chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
23,999 | lean_file_chunk | a3a997d33d9e81106bff9676646f1fb250549e2ef76cb6b5fce0372b5d9e6d1a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Data.Fintype.Lattice
public import Mathlib.Data.Fintype.Sum
public import Mathlib.Topology.Homeomorph.Lemmas
public import Mathlib... |
4,058 | lean_file_chunk | 00c5fdafab5c38b71bbd55cca185f0f324e478ecb96d438054276a13d35be8f7 | lean_source_file | pkg_mathlib_source_file_chunks | @[simp]
theorem _root_.Fin.appendIsometry_toHomeomorph (m n : ℕ) :
(Fin.appendIsometry m n).toHomeomorph = Fin.appendHomeomorph (X := α) m n :=
rfl
/-- The natural `IsometryEquiv` `(Fin m → ℝ) × (Fin l → ℝ) ≃ᵢ (Fin n → ℝ)` when `m + l = n`. -/
@[simps!]
def _root_.Fin.appendIsometryOfEq {n m l : ℕ} (hmln : m + l... |
5,430 | lean_file_chunk | 423e999a9a49ae98792a66440e6d910a23f0a921df4bfc00e8a558b50babbc44 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
module
public import Mathlib.Topology.Perfect
public import Mathlib.Topology.MetricSpace.Polish
public import Mathlib.Topology.MetricSpace.CantorScheme
public impo... |
6,337 | lean_file_chunk | c26b61eed1080b6d9f2ead20088ebff8c0bc8eed36c36b5445fbed6bf68eeb92 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Lp.lpSpace
public import Mathlib.Topology.Sets.Compacts
/-!
# The Kuratowski embedding
Any separable metric spac... |
7,493 | lean_file_chunk | c7c5f9de07826797714e2f50b0146b7ff74329dd6a4f825b1f4926b2d7b84a09 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
module
public import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
public import Mathlib.Topology.EMet... |
18,233 | lean_file_chunk | 8d407d28556c16c1a624fc9d1f1c07bfaa78b833568379af49b93083792f277f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.MetricSpace.Lipschitz
public import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
public import Mathlib.Analysis.Convex.NN... |
10,414 | lean_file_chunk | 7110c083112ac659c6f0091f55aff32efdbf194f0749ef315f79f8698401c3a1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.Algebra.Ring.Real
public import Mathlib.Topology.Algebra.UniformRing
public import Mathlib.Topology.MetricSpace.Algebra
p... |
7,736 | lean_file_chunk | 187236631df63ab7840b40cc40a4b1663823bdbe201134118e472934fc1d9b9c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Data.Rel.Cover
public import Mathlib.Topology.MetricSpace.MetricSeparated
public import Mathlib.Topology.MetricSpace.Thickening
/-!
# Cover... |
7,744 | lean_file_chunk | 96a8f2934297c332c58ade5d2e2810992fed66bf2fd57256bee6f85df1fd819e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
module
public import Mathlib.Algebra.Order.Monoid.Defs
public import Mathlib.Data.Finset.Lattice.Fold
public import Mathlib.Data.Rel
/-!
# Pseudometrics as bundle... |
21,301 | lean_file_chunk | 9c7b4a74d9d5a08c92a2127eded2a0131b77a121ca14c753899a92c629bec3aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Topology.MetricSpace.HausdorffDistance
public import Mathlib.Topology.UniformS... |
15,419 | lean_file_chunk | 33bfee916dc2d114aae146dcca7b100d6a060e3fb0f193342e8ea7df5a9d6a96 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
module
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Data.Setoid.Basic
public ... |
7,007 | lean_file_chunk | dfab6331e010d22aa203b1f420be18679b02c0712f9a442050bc8b9616ae89e8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
module
public import Mathlib.Data.Rel.Separated
public import Mathlib.Topology.EMetricSpace.Defs
public import Mathlib.Topology.MetricS... |
12,074 | lean_file_chunk | 195e99536aa213abe7b0a92c0d8d8966b289b9b3eccf611ad9f7dc7db3764ed6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
module
public import Mathlib.Topology.MetricSpace.Pseudo.Basic
public import Mathlib.Topology.Metri... |
14,026 | lean_file_chunk | 4f9d39fd0ff3566cf0131e89dd7ae8df9593b2cb56d060f29b3d625e2135106a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Order.CompleteLattice.Group
public import Mathlib.Topology.ContinuousMap.Bounded.Basic
public import Mathlib.Topology.ContinuousMap.Compac... |
3,453 | lean_file_chunk | fadcb89e385f6d7e85bb73ed22ba24093ecb0d23bf995a1d712132e0bcd62f0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Field.Basic
public import Mathlib.Topology.MetricSpace.Cauchy
/-!
