Difference between revisions of "Totally bounded"

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==Definition==
 
==Definition==
 
A [[metric space]] {{M|(X,d)}} is ''totally bounded'' if{{rITTGG}}:
 
A [[metric space]] {{M|(X,d)}} is ''totally bounded'' if{{rITTGG}}:

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Demote once content added - MOVE TO OWN PAGE, as there are probably other kinds of totally bounded

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]

References

  1. Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene