Editing Metric space (section)

=== Motivation ===
[[File:Great-circle distance vs straight line distance.svg|thumb|A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points {{mvar|P}} and {{mvar|Q}} on a sphere.]]

To see the utility of different notions of distance, consider the [[surface of the Earth]] as a set of points.  We can measure the distance between two such points by the length of the [[great-circle distance|shortest path along the surface]], "[[as the crow flies]]"; this is particularly useful for shipping and aviation.  We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in [[seismology]], since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points.

The notion of distance encoded by the metric space axioms has relatively few requirements.  This generality gives metric spaces a lot of flexibility.  At the same time, the notion is strong enough to encode many intuitive facts about what distance means.  This means that general results about metric spaces can be applied in many different contexts.

Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways.  A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with [[Wasserstein metric]]s on spaces of [[measure (mathematics)|measure]]s) or the degree of difference between two objects (for example, the [[Hamming distance]] between two strings of characters, or the [[Gromov–Hausdorff convergence|Gromov–Hausdorff distance]] between metric spaces themselves).