##

5.1.1 Metric Spaces

It is straightforward to define Euclidean distance in
. To
define a distance function over any , 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* is a
topological space equipped with a function
such that for any
:

**Nonnegativity:**
.
**Reflexivity:**
if and only if .
**Symmetry:**
.
**Triangle inequality:**
.

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

**Subsections**
Steven M LaValle
2012-04-20