duality between closure and interior

[t6s]

Motivation for this change

This PR adds some lemmas for the closure and interior operators that
are a basic part of the duality between these. This addition will serve
as an initial step towards the theory of regular open/closed sets.

Lemma closureC is modified since its original statement seemed not
conforming to the name.

Checklist

• added corresponding entries in CHANGELOG_UNRELEASED.md

• added corresponding documentation in the headers

Reference: How to document

Reminder to reviewers

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

[affeldt-aist]

Quick thought: could the following definition be useful?

Definition idempotent_fun (U : Type) (f : U -> U) := f \o f =1 f.

[t6s]

Nice. Which file should it be put in?

[zstone1]

(oh wow, mathcomp's existing idempotent f predicate means f x x = x and not f (f x) = f x. I am always surprised by that one.) mathcomp-extra is a suitable place for this, I think

[affeldt-aist]

ssrfun.v? so, I guess for us it means mathcomp_extra.v...

[affeldt-aist]

I am always surprised by that one.

so maybe we could propose to rename it to idempotent_op?

[zstone1]

All looks good. I'm a bit surprised we didn't have some of this fundamental stuff before, so I'm glad to see you adding it.

[affeldt-aist]

Don't hesitate to use ^° when you can. @yoshihiro503 worked hard on making it clickable in the online documentation (https://math-comp.github.io/analysis/htmldoc_1_6_0/index.html) ^_^

[t6s]

Don't hesitate to use ^° when you can. @yoshihiro503 worked hard on making it clickable in the online documentation (https://math-comp.github.io/analysis/htmldoc_1_6_0/index.html) ^_^

The problem is, I cannot easily input that circle symbol. (Do you know any SKK keystroke for that?)
But anyway, I'll do some copy-pasting later.

[t6s]

idempotent_fun is added.
I have also introduced definitions regopen and regclosed to state generalized versions of the idempotent lemmas.
I am afraid that these names are not particularly good, but regular_open seems too long.