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Revision as of 13:06, 27 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
Contents
Definition of a metric space
A metric space is a set [math]X[/math] coupled with a "distance function"[1]:
- [math]d:X\times X\rightarrow\mathbb{R}[/math] or sometimes
- [math]d:X\times X\rightarrow\mathbb{R}_+[/math][2], 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]:
- [math]d(x,y)\ge 0[/math] (This is implicit with the [ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/ilmath] definition)
- [math]d(x,y)=0\iff x=y[/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
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[3] 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.
- 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:
|
See also
Notes
- ↑ Note the strictly greater than 0 requirement for [ilmath]v[/ilmath]
References
- ↑ Introduction to Topology - Bert Mendelson
- ↑ Analysis - Part 1: Elements - Krzysztof Maurin
- ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
- ↑ Functional Analysis - George Bachman and Lawrence Narici