Definition:Totally Bounded Metric Space

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Also known as
A totally bounded space is also referred to as a precompact space.

Equivalence of Definitions
These definitions are shown to be equivalent by Equivalence of Definitions of Total Boundedness.

Also see

 * Totally Bounded Metric Space is Bounded, but the converse is not true. (The simplest example is a countable set with the Standard discrete metric.)


 * Metric Space Compact iff Complete and Totally Bounded.