Difference between revisions of "Discrete metric and topology/Metric space definition"
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| − | Let {{M|X}} be a set. The ''discrete'' | + | Let {{M|X}} be a set. The ''discrete''{{rITTGG}} metric, or ''trivial metric''<ref>Functional Analysis - George Bachman and Lawrence Narici</ref> is the [[Metric space|metric]] defined as follows: |
* {{MM|d:X\times X\rightarrow \mathbb{R}_{\ge 0} }} with {{MM|1=d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right. }} | * {{MM|d:X\times X\rightarrow \mathbb{R}_{\ge 0} }} with {{MM|1=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 {{M|x\ne y}} case. For example we could define {{M|d}} as: | However any strictly positive value will do for the {{M|x\ne y}} case. For example we could define {{M|d}} as: | ||
* {{MM|1=d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right. }} | * {{MM|1=d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right. }} | ||
| − | ** Where {{M|v}} is some arbitrary member of {{M|\mathbb{R}_{> 0} }}<ref group="Note">Note the ''strictly greater than 0'' requirement for {{M|v}}</ref> - traditionally (as mentioned) {{M|1=v=1}} is used. | + | ** Where {{M|v}} is some arbitrary member of {{M|\mathbb{R}_{> 0} }}<ref group="Note">Note the ''strictly greater than 0'' requirement for {{M|v}}</ref> - traditionally (as mentioned) {{M|1=v=1}} is used.<br/> |
| + | '''Note: however in proofs we shall always use the case {{M|1=v=1}} for simplicity''' | ||
<noinclude> | <noinclude> | ||
==Notes== | ==Notes== | ||
Latest revision as of 06:08, 27 November 2015
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
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
