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feat(Topology): closed subspace of proper space is proper #21665

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@erdOne erdOne commented Feb 10, 2025


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@erdOne erdOne added awaiting-CI t-topology Topological spaces, uniform spaces, metric spaces, filters labels Feb 10, 2025
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PR summary a4fa4087d7

Import changes for modified files

Dependency changes

File Base Count Head Count Change
Mathlib.Topology.MetricSpace.Lipschitz 1090 1093 +3 (+0.28%)
Mathlib.Analysis.Complex.Basic 1479 1482 +3 (+0.20%)
Import changes for all files
Files Import difference
57 files Mathlib.Analysis.BoxIntegral.Basic Mathlib.Analysis.BoxIntegral.DivergenceTheorem Mathlib.Analysis.BoxIntegral.Partition.Additive Mathlib.Analysis.BoxIntegral.Partition.Measure Mathlib.Analysis.Calculus.FDeriv.Measurable Mathlib.Analysis.Calculus.LineDeriv.Measurable Mathlib.Analysis.Convex.Measure Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp Mathlib.Dynamics.Ergodic.Action.Basic Mathlib.Dynamics.Ergodic.Action.Regular Mathlib.MeasureTheory.Constructions.BorelSpace.Complex Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap Mathlib.MeasureTheory.Constructions.BorelSpace.Metric Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable Mathlib.MeasureTheory.Constructions.Pi Mathlib.MeasureTheory.Constructions.Polish.Basic Mathlib.MeasureTheory.Constructions.Polish.EmbeddingReal Mathlib.MeasureTheory.Constructions.UnitInterval Mathlib.MeasureTheory.Function.AEEqOfLIntegral Mathlib.MeasureTheory.Function.Egorov Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral Mathlib.MeasureTheory.Function.SimpleFuncDense Mathlib.MeasureTheory.Function.SpecialFunctions.Arctan Mathlib.MeasureTheory.Function.SpecialFunctions.Basic Mathlib.MeasureTheory.Function.SpecialFunctions.Inner Mathlib.MeasureTheory.Function.SpecialFunctions.RCLike Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic Mathlib.MeasureTheory.Function.StronglyMeasurable.ENNReal Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas Mathlib.MeasureTheory.Group.AEStabilizer Mathlib.MeasureTheory.Group.Action Mathlib.MeasureTheory.Group.LIntegral Mathlib.MeasureTheory.Group.Measure Mathlib.MeasureTheory.Group.Prod Mathlib.MeasureTheory.Integral.Indicator Mathlib.MeasureTheory.Integral.LebesgueNormedSpace Mathlib.MeasureTheory.Integral.Marginal Mathlib.MeasureTheory.Integral.MeanInequalities Mathlib.MeasureTheory.Measure.Complex Mathlib.MeasureTheory.Measure.Content Mathlib.MeasureTheory.Measure.ContinuousPreimage Mathlib.MeasureTheory.Measure.Haar.Basic Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace Mathlib.MeasureTheory.Measure.Haar.OfBasis Mathlib.MeasureTheory.Measure.Hausdorff Mathlib.MeasureTheory.Measure.Lebesgue.Basic Mathlib.MeasureTheory.Measure.Lebesgue.Complex Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar Mathlib.MeasureTheory.Measure.Regular Mathlib.MeasureTheory.Measure.RegularityCompacts Mathlib.NumberTheory.Transcendental.Liouville.Measure Mathlib.Probability.Distributions.Geometric Mathlib.Probability.Distributions.Poisson Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes Mathlib.Topology.MetricSpace.HausdorffDimension
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545 files Mathlib.Algebra.Order.AbsoluteValue.Equivalence Mathlib.Analysis.Analytic.Basic Mathlib.Analysis.Analytic.CPolynomial Mathlib.Analysis.Analytic.ChangeOrigin Mathlib.Analysis.Analytic.Composition Mathlib.Analysis.Analytic.Constructions Mathlib.Analysis.Analytic.Inverse Mathlib.Analysis.Analytic.IsolatedZeros Mathlib.Analysis.Analytic.IteratedFDeriv Mathlib.Analysis.Analytic.Linear Mathlib.Analysis.Analytic.OfScalars Mathlib.Analysis.Analytic.Order Mathlib.Analysis.Analytic.Polynomial Mathlib.Analysis.Analytic.RadiusLiminf Mathlib.Analysis.Analytic.Uniqueness Mathlib.Analysis.Analytic.Within Mathlib.Analysis.Asymptotics.AsymptoticEquivalent Mathlib.Analysis.Asymptotics.Lemmas Mathlib.Analysis.Asymptotics.SpecificAsymptotics Mathlib.Analysis.Asymptotics.SuperpolynomialDecay Mathlib.Analysis.Asymptotics.TVS Mathlib.Analysis.Asymptotics.Theta Mathlib.Analysis.CStarAlgebra.Basic Mathlib.Analysis.CStarAlgebra.Classes Mathlib.Analysis.CStarAlgebra.ContinuousLinearMap