Editing Metric space (section)

=== Length spaces ===
[[File:approximate arc length.svg|thumb|One possible approximation for the arc length of a curve.  The approximation is never longer than the arc length, justifying the definition of arc length as a [[supremum]].]]
{{Main|Intrinsic metric}}
A [[curve]] in a metric space {{math|(''M'', ''d'')}} is a continuous function <math>\gamma:[0,T] \to M</math>.  The [[arc length|length]] of {{math|γ}} is measured by
<math display="block">L(\gamma)=\sup_{0=x_0<x_1<\cdots<x_n=T} \left\{\sum_{k=1}^n d(\gamma(x_{k-1}),\gamma(x_k))\right\}.</math>
In general, this supremum may be infinite; a curve of finite length is called ''rectifiable''.{{sfn|Burago|Burago|Ivanov|2001|loc=Definition 2.3.1}}  Suppose that the length of the curve {{math|γ}} is equal to the distance between its endpoints—that is, it is the shortest possible path between its endpoints.  After reparametrization by arc length, {{math|γ}} becomes a ''[[geodesic]]'': a curve which is a distance-preserving function.{{sfn|Margalit|Thomas|2017}}  A geodesic is a shortest possible path between any two of its points.{{efn|This differs from usage in [[Riemannian geometry]], where geodesics are only locally shortest paths.  Some authors define geodesics in metric spaces in the same way.{{sfn|Burago|Burago|Ivanov|2001|loc=Definition 2.5.27}}{{sfn|Gromov|2007|loc=Definition 1.9}}}}

A ''geodesic metric space'' is a metric space which admits a geodesic between any two of its points.  The spaces <math>(\R^2,d_1)</math> and <math>(\R^2,d_2)</math> are both geodesic metric spaces.  In <math>(\R^2,d_2)</math>, geodesics are unique, but in <math>(\R^2,d_1)</math>, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.

The space {{mvar|M}} is a ''[[length space]]'' (or the metric {{mvar|d}} is ''intrinsic'') if the distance between any two points {{mvar|x}} and {{mvar|y}} is the infimum of lengths of paths between them.  Unlike in a geodesic metric space, the infimum does not have to be attained.  An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points {{math|(1, 0)}} and {{math|(-1, 0)}} can be joined by paths of length arbitrarily close to 2, but not by a path of length 2.  An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface.

Given any metric space {{math|(''M'', ''d'')}}, one can define a new, intrinsic distance function {{math|''d''<sub>intrinsic</sub>}} on {{mvar|M}} by setting the distance between points {{mvar|x}} and {{mvar|y}} to be the infimum of the {{mvar|d}}-lengths of paths between them.  For instance, if {{mvar|d}} is the straight-line distance on the sphere, then {{math|''d''<sub>intrinsic</sub>}} is the great-circle distance.  However, in some cases {{math|''d''<sub>intrinsic</sub>}} may have infinite values.  For example, if {{mvar|M}} is the [[Koch snowflake]] with the subspace metric {{mvar|d}} induced from <math>\R^2</math>, then the resulting intrinsic distance is infinite for any pair of distinct points.