The Dimension of a Point that isn’t there

A slice of Swiss cheese has one-dimensional holes;
a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — also sprach Euclid. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

Missing the Point

A point has dimension zero. But what if the point is not there? The real line ${\mathbb{R}}$ is one-dimensional. It is a connected manifold: we can join any two points in ${\mathbb{R}}$ by a path. This is not true of the punctured line ${\mathbb{R}\setminus \{0\}}$ : this manifold falls into two disconnected components, ${\{x\in\mathbb{R}:x<0\}}$ and ${\{x\in\mathbb{R}:x>0\}}$ ; the points ${-1}$ and ${+1}$ cannot be connected by a path within ${\mathbb{R}\setminus \{0\}}$ .

We define the unit ${n}$ -sphere in ${\mathbb{R}^{n+1}}$ as ${\mathbb{S}^n = \{x :|x|=1\}}$ . ${\mathbb{S}^2}$ is the usual sphere in 3-space, ${\mathbb{S}^1}$ is the unit circle in 2-space and ${\mathbb{S}^0 = \{-1,+1\}}$ is the “0-sphere”, a pair of points on the real line. Within the real line ${\mathbb{R}}$ the 0-sphere can be continuously distorted, or shrunk, to a single point. The technical term for this shrinkability is homotopy: we say that ${\mathbb{S}^0 }$ is homotopic to a point in ${\mathbb{R}}$ . Within the punctured line, ${\mathbb{R}\setminus\{0\}}$ , the 0-sphere cannot be shrunk in this way because of the gap or hole. We say that there is a zero-dimensional hole in ${\mathbb{R}\setminus\{0\}}$ .

In the plane ${\mathbb{R}^2}$ , both the 0-sphere ${\{-1,+1\}}$ and the 1-sphere (a circle) can be continuously distorted, or shrunk, to a single point. In the punctured plane, ${\mathbb{R}^2\setminus\{0\}}$ , this is no longer true. The two points of the 0-sphere can be smoothly moved to a single point, but the 1-sphere cannot be shrunk to a point because it surrounds the missing point. We say that ${\mathbb{R}^2\setminus\{0\}}$ has a one-dimensional hole, even thought only a single point is removed. Obviously, this continues: ${\mathbb{R}^3\setminus\{0\}}$ has a 2-dimensional hole, since the unit 2-sphere cannot be shrunk to a point, whereas circles and point-pairs can.

A Hole is not there

We see that the removal of a single point from ${\mathbb{R}^n}$ results in an ${(n-1)}$ -dimensional hole. The point may be zero-dimensional in ${\mathbb{R}^n}$ , but the gap that it leaves in ${\mathbb{R}^n\setminus\{0\}}$ is an ${(n-1)}$ -dimensional hole. The crux is that, in the punctured space ${\mathrm{X}=\mathbb{R}^n\setminus\{0\}}$ , the hole is “not there!” We have to study the properties of the manifold ${\mathrm{X}}$ while remaining within ${\mathrm{X}}$ . This is done by defining algebraic stuctures that enable us to identify and quantify holes of various dimensions in the manifold. Let’s start with a simple example.

Holes usually comprise more than a single point. We compare two manifolds, the plane annulus ${\mathbb{A}^2 = \{x\in\mathbb{R}^2: a \le |x| \le b \}}$ and the spherical shell ${\mathbb{A}^3 = \{x\in\mathbb{R}^3: a \le |x| \le b \}}$ . ${\mathbb{A}^2}$ is like a sheet of paper with a hole punched in it. A loop within ${\mathbb{A}^2}$ that surrounds the hole cannot be contracted to a point within ${\mathbb{A}^2}$ ; there is a one-dimensonal hole in ${\mathbb{A}^2}$ . By contrast, ${\mathbb{A}^3}$ is more like a Swiss cheese. Loops in ${\mathbb{A}^3}$ can all be shrunk to a point, but a sphere surrounding the hole cannot be so shrunk: it is not homotopic to a point. A thin slice of Swiss cheese has one-dimensional holes; a block of Swiss cheese has two-dimensional holes.

A Little History

Enrico Betti (1823–1892).

Bernhard Riemann noticed that holes in a surface could be counted by checking how many cuts could be made while the surface remains in one piece. For a bounded surface like a disk or an annulus, each cut begins and ends on a boundary. Clearly, the disk falls into two pieces once a single cut is made (it has no holes) but the annulus ${\mathbb{A}^2}$ remains connected after a cut from the inner to the outer boundary circle is made.

These ideas were greatly developed by Henri Poincaré in his paper Analysis Situs (1895) and five follow-up papers. He introduced the idea of homology, which generalizes Riemann’s ideas to higher dimensions. Poincaré called the number of holes of dimension ${p}$ the ${p}$ -th Betti number, after Italian mathematician Enrico Betti. The zeroth Betti number, ${b_0}$ , indicates the number of pieces. For a connected manifold, ${b_0 = 1}$ . The first Betti number is the number of one-dimensional holes. For the 2-sphere, ${b_1 = 0}$ ; for the torus, ${b_1 = 2}$ . The second Betti number is the number of cavities in the manifold. For both the 2-sphere and the spherical shell, ${b_2 = 1}$ . In general, ${b_p}$ gives the number of ${p}$ -dimensional holes in the manifold.Poincaré showed that the Betti numbers are topological invariants. They do not change under distortions of a shape as long as it is not cut or broken. Therefore, shapes that have different sets of Betti numbers cannot be homeomorphic or topologically equivalent. For example, the numbers ${(b_0, b_1, b_2)}$ for a sphere are ${(1, 0, 1)}$ while for a torus they are ${(1, 2, 0)}$ . Several other examples are shown in the table above. The invariance greatly helps topologists to categorize topological spaces.