[Merged by Bors] - feat(Topology/Homeomorph): add Equiv.IsHomeomorph_iff and LinearEquiv.IsHomeomorph_iff

[sharky564]

This PR adds two characterisations of IsHomeomorph for bundled equivalences. For a plain Equiv between topological spaces, Equiv.isHomeomorph_iff states that the equivalence is a homeomorphism if and only if it is continuous in both directions. The corresponding statement for a LinearEquiv between topological modules is added as LinearEquiv.isHomeomorph_iff, derived from the Equiv version.

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The motivation comes from work on topological complements of submodules in PR #38547, where one wants to upgrade a LinearEquiv to a ContinuousLinearEquiv and the cleanest characterisation is in terms of continuity in both directions. The existing API offers isHomeomorph_iff_exists_inverse, but for equivalences this requires manually identifying the existential inverse with e.symm, which is unnecessarily indirect.

[github-actions]

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[github-actions]

PR summary 606efcb73e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ Equiv.isHomeomorph_iff
+ LinearEquiv.isHomeomorph_iff

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to technical debt.

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[scholzhannah]

Thank you for your PR!

Two general comments:

Could you move the first paragraph of your PR description above the line? That way it ends up in the commit.

Could you explain in more detail what you need this for? Typically you want to use IsHomeomorph when the inverse map isn't nice or easy to describe and otherwise use Homeomorph. It seems to me that if you already have an Equiv you should always be in the second case. Could you explain why it is more comfortable to use IsHomeomorph in your application?

[ADedecker]

Typically you want to use IsHomeomorph when the inverse map isn't nice or easy to describe and otherwise use Homeomorph.

I think the point here is that we have a criterion of the form "this property is satisfied iff a certain equiv is an homeomorphism". Of course we then bundle it to an Homeomorph, but in order to state the iff the most convenient is really to use IsHomeomorph.

[sharky564]

Thanks for the review! I have moved the description paragraph.

On the motivation: ADedecker captured the use case exactly. The pattern in #38547 is a criterion of the shape "the submodules p and q are topological complements iff p.prodEquivOfIsCompl q h is a homeomorphism" and stating the iff directly in terms of IsHomeomorph is cleaner than constructing a Homeomorph inside the iff. The bundled Homeomorph is then built once on the forward direction.

[sharky564]

-awaiting-author

[scholzhannah]

Okay, thank you for explaining! This seems good to me then.

[robin-carlier]

Thanks!

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by robin-carlier.

[ADedecker]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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