Difference between revisions of "Metric space"

Revision as of 20:22, 23 July 2015 (view source) Alec (Talk | contribs) m (→‎Notes: Typo) ← Older edit		Latest revision as of 06:07, 27 November 2015 (view source) Alec (Talk | contribs) m
(2 intermediate revisions by the same user not shown)
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]]		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==
−	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>{{rITTGG}}:
	* <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>, Note that here I prefer the notation <math>d:X\times X\rightarrow\mathbb{R}_{\ge 0}</math>		* <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>
Line 27:		Line 27:
	{{:Discrete metric and topology/Metric space definition}}		{{:Discrete metric and topology/Metric space definition}}
	====Notes====		====Notes====
−	{| class="wikitable" border="1"	+	{{:Discrete metric and topology/Summary}}
−	|-	+
−	! 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.	+
−	|}	+
	==See also==		==See also==

---

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

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

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^[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:

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

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