feat(Topology/Homeomorph): add Equiv.IsHomeomorph_iff and LinearEquiv.IsHomeomorph_iff #38660
feat(Topology/Homeomorph): add Equiv.IsHomeomorph_iff and LinearEquiv.IsHomeomorph_iff #38660sharky564 wants to merge 3 commits intoleanprover-community:masterfrom
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PR summary b9cac30121Import changes for modified filesNo significant changes to the import graph Import changes for all files
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This PR adds two characterisations of
IsHomeomorphfor bundled equivalences. For a plainEquivbetween topological spaces,Equiv.isHomeomorph_iffstates that the equivalence is a homeomorphism if and only if it is continuous in both directions. The corresponding statement for aLinearEquivbetween topological modules is added asLinearEquiv.isHomeomorph_iff, derived from theEquivversion.The motivation comes from work on topological complements of submodules in PR #38547, where one wants to upgrade a
LinearEquivto aContinuousLinearEquivand the cleanest characterisation is in terms of continuity in both directions. The existing API offersisHomeomorph_iff_exists_inverse, but for equivalences this requires manually identifying the existential inverse withe.symm, which is unnecessarily indirect.