Original sentence: The term precompact (or pre-compact) is also used with the same meaning; but this terms is also used to mean relatively bounded (which is equivalent in a complete metric space); see also use of the axiom of choice below
Edited sentence: A metric space  is totally bounded.Each totally bounded space is bounded, but the converse is not true in general.For example, let M be the natural numbers and define d by setting.If M is Euclidean space and d is Euclidean distance, then.If the the diameter of the region is finite, then you need only finitely many towers to solve the problem.In a complete metric space.See also use of the axiom of choice below.The term precompact (or pre-compact) is sometimes used with the same meaning, but `pre-compact'  is also used to mean relatively bounded.This latter fact is related to a problem in telecommunications: In your warehouse, you have cell phone towers with transmitters/receivers whose range is.You wish to set up the towers in various locations so that each cell phone in a particular region can communicate with some tower.a finite subset F of M so that for each m in M we have.a subset (with the subspace topology) is totally bounded if and only if it is bounded.if and only if for every real number  there exist.if m and n are distinct  and.regardless of how small  may be.these meanings coincide but in general they do not