# Completeness in terms of `Cauc... |
12,645 | lean_file_chunk | 89a3221f3c20bf77fcb05735077f4a3c0f82fd983979cf8c9071e32b08710504 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jovan Gerbscheid. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jovan Gerbscheid, Newell Jensen
-/
module
public import Mathlib.Topology.MetricSpace.Congruence
public import Mathlib.Topology.MetricSpace.Dilation
public import Mathlib.Tactic.FinC... |
11,062 | lean_file_chunk | 78b765391596d5a5ecef3d75746fc79cd9c1d56b00e4a1b9f4d9f6f8c1eb9a97 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.UniformSpace.CompleteSeparated
public import Mathlib.Topology.EMetricSpace.Lipschitz
public import Mathlib.Topology.MetricSpa... |
23,962 | lean_file_chunk | a4a449827993d878182bc0b4ce84505b899a4f383a6a50fce40656f6409c8099 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Order.ConditionallyCompleteLattice.Group
public import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing tw... |
5,425 | lean_file_chunk | 1850e8af2d99685b59f95e4726a35cc65b152bebdc4ea019f69f5070d9221553 | lean_source_file | pkg_mathlib_source_file_chunks | def inductiveLimitDist (f : ∀ n, X n → X (n + 1)) (x y : Σ n, X n) : ℝ :=
dist (leRecOn (le_max_left x.1 y.1) (f _) x.2 : X (max x.1 y.1))
(leRecOn (le_max_right x.1 y.1) (f _) y.2 : X (max x.1 y.1))
/-- The predistance on the disjoint union `Σ n, X n` can be computed in any `X k` for large
enough `k`. -/
theore... |
9,771 | lean_file_chunk | 116cdba1631553f31efaf0866917dcd4d41153240f0e1c9bd4fb7216f6b7c879 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
module
public import Mathlib.Topology.MetricSpace.Pseudo.Defs
/-!
# Metric spaces
This file defin... |
19,405 | lean_file_chunk | a63f6d8457f3a1fff80383179235334abb8bc2ceca306419972db5f5f1c0a881 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
public import Mathlib.Topology.Algebra.ConstMulAction
public import Mathlib.Topology.MetricS... |
15,295 | lean_file_chunk | e7940aecbf1c42033464282880be7dc59f1c52f16700e6164fd7af9e4dcfad76 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.MetricSpace.PiNat
public import Mathlib.Topology.Metrizable.CompletelyMetrizable
public import Mathlib.Topology.Sets.Open... |
8,458 | lean_file_chunk | c963e12e09672f1a907c9518780f74fdeee46ebf74c204b56fbc6eeb069a7ee4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.MetricSpace.Dilation
/-!
# Dilation equivalence
In this file we define `DilationEquiv X Y`, a type of bundled equivalences ... |
15,980 | lean_file_chunk | 0cde2d04fe98160067a80451e49e148f6176aeef1d032fe6578a4ac5c7bd39ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
module
public import Mathlib.Order.Interval.Set.ProjIcc
public import Mathlib.Topology.Bornology.Hom
publi... |
23,947 | lean_file_chunk | 1514e211fb3b5a583259b227f0b846d2dcff785b888cf8a2a8bf4ff27d961a16 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli
public import Mathlib.Topology.ContinuousMap.Bounded.Normed
public import Mathlib.Topo... |
1,903 | lean_file_chunk | aae7af523f43f082776a13444420efb8c6d36feaa4d863dca5e77b2a174b7113 | lean_source_file | pkg_mathlib_source_file_chunks | optimal coupling. This follows from the fact that `HD` of the optimal candidate is exactly
the Hausdorff distance in the optimal coupling, although we only prove here the inequality
we need. -/
theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
hausdorffDist (range (optimalGHInjl X Y)) (range (opti... |
6,235 | lean_file_chunk | 4ee08f7234d715d6febc7814c6208dd9e2c4eb685ef7bb5a03db1f65b9afa29f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.EMetricSpace.Paracompact
public import Mathlib.Topology.MetricSpace.Basic
public import Mathlib.Topology.MetricSpace.ProperSp... |
6,374 | lean_file_chunk | 15de1bf0e7b395819d3e968a9652b3ffd6f07ed103a3988b44e4acdab3beca45 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
module
public import Mathlib.Topology.MetricSpace.PiNat
/-!
