Extensions of (pseudo)metric spaces

[malarbol]

This PR introduces the following concepts:

• extension-Pseudometric-Space P: a pseudometric space Q with an isometry i : P → Q;
• extension-Metric-Space M: a metric space N with an isometry i : M → N;
• actions on Cauchy approximations of extensions of (pseudo)metric spaces;
• isometries between extensions of a metric space.

[fredrik-bakke]

@malarbol this pull request is very large. Is there any possibility of splitting it into smaller chunks, preferrably of less than 1000 lines of code each?

[malarbol]

@malarbol this pull request is very large. Is there any possibility of splitting it into smaller chunks, preferrably of less than 1000 lines of code each?

Should be better now.

[malarbol]

I'm re-drafting this to start over, again. The end game is still #1458 and Cauchy precompletions (metric quotients of Cauchy pseudocompletions). I found a way to formalize some of what I wanted, i.e. "any isometry from a metric space into a complete metric space uniquely extends to the Cauchy precompletion" or "any Cauchy-dense complete extension is isometrically equivalent to the Cauchy precompletion" (see #1776 for those interested) but maybe I don't actually need the concepts of extension of metric spaces to state them. I'll try another road; at least I know a little bit better where I go. And I'll do my best to keep my PRs under 1k LOC 😉.