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

Latest revision as of 06:08, 27 November 2015

Let $X$ be a set. The discrete^[1] metric, or trivial metric^[2] 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)↦{0if x=yvotherwise
- 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

@@ 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:
 * {{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>
 ==Notes==