# (Topological) Schemes and their induced maps
In topology, and especially descriptive set theory, on... |
23,959 | lean_file_chunk | 353944c0a354c908d90aad5f291c605e6bcd643240d3833fb2a4b1e8fcd999dd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
module
public import Mathlib.Topology.Order.Bornology
public import Mathlib.Topology.Order.Compact
... |
5,429 | lean_file_chunk | 0ce2c50b55140963243e2b884fd3c8b5e72679edd3b0fef69c4aca3f16f04463 | lean_source_file | pkg_mathlib_source_file_chunks | diam_le_of_subset_closedBall h ball_subset_closedBall
/-- If a family of complete sets with diameter tending to `0` is such that each finite intersection
is nonempty, then the total intersection is also nonempty. -/
theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
(h0 : IsComplete (s 0... |
17,405 | lean_file_chunk | d49feab0c4288b33876445a084eed9af26d5ea24e2fffa9eb66e7abfb9ea1dbb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Markus Himmel, Lorenzo Luccioli, Alessio Rondelli, Etienne Marion
-/
module
public import Mathlib.Data.ENat.Lattice
public import Mathlib.Data.Set.Card
public import Mathl... |
5,467 | lean_file_chunk | bc4725893991853d701043b1b2fb86023eed532351e061d92072908055cf527b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.UniformSpace.Equicontinuity
public import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
/-!
# Equicontinuity in metric spaces... |
23,918 | lean_file_chunk | c8b28e66f7532d1ba80c7ca88e490c76ac3e079053174bea324e02f6f2bc6fdc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.SpecificLimits.Basic
public import Mathlib.Topology.MetricSpace.IsometricSMul
public import Mathlib.Tactic.Finiteness
/-... |
20,320 | lean_file_chunk | a934f739463cd922d06cdfcc143420ccb342f8424b43d44c4d304279f66969c5 | lean_source_file | pkg_mathlib_source_file_chunks | @[deprecated (since := "2026-01-08")] alias hausdorffEdist_image := hausdorffEDist_image
@[deprecated (since := "2026-01-08")] alias hausdorffEdist_le_ediam := hausdorffEDist_le_ediam
@[deprecated (since := "2026-01-08")] alias hausdorffEdist_triangle := hausdorffEDist_triangle
@[deprecated (since := "2026-01-08")]
al... |
7,702 | lean_file_chunk | b46d7f48d65e3a1d32e5b49b37912a4aabcc26c4fb2499c832a276e20fe993d4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jovan Gerbscheid. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jovan Gerbscheid, Newell Jensen
-/
module
public import Mathlib.Topology.MetricSpace.Pseudo.Defs
public import Mathlib.Topology.MetricSpace.Isometry
public import Mathlib.Topology.M... |
15,189 | lean_file_chunk | 37aca9afcfc77b72471b9105c70721006300224e0902d0da30278d231aef74ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
module
public import Mathlib.Topology.Algebra.MulAction
public import Mathlib.Topology.Algebra.SeparationQuotient.Basic
public import Mathlib.Topology.Algebra.Unif... |
8,911 | lean_file_chunk | 1e49c100ee024baeaa98d9b215800be3e199d7a1b4a9b666a2038a2fcabc94eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.EMetricSpace.Paracompact
public import Mathlib.Topology.Instances.ENNReal.Lemmas
public import Mathlib.Analysis.Convex.Partit... |
23,985 | lean_file_chunk | f349a600001a43dcacf18e7f85dadb071a1d3d33cd632716f682483ba57983ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Normed.Group.FunctionSeries
public import Mathlib.Topology.Algebra.MetricSpace.Lipschitz
public import Mathlib.Topology.M... |
23,990 | lean_file_chunk | e6fedf9f31ebb66ad54dbf10cb5ad8cb20851dab38359ccfd75899a76094737f | lean_source_file | pkg_mathlib_source_file_chunks | /-- Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a Lipschitz retraction
onto this set, i.e., a Lipschitz map with range equal to `s`, equal to the identity on `s`. -/
theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
(hne : s.Nonempty) :
∃ f : (∀ n, E... |
3,902 | lean_file_chunk | ad100bb20762d5280fc865ffbae0017e921e0994103147de85c37594167864e1 | lean_source_file | pkg_mathlib_source_file_chunks | /-- If `X` is compact, and there exists a sequence of continuous functions `f i : X → Y i` to
metric spaces `Y i` that separate points on `X`, then `X` is metrizable. -/
lemma TopologicalSpace.MetrizableSpace.of_countable_separating (f : ∀ i, X → Y i)
(continuous_f : ∀ i, Continuous (f i)) (separating_f : Pairwise ... |
1,557 | lean_file_chunk | 2eb2109f29665a64480b25cb244e629f4af67aec74799f60e8e9126719ace030 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Topology.Sequences
public import Mathlib.Topology.MetricSpace.Bounded
/-!
