Difference between revisions of "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 $X$ coupled with a "distance function"^[1]^[2]:

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

With the properties that for $x,y,z\in X$ :

(This is implicit with the $d:X\times X\rightarrow\mathbb{R}_{\ge 0}$ definition)
$d(x,y)=0\iff x=y$
$d(x,y)=d(y,x)$ - Symmetry
$d(x,z)\le d(x,y)+d(y,z)$ - the Triangle inequality

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

Examples of metrics

Euclidian Metric

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

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

[Expand]
Proof that this is a metric

Discrete Metric

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

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

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

d:(x,y)↦{0if x=yvotherwise
- Where $v$ is some arbitrary member of $\mathbb{R}_{> 0}$ ^{[Note 1]} - traditionally (as mentioned) $v=1$ is used.

Note: however in proofs we shall always use the case $v=1$ for simplicity

Notes

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

Notes

Jump up ↑ Note the strictly greater than 0 requirement for $v$

References

Jump up ↑ Introduction to Topology - Bert Mendelson
↑ ^{Jump up to: 2.0} ^2.1 Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici

[5] Jump up ↑ Note the strictly greater than 0 requirement for $v$

[Topology-1] Jump up ↑ Introduction to Topology - Bert Mendelson

[ITTGG-2] {Jump up to: 2.0} ^2.1 Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene

[Analysis-3] Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin

[4] Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici

[1]

[2]

[3]

[4]

[Note 1]

@@ Line 1: / Line 1: @@
+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==
+A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>{{rITTGG}}:
-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>)
+* <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes
+* <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>
-# <math>d(x,y)\ge 0</math>
+With the properties that for <math>x,y,z\in X</math>:
+# <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)=d(y,x)</math>
+# <math>d(x,y)=d(y,x)</math> - Symmetry
-# <math>d(x,z)\le d(x,y)+d(y,z)</math> - the [[Triangle Inequality]]
+# <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})</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>
+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 on <math>\mathbb{R}^n</math>===
+===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{\prod^n_{i=1}(x_i^2+y_i^2)}</math>
+<math>d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}</math>
+{{Begin Theorem}}
-====Proof it is a metric====
+Proof that this is a metric
-{{Todo|Proof this is a metric}}
+{{Begin Proof}}
+{{Todo}}
+{{End Proof}}{{End Theorem}}
-===Discreet Metric===
+===[[Discrete metric and topology|Discrete Metric]]===
-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>.
+{{:Discrete metric and topology/Metric space definition}}
+====Notes====
+{{:Discrete metric and topology/Summary}}
-<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
+==See also==
-& x=y\\
+* [[Topology induced by a metric]]
-& \text{otherwise}
+* [[Connected space]]
-     \end{array}\right.</math>
+* [[Topological space]]
+==Notes==
+<references group="Note"/>
+==References==
+<references/>
 {{Definition|Topology|Metric Space}}

Difference between revisions of "Metric space"

Latest revision as of 06:07, 27 November 2015

Contents

Definition of a metric space

Examples of metrics

Euclidian Metric

Discrete Metric

Notes

See also

Notes

References

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