complete metric space
A complete metric space is one with no holes: whenever a sequence bunches up so tightly that its terms become arbitrarily close to each other, there is actually a point waiting there for it to converge to. Nothing that ought to have a limit is left stranded.
Formally, a metric space (X, d) is complete if every Cauchy sequence in X converges to a point of X. A Cauchy sequence is one whose terms eventually cluster: for every epsilon > 0 there is an N with d(x_m, x_n) < epsilon for all m, n >= N. Completeness asserts that this internal tightening is enough to guarantee an actual limit inside the space.
The real numbers R are complete — this is what makes calculus work — but the rationals Q are not: the sequence 3, 3.1, 3.14, 3.141, ... of decimal approximations to pi is Cauchy in Q yet has no rational limit, because pi is irrational. Completeness is therefore not automatic; it is a strong and valuable property, and it is precisely what completion repairs.
Every convergent sequence is Cauchy in any metric space; completeness is the converse, that every Cauchy sequence converges. Completeness is not a topological property: it can be destroyed by switching to a different but equivalent metric. R and the open interval (-1, 1) are homeomorphic, yet R is complete and (-1, 1) with its usual metric is not.