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$.

Steven M LaValle 2012-04-20