Abstract theory of Hausdorff completions of uniform spaces #

This file characterizes Hausdorff completions of a uniform space α as complete Hausdorff spaces equipped with a map from α which has dense image and induce the original uniform structure on α. Assuming these properties we "extend" uniformly continuous maps from α to complete Hausdorff spaces to the completions of α. This is the universal property expected from a completion. It is then used to extend uniformly continuous maps from α to α' to maps between completions of α and α'.

This file does not construct any such completion, it only study consequences of their existence. The first advantage is that formal properties are clearly highlighted without interference from construction details. The second advantage is that this framework can then be used to compare different completion constructions. See topology/uniform_space/compare_reals for an example. Of course the comparison comes from the universal property as usual.

A general explicit construction of completions is done in uniform_space/completion, leading to a functor from uniform spaces to complete Hausdorff uniform spaces that is left adjoint to the inclusion, see uniform_space/UniformSpace for the category packaging.

Implementation notes #

A tiny technical advantage of using a characteristic predicate such as the properties listed in abstract_completion instead of stating the universal property is that the universal property derived from the predicate is more universe polymorphic.

References #

We don't know any traditional text discussing this. Real world mathematics simply silently identify the results of any two constructions that lead to something one could reasonnably call a completion.

Tags #

uniform spaces, completion, universal property