metric structure

[affeldt-aist]

Motivation for this change

@zstone1

Checklist

• added corresponding entries in CHANGELOG_UNRELEASED.md

• added corresponding documentation in the headers

Reference: How to document

Merge policy

As a rule of thumb:

• PRs with several commits that make sense individually and that
  all compile are preferentially merged into master.
• PRs with disorganized commits are very likely to be squash-rebased.

Reminder to reviewers

• Read this Checklist
• Put a milestone if possible
• Check labels

[affeldt-aist]

@zstone1 could this be the first step towards addressing issue #1437 ?
This PR introduces a definition of metric space, move the lemma cvg_nbhsP from realfun.v to the directory topology_theory, incidentally together with at_left/at_right, which are now easier to move that normedtype.v has been split and reordered.

I propose to generalize a few more lemmas in the same manner, eventually moving a few things from normedtype_theory to topology_theory, so that we can arrive at a minimal PR introducing the metric structure without much disturbance for users. A complete generalization work can be carried incrementally. I am afraid that doing all the generalizations as a condition for merging will create a monster PR.

[CohenCyril]

One should add a factory with just the metric (with the triangular inequality) that builds balls from there.

[zstone1]

Makes sense to add this. I'll note that there are a handful of places (itll take me a minute to find them all) where we use pseudometric + hausdorff instead of this. Might be nice for a followup PR to add a factory for hausdorff + pseudometric => metric too, and upgrade those places.

[affeldt-aist]

Many thanks for the comments, I should be able to address them later this week.

[affeldt-aist]

One should add a factory with just the metric (with the triangular inequality) that builds balls from there.

I have added this factory. 118c342

Might be nice for a followup PR to add a factory for hausdorff + pseudometric => metric too, and upgrade those places.

I couldn't figure it right away so we can maybe postpone to the next release (I'll create an issue about it to remember it).