# Sequential compacts ... |
4,986 | lean_file_chunk | f748ce5948c5b60a51bb459552d2db40f906f29c5221c554a6ba8d992f5905a2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Vasilii Nesterov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vasilii Nesterov
-/
module
public import Mathlib.Topology.Instances.CantorSet
public import Mathlib.Topology.MetricSpace.PiNat
/-!
# Hausdorff–Alexandroff Theorem
In this file, we... |
19,060 | lean_file_chunk | dfb48c1b7bf50778268e23c253bb5b329e919d0fe87be9e0f508d069765008e7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
module
public import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
public import Mathlib.Topology.EMetricSpace.Paracompact
public import Mathlib.Topology.Sepa... |
4,731 | lean_file_chunk | f560727574480ce1555057a9cfe594c89ca7a1c86328aa46bc67e96ec0cfa9aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.Semicontinuity.Basic
public import Mathlib.Topology.Compactness.Lindelof
/-!
# Envelopes of Semicontinuous functions on He... |
23,908 | lean_file_chunk | 1f42193b94b995f2090e9a4b6a3ebb92e149d2f8aafbb902dcc1233f4b767875 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Antoine Chambert-Loir, Anatole Dedecker
-/
module
public import Mathlib.Topology.Semicontinuity.Defs
public import Mathlib.Algebra.GroupWithZero.Indicator
public... |
23,968 | lean_file_chunk | 5b45b5a070ab588a161a2ac1481cddbdea51d627c01ebd52ea51b7a21b9c716f | lean_source_file | pkg_mathlib_source_file_chunks | /-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with
`[ContinuousAdd]`. The primed version of the lemma uses an explicit continuity assumption on
addition, for application to `EReal`. -/
theorem LowerSemicontinuous.add {f g : α → γ} (hf : LowerSemicontinuous f)
(hg : LowerSemic... |
9,255 | lean_file_chunk | 1bcbdea68c1d49f5d24bb40d80c38c3dd135ef488536658288063159b0b8e167 | lean_source_file | pkg_mathlib_source_file_chunks | addition, for application to `EReal`. -/
theorem UpperSemicontinuousAt.add {f g : α → γ} (hf : UpperSemicontinuousAt f x)
(hg : UpperSemicontinuousAt g x) : UpperSemicontinuousAt (fun z => f z + g z) x :=
hf.add' hg continuous_add.continuousAt
/-- The sum of two upper semicontinuous functions is upper semicontin... |
23,921 | lean_file_chunk | 73d0ff2bb8d12eb1183e43896dc1f68df62479e2686aabb6354ea58f9fb8b89e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Antoine Chambert-Loir, Anatole Dedecker, Jireh Loreaux
-/
module
public import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.ContinuousOn
/-!
