Editing Metric space (section)

===Simple examples===
====The real numbers====
The [[real number]]s with the distance function <math>d(x,y) = | y - x |</math> given by the [[absolute difference]] form a metric space.  Many properties of metric spaces and functions between them are generalizations of concepts in [[real analysis]] and coincide with those concepts when applied to the real line.

====Metrics on Euclidean spaces====
[[File:Minkowski_distance_examples.svg|thumb|Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard]]
The Euclidean plane <math>\R^2</math> can be equipped with many different metrics.  The [[Euclidean distance]] familiar from school mathematics can be defined by
<math display="block">d_2((x_1,y_1),(x_2,y_2))=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.</math>

The [[taxicab geometry|''taxicab'' or ''Manhattan'' distance]] is defined by
<math display="block">d_1((x_1,y_1),(x_2,y_2))=|x_2-x_1|+|y_2-y_1|</math>
and can be thought of as the distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article.

The ''maximum'', <math>L^\infty</math>, or ''[[Chebyshev distance]]'' is defined by
<math display="block">d_\infty((x_1,y_1),(x_2,y_2))=\max\{|x_2-x_1|,|y_2-y_1|\}.</math>
This distance does not have an easy explanation in terms of paths in the plane, but it still satisfies the metric space axioms. It can be thought of similarly to the number of moves a [[King (chess)|king]] would have to make on a [[chess]] [[Board game|board]] to travel from one point to another on the given space. 

In fact, these three distances, while they have distinct properties, are similar in some ways.  Informally, points that are close in one are close in the others, too.  This observation can be quantified with the formula
<math display="block">d_\infty(p,q) \leq d_2(p,q) \leq d_1(p,q) \leq 2d_\infty(p,q),</math>
which holds for every pair of points <math>p, q \in \R^2</math>.

A radically different distance can be defined by setting
<math display="block">d(p,q)=\begin{cases}0, & \text{if }p=q, \\ 1, & \text{otherwise.}\end{cases}</math>
Using [[Iverson bracket]]s,
<math display="block">d(p,q) = [p\ne q]</math>
In this ''discrete metric'', all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either.  Intuitively, the discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points.

All of these metrics make sense on <math>\R^n</math> as well as <math>\R^2</math>.

====Subspaces====
Given a metric space {{math|(''M'', ''d'')}} and a [[subset]] <math>A \subseteq M</math>, we can consider {{mvar|A}} to be a metric space by measuring distances the same way we would in {{mvar|M}}.  Formally, the ''induced metric'' on {{mvar|A}} is a function <math>d_A:A \times A \to \R</math> defined by
<math display="block">d_A(x,y)=d(x,y).</math>
For example, if we take the two-dimensional sphere {{math|S<sup>2</sup>}} as a subset of <math>\R^3</math>, the Euclidean metric on <math>\R^3</math> induces the straight-line metric on {{math|S<sup>2</sup>}} described above.  Two more useful examples are the open interval {{open-open|0, 1}} and the closed interval {{closed-closed|0, 1}} thought of as subspaces of the real line.