5.1.1 Metric Spaces

It is straightforward to define Euclidean distance in ${\mathbb{R}}^n$. To define a distance function over any ${\cal C}$, however, certain axioms will have to be satisfied so that it coincides with our expectations based on Euclidean distance.

The following definition and axioms are used to create a function that converts a topological space into a metric space.^5.1 A metric space $(X,\rho)$ is a topological space $X$ equipped with a function $\rho : X \times X \rightarrow {\mathbb{R}}$ such that for any $a,b,c \in X$:

1. Nonnegativity: $\rho(a,b) \geq 0$.
2. Reflexivity: $\rho(a,b) = 0$ if and only if $a = b$.
3. Symmetry: $\rho(a,b) = \rho(b,a)$.
4. Triangle inequality: $\rho(a,b) + \rho(b,c) \geq \rho(a,c)$.

The function $\rho$ defines distances between points in the metric space, and each of the four conditions on $\rho$ agrees with our intuitions about distance. The final condition implies that $\rho$ is optimal in the sense that the distance from $a$ to $c$ will always be less than or equal to the total distance obtained by traveling through an intermediate point $b$ on the way from $a$ to $c$.