The space is called precompact or totally bounded if for every there is a finite cover of by open balls of radius . Every totally bounded space is bounded. To see this, start with a finite cover by -balls for some arbitrary . Since the subset of consisting of the centers of these balls is finite, it has finite diameter, say . By the triangle inequality, the diameter of the whole space is at most . The converse does not hold: an example of a metric space that is bounded but not totally bounded is (or any other infinite set) with the discrete metric.
The space is called precompact or totally bounded if for every there is a finite cover of by open balls of radius . Every ?totally bounded space is bounded. To see this, start with a finite cover by -balls for some arbitrary . Since the subset of consisting of the centers of these balls is finite, it has finite diameter, say . By the triangle inequ,ality, the diameter of the whole space is at most . The converse does not hold: an example of a metric space that is bounded but not totally bounded is (or any other infinite set) with the discrete metric.