What is upper completion?

In the field of mathematics, an upper completion is a way to construct a complete metric space by adding infinitely many points "at infinity" or "points of accumulation" to a given metric space. The resulting space is called the upper completion of the original space.

Formally, given a metric space (X, d), the upper completion is defined as the set of Cauchy sequences with values in X, modulo the equivalence relation that identifies two sequences if they have the same limit in X. The distance between two equivalence classes is defined as the limit of the distance between any two representatives of the classes.

The upper completion of a space is always a complete metric space and contains the original space as a dense subspace. It is unique up to isometry, meaning that any two upper completions of the same space are isometrically equivalent.

The upper completion is useful in various branches of mathematics, such as analysis, topology, and geometry, where complete spaces often play an important role. It also has applications in physics and engineering, where it provides a way to model phenomena that involve infinite or asymptotic behavior.