[stracer]

This is standard mathematical analysis. Infinite sequence of elements may look like it converges to some target element, judging by mutual distances converging to zero. Such sequence is a Cauchy sequence. When the target element actually exists, then the sequence is also convergent. A space where every Cauchy sequence is convergent, is called complete.

Example: if the space is all real numbers except 0, then any sequence of real numbers accumulating around 0 (for howsoever small a distance, there is always infinite number of points closer to 0), the sequence is a Cauchy sequence, but not convergent (because 0 is not present). So that space is not complete (has a hole).

If the space is all real numbers, then the same sequence is also convergent, and the space is complete (no holes).