completion

The completion of a metric space fills in its holes. You take a space with missing limits — sequences that bunch up but converge to nothing inside — and you formally add exactly the points they were reaching for, producing a complete space that contains the original as a faithful, dense copy.

Precisely, the completion of a metric space (X, d) is a complete metric space (X-hat, d-hat) together with an isometric embedding of X into X-hat whose image is dense. Such a completion always exists and is unique up to isometry. One standard construction takes the new points to be equivalence classes of Cauchy sequences in X, where two Cauchy sequences are identified when the distance between corresponding terms tends to 0.

The most famous completion is that of the rationals: completing Q under the usual metric yields the real numbers R. The original space sits densely inside its completion, so every new point is a limit of old ones; in this sense the completion adds nothing that was not already approached, it merely supplies the limits that were missing.