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feat(Topology/Homeomorph): add Equiv.IsHomeomorph_iff and LinearEquiv.IsHomeomorph_iff #38660

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feat(Topology/Homeomorph): add Equiv.IsHomeomorph_iff and LinearEquiv.IsHomeomorph_iff #38660
sharky564 wants to merge 3 commits intoleanprover-community:masterfrom
sharky564:SK_IsHomeomorphism_Gaps

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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.
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 github-actions Bot added the new-contributor This PR was made by a contributor with at most 5 merged PRs. Welcome to the community! label Apr 28, 2026
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Welcome new contributor!

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github-actions Bot commented Apr 28, 2026

PR summary b9cac30121

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

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).

@github-actions github-actions Bot added the t-topology Topological spaces, uniform spaces, metric spaces, filters label Apr 28, 2026
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