Totally bounded
From Maths
Revision as of 10:58, 1 December 2015 by Alec (Talk | contribs) (Created page with "==Definition== A metric space {{M|(X,d)}} is ''totally bounded'' if{{rITTGG}}: * {{M|1=\forall\epsilon>0\exists n\in\mathbb{N}\exists\{B_i\}_{i=1}^n\text{ of} }} open ba...")
Definition
A metric space [ilmath](X,d)[/ilmath] is totally bounded if[1]:
- [ilmath]\forall\epsilon>0\exists n\in\mathbb{N}\exists\{B_i\}_{i=1}^n\text{ of}[/ilmath] open balls[ilmath]\text{ of radius }\epsilon[X\subseteq\cup_{i=1}^n B_i][/ilmath], that is:
- [ilmath]\forall\epsilon>0[/ilmath] there exists a finite collection of open balls, each of radius [ilmath]\epsilon[/ilmath], such that the family of balls cover [ilmath]X[/ilmath]