Editing Metric space (section)

{{Short description|Mathematical space with a notion of distance}}
{{use dmy dates|date=December 2020|cs1-dates=y}}
[[File:Manhattan distance.svg|thumb|200px|The [[Two-dimensional Euclidean space|plane]] (a set of points) can be equipped with different metrics.  In the [[Taxicab geometry|taxicab metric]] the red, yellow and blue paths have the same [[Arc length|length]] (12), and are all shortest paths. In the [[Euclidean metric]], the green path has length <math>6 \sqrt{2} \approx 8.49</math>, and is the unique shortest path, whereas the red, yellow, and blue paths still have length 12.]]

In [[mathematics]], a '''metric space''' is a [[Set (mathematics)|set]] together with a notion of ''[[distance]]'' between its [[Element (mathematics)|elements]], usually called [[point (geometry)|points]].  The distance is measured by a [[function (mathematics)|function]] called a '''metric''' or '''distance function'''.{{sfn|Čech|1969|p=42}}  Metric spaces are a general setting for studying many of the concepts of [[mathematical analysis]] and [[geometry]].

The most familiar example of a metric space is [[3-dimensional Euclidean space]] with its usual notion of distance.  Other well-known examples are a [[sphere]] equipped with the [[angular distance]] and the [[hyperbolic plane]].  A metric may correspond to a [[Conceptual metaphor |metaphorical]], rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the [[Hamming distance]], which measures the number of characters that need to be changed to get from one string to another.

Since they are very general, metric spaces are a tool used in many different branches of mathematics.  Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including [[Riemannian manifold]]s, [[normed vector space]]s, and [[graph (discrete mathematics)|graph]]s.  In [[abstract algebra]], the [[p-adic numbers|''p''-adic numbers]] arise as elements of the [[completion (metric space)|completion]] of a metric structure on the [[rational numbers]].  Metric spaces are also studied in their own right in '''metric geometry'''{{sfn|Burago|Burago|Ivanov|2001}} and '''analysis on metric spaces'''.{{sfn|Heinonen|2001}}

Many of the basic notions of [[mathematical analysis]], including [[ball (mathematics)|ball]]s, [[Complete metric space|completeness]], as well as [[uniform continuity|uniform]], [[Lipschitz continuity|Lipschitz]], and [[Hölder continuity]], can be defined in the setting of metric spaces.  Other notions, such as [[continuous function|continuity]], [[compactness]], and [[open set|open]] and [[closed set]]s, can be defined for metric spaces, but also in the even more general setting of [[topological space]]s.