Difference between revisions of "Discrete metric and topology/Metric space definition"

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.

Note: however in proofs we shall always use the case [ilmath]v=1[/ilmath] for simplicity

@@ Line 1: / Line 1: @@
-Let {{M|X}} be a set. The ''discrete''<ref>Todo - this is how I was taught, can't find source</ref> metric, or ''trivial metric''<ref>Functional Analysis - George Bachman and Lawrence Narici</ref> is the [[Metric space|metric]] defined as follows:
+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. }}
 However any strictly positive value will do for the {{M|x\ne y}} case. For example we could define {{M|d}} as: