nLab
metric space

Idea

A metric space is a set which comes equipped with a function which measures distance between points, called a metric. The metric can be used to generate a topology on the set, and a topological space whose topology comes from some metric is said to be metrizable.

Definition

Traditionally, a metric space is defined to be a set $X$ equipped with a function

$$d:X\times X\to [0,\mathrm{\infty })$$d: X \times X \to [0, \infty)

(valued in nonnegative real numbers) satisfying the following axioms:

• Triangle inequality: $d(x,y)+d(y,z)\ge d(x,z)$;
• Point inequality: $0\ge d(x,x)$ (so $0=d(x,x)$);
• Separation: $x=y$ if $d(x,y)=0$ (so $x=y$ iff $d(x,y)=0$);
• Symmetry: $d(x,y)=d(y,x)$.

Given a metric space $(X,d)$ and a point $x\in X$, the open ball centered at $x$ of radius $r$ is

$$B_r(x)=\{y\in X:d(x,y)<r\}$$B_r(x) = \{y \in X: d(x, y) \lt r\}

and it may be shown that the open balls form a basis for a topology on $X$. In fact, metric spaces are examples of uniform spaces, and much of the general theory of metric spaces, including for example the notion of completion of a metric space, can be extrapolated to uniform spaces and even Cauchy spaces.

As topological spaces, metric spaces enjoy a number of separation properties: they are Hausdorff, regular, and even normal. They are also paracompact.

Variations

If we allow $d$ to take values in $[0,\mathrm{\infty }]$ instead of just in $[0,\mathrm{\infty })$, then we get extended metric spaces. If we drop separation, then we get pseudometric spaces. If we drop the symmetry condition, then we get quasimetric spaces. Thus the most general notion is that of an extended quasipseudometric space, which are also called Lawvere metric spaces for the reasons below.

On the other hand, if we strengthen the triangle inequality to

$$\max (d(x,y),d(y,z))\ge d(x,z),$$max(d(x,y), d(y,z)) \geq d(x,z) ,

then we get ultrametric spaces, a more restricted concept. (This include for example $p$-adic completions of number fields.) Extended quasipseudoultrametric spaces can also be called Lawvere ultrametric spaces.

Lawvere metric spaces

Lawvere has pointed out that Lawvere metric spaces are precisely categories enriched in the monoidal poset $([0,\mathrm{\infty }],\ge )$, where the monoidal product is taken to be addition. Taking the monoidal product to be supremum instead, enriched categories amount to Lawvere ultrametric spaces.

Thus generalized, many constructions and results on metric spaces turn out to be special cases of yet more general constructions and results of enriched category theory. This includes for example the notion of Cauchy completion, which in general enriched category theory is related to Karoubi envelopes and Morita equivalence.

One would like to say that imposing the symmetry axiom gives us enriched groupoids?, which is correct for ultrametric spaces. It's not clear, however, what this means for more general metric spaces, since the enriching category is then not cartesian.

Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched $†$-categories?

Toby: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched $†$-categories.

Motivation for the axioms

The triangle axiom is the fundamental idea behind a metric space; it goes back (at least) to Euclid and captures the idea that we are discussing the shortest distance between two points. Given the triangle inequality, we have the polygon inequality

$$d(x_0,x_1)+\cdots +d(x_{n-1},x_n)\ge d(x_0,x_n)$$d(x_0,x_1) + \cdots + d(x_{n-1},x_n) \geq d(x_0,x_n)

for all $n>0$; the point inequality extends this to $n=0$.

Besides extended metric spaces (where distances may be infinite), one might consider spaces where distances may be negative. But in fact this gives us nothing new, at least if we have symmetry. First,

$$d(x,x)+d(x,x)\ge d(x,x)$$d(x,x) + d(x,x) \geq d(x,x)

forces $d(x,x)\ge 0$, so $d(x,x)=0$; then

$$2d(x,y)=d(x,y)+d(y,x)\ge d(x,x)$$2 d(x,y) = d(x,y) + d(y,x) \geq d(x,x)

forces $d(x,y)\ge 0$. A generalisation to negative distances is possible for quasimetric spaces, however; the simplest example has $2$ elements, with $d(x,y)=-d(y,x)$.

We can define a partial order $\le$ on the points of a Lawvere metric space:

$$x\le y\iff d(x,y)=0.$$x \leq y \;\Leftrightarrow\; d(x,y) = 0 .

Then the symmetry axiom implies that this relation is symmetric and hence an equivalence relation. The quotient set under this equivalence relation satisfies separation; in this way, every pseudometric space is equivalent (as an enriched category) to a metric space. Even for quasimetric spaces, we can still define an equivalence relation:

$$x\equiv y\iff d(x,y)=0\wedge d(y,x)=0.$$x \equiv y \;\Leftrightarrow\; d(x,y) = 0 \;\wedge\; d(y,x) = 0 .

In constructive mathematics, it works better to use $<$:

$$x<y\iff d(x,y)>0;$$x \lt y \;\Leftrightarrow\; d(x,y) \gt 0 ;

then the symmetry axiom implies that this is an apartness relation, which (for quasimetric spaces) we can also define directly:

$$x\#y\iff d(x,y)>0\vee d(y,x)>0.$$x \# y \;\Leftrightarrow\; d(x,y) \gt 0 \;\vee\; d(y,x) \gt 0 .

Examples

• Every set carries the discrete metric given by

  $$d(x,y)=\{\begin{array}{cc}0& \mathrm{if}x=y\\ 1& \mathrm{otherwise}\end{array}$$d(x,y) = \left\{\array{ 0 & if x = y \\ 1 & otherwise }\right.

  For certain purposes, it makes more sense to make most the non-zero distance $\mathrm{\infty }$ instead of $1$; then one has an extended metric space.

• Every normed vector space is a metric space by $d(x,y):=\parallel x-y\parallel$; a pseudonormed vector space is a pseudometric space.

• Every connected Riemannian manifold becomes a pseudometric space (or a metric space if, as is often assumed, the manifold is Hausdorff) by taking the distance between two points to be infimum of the Riemannian length of all continuously differentiable paths connecting them

  $$d(x,y):={\inf }_{x\stackrel{\gamma }{\to }y}\mathrm{len}(\gamma )$$d(x,y) := inf_{x \stackrel{\gamma}{\to} y} len(\gamma)

  If the manifold might not be connected, then it still becomes an extended metric space.