Difference between revisions of "Metric space"

Revision as of 22:26, 11 July 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 20:18, 23 July 2015 (view source) Alec (Talk | contribs) m Newer edit →
Line 3:		Line 3:
	A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>:		A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>:
	* <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes		* <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>	+	* <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>
	With the properties that for <math>x,y,z\in X</math>:		With the properties that for <math>x,y,z\in X</math>:
−	# <math>d(x,y)\ge 0</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)=0\iff x=y</math>
	# <math>d(x,y)=d(y,x)</math> - Symmetry		# <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==		==Examples of metrics==
Line 24:		Line 24:
	{{End Proof}}{{End Theorem}}		{{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====
		+	{| class="wikitable" border="1"
		+	|-
		+	! Property
		+	! Comment
		+	|-
		+	! [[Topology induced by a metric|induced topology]]
		+	| [[Discrete topology|discrete topology]] - which is the topology {{M|(X,\mathcal{P}(X))}} (where {{M|\mathcal{P} }} denotes [[Power set|power set]])
		+	|-
		+	! [[Open ball]]
		+	| {{M|1=B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r>1 \\ X & \text{otherwise}\end{array}\right. }}
		+	|-
		+	! [[Open set|Open sets]]
		+	| Every subset of {{M|X}} is open.<br/>'''Proof outline:''' as for a subset {{M|A\subseteq X}} we can show {{M|1=\forall x\in A\exists r[B_r(x)\subseteq A] }} by choosing {{M|r=\tfrac{1}{2} }} say, that is {{M|A}} contains an ''[[Open ball|open ball]]'' centred at each point in {{M|A}}.
		+	|-
		+	! [[Connected space|Connected]]
		+	| The [[Topological space|topology]] generated by {{M|(X,d_\text{discrete})}} is '''not''' connected if {{M|X}} has more than one point.<br/>'''Proof outline:'''
		+	:* Let {{M|A}} be any non empty subset of {{M|X}}, then define {{M|1=B:=A^c}} which is also a subset of {{M|X}}, thus {{M|B}} is open. Then {{M|1=A\cap B=\emptyset}} and {{M|1=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 {{M|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.
		+	|}
		+
−	It is given by:
−	* <math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
−	0 & x=y\\
−	1 & \text{otherwise}
−	\end{array}\right.</math>
−	'''Note:''' it is sometimes called the ''trivial'' metric<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>
−	{{Begin Theorem}}
−	Proof that this is a metric
−	{{Begin Proof}}
−	{{Todo|Really easy though}}
−	{{End Proof}}{{End Theorem}}
	==See also==		==See also==
		+	* [[Topology induced by a metric]]
		+	* [[Connected space]]
	* [[Topological space]]		* [[Topological space]]
−		+	==Notes==
		+	<references group="Note"/>
	==References==		==References==
	<references/>		<references/>
	{{Definition|Topology|Metric Space}}		{{Definition|Topology|Metric Space}}

---

Revision as of 20:18, 23 July 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]:

• $$d:X\times X\rightarrow\mathbb{R}$$ or sometimes
• $$d:X\times X\rightarrow\mathbb{R}_+$$ ^[2], Note that here I prefer the notation $$d:X\times X\rightarrow\mathbb{R}_{\ge 0}$$

With the properties that for

$$x,y,z\in X$$ :

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

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

Discrete Metric

Let $X$ be a set. The discrete^[3] 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:

• $$d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.$$
  • 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	$B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r>1 \\ X & \text{otherwise}\end{array}\right.$
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.

See also

• Topology induced by a metric
• Connected space
• Topological space

Notes

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

References

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