Difference between revisions of "Discrete metric and topology"
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==Metric space definition== | ==Metric space definition== | ||
{{:Discrete metric and topology/Metric space definition}} | {{:Discrete metric and topology/Metric space definition}} | ||
| + | ==Open balls== | ||
| + | The [[Open ball|open balls]] of {{M|X}} with the discrete topology are entirely {{M|X}} or a single point, that is: | ||
| + | {{Begin Theorem}} | ||
| + | * {{MM|1=B_r(x):=\{p\in X\vert\ d(x,p)<r\}=\left\{\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}\right.}} | ||
| + | {{Begin Proof}} | ||
| + | : By definition {{MM|1=B_r(x):=\{p\in X\vert\ d(x,p)<r\} }} note that for: | ||
| + | :* {{M|1=r\le 1}} we have {{M|1=B_r(x)=\{x\} }} as | ||
| + | :** {{M|1=d(x,p)< r \le 1}} so {{M|1=d(x,p)<1}} only when {{M|1=x=p}}, as if {{M|x\ne p}} then {{M|1=d(x,p)=1\not<1}} (proof by [[Contrapositive|contrapositive]]) | ||
| + | :* {{M|1=r> 1}} we have {{M|1=B_r(X)=X}} as | ||
| + | :** {{M|d(x,y)\le 1}} always, so we have {{M|\forall x,y\in X[d(x,y)\le 1< r]}} so {{M|\forall x,y\in X[d(x,y)< r]}} thus the ball contains every point in {{M|X}} | ||
| + | This completes the proof | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | ==Open sets== | ||
| + | The [[Open set|open sets]] of {{M|(X,d_\text{discrete})}} consist of every subset of {{M|X}} (the [[Power set|power set]] of {{M|X}}) - this is how the [[Topology induced by a metric|topology induced by the metric]] may be denoted {{M|(X,\mathcal{P}(X))}} | ||
| + | {{Begin Theorem}} | ||
| + | Every subset of {{M|X}} is an open set | ||
| + | {{Begin Proof}} | ||
| + | : Let {{M|A}} be a subset of {{M|X}}, we will show that {{M|\forall x\in A\exists r>0[B_r(x)\subseteq A]}} | ||
| + | {{Todo|Do this, it's easy enough see [[Metric space]] page for outline}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | |||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology|Metric Space}} | {{Definition|Topology|Metric Space}} | ||
Revision as of 20:40, 23 July 2015
Metric space definition
Let [ilmath]X[/ilmath] be a set. The discrete[1] metric, or trivial metric[2] 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
Open balls
The open balls of [ilmath]X[/ilmath] with the discrete topology are entirely [ilmath]X[/ilmath] or a single point, that is:
- [math]B_r(x):=\{p\in X\vert\ d(x,p)<r\}=\left\{\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}\right.[/math]
- By definition [math]B_r(x):=\{p\in X\vert\ d(x,p)<r\}[/math] note that for:
- [ilmath]r\le 1[/ilmath] we have [ilmath]B_r(x)=\{x\}[/ilmath] as
- [ilmath]d(x,p)< r \le 1[/ilmath] so [ilmath]d(x,p)<1[/ilmath] only when [ilmath]x=p[/ilmath], as if [ilmath]x\ne p[/ilmath] then [ilmath]d(x,p)=1\not<1[/ilmath] (proof by contrapositive)
- [ilmath]r> 1[/ilmath] we have [ilmath]B_r(X)=X[/ilmath] as
- [ilmath]d(x,y)\le 1[/ilmath] always, so we have [ilmath]\forall x,y\in X[d(x,y)\le 1< r][/ilmath] so [ilmath]\forall x,y\in X[d(x,y)< r][/ilmath] thus the ball contains every point in [ilmath]X[/ilmath]
- [ilmath]r\le 1[/ilmath] we have [ilmath]B_r(x)=\{x\}[/ilmath] as
This completes the proof
Open sets
The open sets of [ilmath](X,d_\text{discrete})[/ilmath] consist of every subset of [ilmath]X[/ilmath] (the power set of [ilmath]X[/ilmath]) - this is how the topology induced by the metric may be denoted [ilmath](X,\mathcal{P}(X))[/ilmath]
Every subset of [ilmath]X[/ilmath] is an open set
- Let [ilmath]A[/ilmath] be a subset of [ilmath]X[/ilmath], we will show that [ilmath]\forall x\in A\exists r>0[B_r(x)\subseteq A][/ilmath]
TODO: Do this, it's easy enough see Metric space page for outline
Notes
- ↑ Note the strictly greater than 0 requirement for [ilmath]v[/ilmath]
References
- ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
- ↑ Functional Analysis - George Bachman and Lawrence Narici
