Metric space: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Turms
(discrete metric + minor edits)
imported>Wlodzimierz Holsztynski
(→‎Metric on a set: non-negativity of a metric follows from the other axioms)
Line 4: Line 4:


== Metric on a set==
== Metric on a set==
Let <math>X\,</math> be an arbitrary set. A '''metric''' <math>d\,</math> on <math>X\,</math> is a function <math>d: X \times X \rightarrow \mathbb{R}</math> with the following properties:
Let <math>X\,</math> be an arbitrary set. A '''metric''' <math>d\,</math> on <math>X\,</math> is a function <math>d: X \times X \rightarrow \mathbb{R}</math> &nbsp; with the following properties:


#<math>d(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X</math> (non-negativity)
#<math>d(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X</math> &nbsp; (symmetry)
#<math>d(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X</math> (symmetry)
#<math>d(x_1,x_2)\leq d(x_1,x_3)+d(x_3,x_2) \quad \forall x_1,x_2,x_3 \in X</math> &nbsp; (triangular inequality)  
#<math>d(x_1,x_2)\leq d(x_1,x_3)+d(x_3,x_2) \quad \forall x_1,x_2,x_3 \in X</math>(triangular inequality)  
#<math>d(x_1,x_2)=0\ \Leftrightarrow \ x_1=x_2\,</math>
#<math>d(x_1,x_2)=0\,</math> if and only if <math>x_1=x_2\,</math>
 
It follows from the above three axioms of a metric (also called '''distance function''') that:
 
::<math>d(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X</math> &nbsp; (non-negativity)


== Formal definition of metric space ==
== Formal definition of metric space ==

Revision as of 09:17, 17 December 2007

This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, a metric space is, roughly speaking, an abstract mathematical structure that generalizes the notion of a Euclidean space which has been equipped with the Euclidean distance, to more general classes of sets such as a set of functions. A metric space consists of two components, a set and a metric on that set. On a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its associated set (for example, as an abstract distance between two functions in a set of functions) and induces a topology on the set called the metric topology.

Metric on a set

Let be an arbitrary set. A metric on is a function   with the following properties:

  1.   (symmetry)
  2.   (triangular inequality)

It follows from the above three axioms of a metric (also called distance function) that:

  (non-negativity)

Formal definition of metric space

A metric space is an ordered pair where is a set and is a metric on .

For shorthand, a metric space is usually written simply as once the metric has been defined or is understood.

Metric topology

A metric on a set induces a particular topology on called the metric topology. For any , let the open ball of radius around the point be defined as . Define the collection of subsets of (meaning that ) consisting of the empty set and all sets of the form:

where is an arbitrary index set (can be uncountable) and and for all . Then the set satisfies all the requirements to be a topology on and is said to be the topology induced by the metric . Any topology induced by a metric is said to be a metric topology.

Examples

  1. The "canonical" example of a metric space, and indeed what motivated the general definition of such a space, is the Euclidean space endowed with the Euclidean distance defined by for all .
  2. Consider the set of all real valued continuous functions on the interval with . Define the function by for all . This function is a metric on and induces a topology on often known as the norm topology or uniform topology.
  3. Let be any nonempty set. The discrete metric on is defined as if and otherwise. In this case the induced topology is the so called discrete topology.

See also

Topology

Topological space

Normed space


References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980