensional Euclidean space ${\mathbb{R}^n}$ space are compact. But these spaces are locally compact: every point has a compact neighbourhood.

When we study a non-compact topological space ${X}$ , it is often convenient to embed ${X}$ in a space ${X^*}$ 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, ${+\infty}$ and ${-\infty}$ . We may denote the extended line as ${\mathbb{R}^\sharp}$ . Many results are simpler for ${\mathbb{R}^\sharp}$ than for ${\mathbb{R}}$ .

Adding a Point

One-point compactification of the real line. Top: ${\mathbb{R}}$ . Bottom: extremities folded around to meet at point ${\infty}$ .

The simplest way to extend a locally compact space to a compact space is to adjoin a single point, usually denoted ${\infty}$ . 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 ${\infty}$ (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.

Stereeographic projection between the Euclidean plane and the unit sphere. Note that no point in ${\mathbb{R}^2}$ maps to the point ${P}$ in ${S^2}$ . This becomes the point at infinity in the one-point compactification of ${\mathbb{R}^2}$ .

The same idea works in higher dimensions. The one-point compactification of Euclidean ${n}$ -space ${\mathbb{R}^n}$ is the ${n}$ -sphere:

$\displaystyle R^n \cup \{ \infty \} \cong S^n \,.$

An ${n}$ -sphere is the set of points in ${\mathbb{R}^{n+1}}$ that are at a distance ${r}$ (the radius) from a fixed point, the centre. Taking a unit radius and the centre at the origin, we can write the equations

$\displaystyle \begin{array}{rcl} x^2 &=& 1 \,, \quad S^0, \mbox{\ two points, boundary of interval.} \\ x^2 + y^2 &=& 1 \,, \quad S^1, \mbox{\ a circle, boundary of a disk.} \\ x^2 + y^2 + z^2 &=& 1 \,, \quad S^2, \mbox{\ a sphere, surface of a ball.} \\ x^2 + y^2 + z^2 + w^2 &=& 1 \,, \quad S^3, \mbox{\ a 3-sphere, boundary of a 4-ball.} \\ \end{array}$

An especially interesting case arises when ${n=3}$ . One way to construct the 3-sphere is as the one-point compactification of Euclidean three-space: just add ${\infty}$ to ${\mathbb{R}^3}$ and define neighbourhoods of ${\infty}$ to be sets that are complements of compact sets in ${\mathbb{R}^3}$ . 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.

ThatsMaths