isolated points

[affeldt-aist]

Motivation for this change

fixes #1775

define isolated points and properties and refine a lemma about closure

@IshiguroYoshihiro

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.

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[affeldt-aist]

The last commit adds a proof that there is a countable number of isolated points, this is need for a proof by @IshiguroYoshihiro

[affeldt-aist]

It looks like the introduction of isolated points allowed for a strict improvement of the lemma closure_isolated_limit_point since it replaces a union with a disjoint union and we also have a used for the new lemma countable_isolated (work by @IshiguroYoshihiro ). Do you (@zstone1 , @t6s ) think we are missing something?
(Also, CI errors are unrelated.)

[t6s]

I like these magic numbers. Looks like optimal choices.

[t6s]

infinite_prod_rat and infinite_prod_nat can be unified by adding following lemmas:

Lemma finite_setX_or T T' (A : set T) (B : set T') :
  finite_set (A `*` B) -> finite_set A \/ finite_set B.
Proof.
have [->|/set0P[a Aa]] := eqVneq A set0; first by left.
have: [set a] `*` B `<=` A `*` B by move=> x/= [] -> ?; split.
move/sub_finite_set/[apply]/(finite_image snd).
rewrite (_ : _ @` _ = B); first by right.
apply/seteqP; split=> b /=.
  by case=> -[] /= ? ? [] ? ? <-.
by move=> ?; exists (a, b).
Qed.

Lemma infinite_setX {T} {A B : set T} :
  infinite_set A -> infinite_set B -> infinite_set (A `*` B).
Proof.
by move=> iA iB; have /not_orP := conj iA iB; exact/contra_not/finite_setX_or.
Qed.

[t6s]

Looks good, modulo the refactoring comment.