Mathlib.Analysis.CStarAlgebra.Exponential Mathlib.Analysis.CStarAlgebra.Matrix Mathlib.Analysis.CStarAlgebra.Multiplier Mathlib.Analysis.CStarAlgebra.Unitization Mathlib.Analysis.CStarAlgebra.lpSpace Mathlib.Analysis.Calculus.AddTorsor.AffineMap Mathlib.Analysis.Calculus.AddTorsor.Coord Mathlib.Analysis.Calculus.BumpFunction.Basic Mathlib.Analysis.Calculus.BumpFunction.InnerProduct Mathlib.Analysis.Calculus.Conformal.InnerProduct Mathlib.Analysis.Calculus.Conformal.NormedSpace Mathlib.Analysis.Calculus.ContDiff.Basic Mathlib.Analysis.Calculus.ContDiff.Bounds Mathlib.Analysis.Calculus.ContDiff.CPolynomial Mathlib.Analysis.Calculus.ContDiff.Defs Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno Mathlib.Analysis.Calculus.ContDiff.FiniteDimension Mathlib.Analysis.Calculus.ContDiff.RCLike Mathlib.Analysis.Calculus.ContDiff.WithLp Mathlib.Analysis.Calculus.DSlope Mathlib.Analysis.Calculus.Darboux Mathlib.Analysis.Calculus.Deriv.Abs Mathlib.Analysis.Calculus.Deriv.Add Mathlib.Analysis.Calculus.Deriv.AffineMap Mathlib.Analysis.Calculus.Deriv.Basic Mathlib.Analysis.Calculus.Deriv.Comp Mathlib.Analysis.Calculus.Deriv.Inv Mathlib.Analysis.Calculus.Deriv.Inverse Mathlib.Analysis.Calculus.Deriv.Linear Mathlib.Analysis.Calculus.Deriv.Mul Mathlib.Analysis.Calculus.Deriv.Pi Mathlib.Analysis.Calculus.Deriv.Polynomial Mathlib.Analysis.Calculus.Deriv.Pow Mathlib.Analysis.Calculus.Deriv.Prod Mathlib.Analysis.Calculus.Deriv.Shift Mathlib.Analysis.Calculus.Deriv.Slope Mathlib.Analysis.Calculus.Deriv.Star Mathlib.Analysis.Calculus.Deriv.Support Mathlib.Analysis.Calculus.Deriv.ZPow Mathlib.Analysis.Calculus.DiffContOnCl Mathlib.Analysis.Calculus.FDeriv.Add Mathlib.Analysis.Calculus.FDeriv.Analytic Mathlib.Analysis.Calculus.FDeriv.Basic Mathlib.Analysis.Calculus.FDeriv.Bilinear Mathlib.Analysis.Calculus.FDeriv.Comp Mathlib.Analysis.Calculus.FDeriv.Equiv Mathlib.Analysis.Calculus.FDeriv.Extend Mathlib.Analysis.Calculus.FDeriv.Linear Mathlib.Analysis.Calculus.FDeriv.Mul Mathlib.Analysis.Calculus.FDeriv.Norm Mathlib.Analysis.Calculus.FDeriv.Pi Mathlib.Analysis.Calculus.FDeriv.Prod Mathlib.Analysis.Calculus.FDeriv.RestrictScalars Mathlib.Analysis.Calculus.FDeriv.Star Mathlib.Analysis.Calculus.FDeriv.Symmetric Mathlib.Analysis.Calculus.FDeriv.WithLp Mathlib.Analysis.Calculus.FirstDerivativeTest Mathlib.Analysis.Calculus.FormalMultilinearSeries Mathlib.Analysis.Calculus.Gradient.Basic Mathlib.Analysis.Calculus.Implicit Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv Mathlib.Analysis.Calculus.InverseFunctionTheorem.FiniteDimensional Mathlib.Analysis.Calculus.IteratedDeriv.Defs Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas Mathlib.Analysis.Calculus.LHopital Mathlib.Analysis.Calculus.LagrangeMultipliers 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3

Declarations diff

+ ProperSpace.of_isClosed
+ instance : ProperSpace ℂ := lipschitz_equivRealProd.properSpace
- instance : ProperSpace ℂ

You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.


No changes to technical debt.

You can run this locally as

./scripts/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).

Comment on lines +126 to +131
lemma ProperSpace.of_isClosed {X : Type*} [PseudoMetricSpace X] [ProperSpace X]
{s : Set X} (hs : IsClosed s) :
ProperSpace s :=
⟨fun x r ↦ Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr
((isCompact_closedBall x.1 r).of_isClosed_subset
(hs.isClosedMap_subtype_val _ isClosed_ball) (Set.image_subset_iff.mpr subset_rfl))⟩
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I think I would prove this using LipschitzWith.properSpace, by first specializing it to closed isometric maps (which are automatically lipschitz and proper, and I think this intermediate statement is convenient for discoverability), and then apply it to the case of actual subtype inclusions. Is that too much of a mess for the imports?

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Yes, this is precisely what happened. I needed the result for closed subspaces, generalized the proof to proper lipschitz maps, proceeded to use the general lemma to prove it but then discovered that it will need to live in the lipschiz file and import isometry so I gave up.

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I added a comment to point to the general lemma.

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