Difference between revisions of "Metric space"

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A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see: [[Subtypes of topological spaces]]
 
==Definition of a metric space==
 
==Definition of a metric space==
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A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>{{rITTGG}}:
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A metric space is a set <math>X</math> coupled with a "distance function" <math>d:X\times X\rightarrow\mathbb{R}</math> with the properties (for <math>x,y,z\in X</math>)
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* <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes
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* <math>d:X\times X\rightarrow\mathbb{R}_+</math><ref name="Analysis">Analysis - Part 1: Elements - Krzysztof Maurin</ref>, Note that here I prefer the notation <math>d:X\times X\rightarrow\mathbb{R}_{\ge 0}</math>
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# <math>d(x,y)\ge 0</math>
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With the properties that for <math>x,y,z\in X</math>:
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# <math>d(x,y)\ge 0</math> (This is implicit with the {{M|1=d:X\times X\rightarrow\mathbb{R}_{\ge 0} }} definition)
 
# <math>d(x,y)=0\iff x=y</math>
 
# <math>d(x,y)=0\iff x=y</math>
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# <math>d(x,y)=d(y,x)</math>
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# <math>d(x,y)=d(y,x)</math> - Symmetry
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# <math>d(x,z)\le d(x,y)+d(y,z)</math> - the [[Triangle Inequality]]
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# <math>d(x,z)\le d(x,y)+d(y,z)</math> - the [[Triangle inequality]]
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We will denote a metric space as <math>(X,d)</math> (as <math>(X,d:X\times X\rightarrow\mathbb{R})</math> is too long and [[Mathematicians are lazy]]) or simply <math>X</math> if it is obvious which metric we are talking about on <math>X</math>
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We will denote a metric space as <math>(X,d)</math> (as <math>(X,d:X\times X\rightarrow\mathbb{R}_{\ge 0})</math> is too long and [[Mathematicians are lazy]]) or simply <math>X</math> if it is obvious which metric we are talking about on <math>X</math>
  
 
==Examples of metrics==
 
==Examples of metrics==
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For <math>x=(x_1,...,x_n)\in\mathbb{R}^n</math> and <math>y=(y_1,...,y_n)\in\mathbb{R}^n</math> we define the Euclidian metric by:
 
For <math>x=(x_1,...,x_n)\in\mathbb{R}^n</math> and <math>y=(y_1,...,y_n)\in\mathbb{R}^n</math> we define the Euclidian metric by:
  
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<math>d_{\text{Euclidian}}(x,y)=\sqrt{\prod^n_{i=1}((x_i-y_i)^2)}</math>
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<math>d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}</math>
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{{Begin Theorem}}
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====Proof it is a metric====
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Proof that this is a metric
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{{Todo|Proof this is a metric}}
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{{Begin Proof}}
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{{Todo}}
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{{End Proof}}{{End Theorem}}
  
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===Discreet Metric===
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===[[Discrete metric and topology|Discrete Metric]]===
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This is a useless metric, but is a metric and induces the Discreet [[Topological space|Topology]] on X, where the topology is the powerset of <math>X</math>, <math>\mathcal{P}(X)</math>.
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{{:Discrete metric and topology/Metric space definition}}
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====Notes====
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{{:Discrete metric and topology/Summary}}
  
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<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
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==See also==
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      1 & x=y\\
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* [[Topology induced by a metric]]
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      0 & \text{otherwise}
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* [[Connected space]]
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    \end{array}\right.</math>
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* [[Topological space]]
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
  
 
{{Definition|Topology|Metric Space}}
 
{{Definition|Topology|Metric Space}}

Latest revision as of 06:07, 27 November 2015

A normed space is a special case of a metric space, to see the relationships between metric spaces and others see: Subtypes of topological spaces

Definition of a metric space

A metric space is a set [math]X[/math] coupled with a "distance function"[1][2]:

  • [math]d:X\times X\rightarrow\mathbb{R}[/math] or sometimes
  • [math]d:X\times X\rightarrow\mathbb{R}_+[/math][3], Note that here I prefer the notation [math]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/math]

With the properties that for [math]x,y,z\in X[/math]:

  1. [math]d(x,y)\ge 0[/math] (This is implicit with the [ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/ilmath] definition)
  2. [math]d(x,y)=0\iff x=y[/math]
  3. [math]d(x,y)=d(y,x)[/math] - Symmetry
  4. [math]d(x,z)\le d(x,y)+d(y,z)[/math] - the Triangle inequality

We will denote a metric space as [math](X,d)[/math] (as [math](X,d:X\times X\rightarrow\mathbb{R}_{\ge 0})[/math] is too long and Mathematicians are lazy) or simply [math]X[/math] if it is obvious which metric we are talking about on [math]X[/math]

Examples of metrics

Euclidian Metric

The Euclidian metric on [math]\mathbb{R}^n[/math] is defined as follows: For [math]x=(x_1,...,x_n)\in\mathbb{R}^n[/math] and [math]y=(y_1,...,y_n)\in\mathbb{R}^n[/math] we define the Euclidian metric by:

[math]d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}[/math]

Proof that this is a metric


TODO:


Discrete Metric

Let [ilmath]X[/ilmath] be a set. The discrete[2] metric, or trivial metric[4] is the metric defined as follows:

  • [math]d:X\times X\rightarrow \mathbb{R}_{\ge 0} [/math] with [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right.[/math]

However any strictly positive value will do for the [ilmath]x\ne y[/ilmath] case. For example we could define [ilmath]d[/ilmath] as:

  • [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.[/math]
    • Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath][Note 1] - traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.

Note: however in proofs we shall always use the case [ilmath]v=1[/ilmath] for simplicity

Notes

Property Comment
induced topology discrete topology - which is the topology [ilmath](X,\mathcal{P}(X))[/ilmath] (where [ilmath]\mathcal{P} [/ilmath] denotes power set)
Open ball [ilmath]B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right.[/ilmath]
Open sets Every subset of [ilmath]X[/ilmath] is open.
Proof outline: as for a subset [ilmath]A\subseteq X[/ilmath] we can show [ilmath]\forall x\in A\exists r[B_r(x)\subseteq A][/ilmath] by choosing say, that is [ilmath]A[/ilmath] contains an open ball centred at each point in [ilmath]A[/ilmath].
Connected The topology generated by [ilmath](X,d_\text{discrete})[/ilmath] is not connected if [ilmath]X[/ilmath] has more than one point.
Proof outline:
  • Let [ilmath]A[/ilmath] be any non empty subset of [ilmath]X[/ilmath], then define [ilmath]B:=A^c[/ilmath] which is also a subset of [ilmath]X[/ilmath], thus [ilmath]B[/ilmath] is open. Then [ilmath]A\cap B=\emptyset[/ilmath] and [ilmath]A\cup B=X[/ilmath] thus we have found a separation, a partition of non-empty disjoint open sets, that separate the space. Thus it is not connected
  • if [ilmath]X[/ilmath] has only one point then we cannot have a partition of non empty disjoint sets. Thus it cannot be not connected, it is connected.

See also

Notes

  1. ↑ Note the strictly greater than 0 requirement for [ilmath]v[/ilmath]

References

  1. ↑ Introduction to Topology - Bert Mendelson
  2. ↑ 2.0 2.1 Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
  3. ↑ Analysis - Part 1: Elements - Krzysztof Maurin
  4. ↑ Functional Analysis - George Bachman and Lawrence Narici
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