# Semicont... |
12,023 | lean_file_chunk | 014140bd1e6c7d555c029567e95c4b3a36ebaa374d482a7bfb84afebe4407608 | lean_source_file | pkg_mathlib_source_file_chunks | /-- A function `f : α → Set β` is upper hemicontinuous at `x` if, whenever `t` is a neighborhood of
`f x`, then `t` is a neighborhood of `f x'` for all `x'` sufficiently close to `x`. -/
abbrev UpperHemicontinuousAt (f : α → Set β) (x : α) :=
SemicontinuousAt (fun x t ↦ t ∈ 𝓝ˢ (f x)) x
/-- A function `f : α → Set β... |
15,475 | lean_file_chunk | 502d59f2045fc3ed9568c19fcbcead5441e39ecb003dde2e8cc5e5a55d60da0a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Topology.Semicontinuity.Defs
public import Mathlib.Topology.NhdsWithin
public import Mathlib.Topology.Separation.Regular
public import Mat... |
23,991 | lean_file_chunk | fae7869cd27f5d9eaa05ed26b495e49902c0051ff5f15a90103d79436dfcdc18 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.InnerProductSpace.LinearMap
public import Mathlib.Topology.VectorBundle.Constructions
public import Mathlib.Topology.Vect... |
2,344 | lean_file_chunk | 05d3a91b67c0d8c85be2db5a0b88f6dcf5102cbe1622302a55388dd5bf468c59 | lean_source_file | pkg_mathlib_source_file_chunks | InnerProductSpace ℝ (E b) := fast_instance%
.ofCoreOfTopology (h.g.toCore b) (h.g.continuousAt b) (h.g.isVonNBounded b)
variable (F E) in
/-- A family of inner product space structures on the fibers of a fiber bundle, defining the same
topology as the already existing one, and varying continuously with the base poin... |
23,967 | lean_file_chunk | ee2dcf764bfe48652011296c4de8050c5f2e616a4ac3999e8e34e347196de05f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
module
public import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
public import Mathl... |
18,467 | lean_file_chunk | c71357938b7afbb820f891cdea45e2395a1dd89fdd433d21eedaf5f0d412820b | lean_source_file | pkg_mathlib_source_file_chunks | ⟨trivialVectorBundleCore R B F ι⟩
namespace VectorBundleCore
variable {R B F} {ι : Type*}
variable (Z : VectorBundleCore R B F ι)
/-- Natural identification to a `FiberBundleCore`. -/
@[simps (attr := mfld_simps) -fullyApplied]
def toFiberBundleCore : FiberBundleCore ι B F :=
{ Z with
coordChange := fun i j b ... |
8,191 | lean_file_chunk | 6bd73cd23ce8b921a546c7bb81ff67e15fb8ef7c2528d699c0901267cdb9551f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
module
public import Mathlib.Topology.FiberBundle.Constructions
public import Mathlib.Topology.VectorBundle... |
23,999 | lean_file_chunk | f8df6cc4e553a036343be7131f8b0f6bfd229bb97d3bc229bdac78ef4222366c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Floris van Doorn
-/
module
public import Mathlib.Topology.VectorBundle.Basic
/-!
# The vector bundle of continuous (semi)linear maps
We define the (topological) ve... |
1,992 | lean_file_chunk | 351e5f112ddbd879e9160616387c2f7e90478f92b3aad91057dcb53feabf918d | lean_source_file | pkg_mathlib_source_file_chunks | One can apply `ψ m` to `v m` and `w m`, and the resulting map is `C^n`. -/
lemma ContinuousOn.clm_bundle_apply₂
(hψ : ContinuousOn
(fun m ↦ TotalSpace.mk' (F₁ →L[𝕜] F₂ →L[𝕜] F₃)
(E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) s)
(hv : ContinuousOn (fun m ↦ TotalSpace.mk' F₁ (b m)... |
1,031 | lean_file_chunk | 8c7a00c7660bb6042466992df89196a8af3fa022c287591c144ed4fb02ba42ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Topology.VectorBundle.Basic
public import Mathlib.LinearAlgebra.FiniteDimensional.Defs
/-! # Finite-rank vector bundles -/
@[expose] pu... |
1,470 | lean_file_chunk | 5f804be4c3dfd257a58949700dee454c2b2158a664906062fd31a7563371fa86 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Heather Macbeth
-/
module
public import Mathlib.LinearAlgebra.Matrix.Defs
public import Mathlib.Topology.UniformSpace.Pi
public import Mathlib.Topology.Algebra.IsUniformGrou... |
4,282 | lean_file_chunk | 576e4bd84720cb65f0d69827f97c69104c22cd67aa88b88e89ab2586e0489a3f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Topology.Separation.Regular
public import Mathlib.Topology.UniformSpace.Defs
public import Mathlib.Tactic.TautoSet
/-!... |
11,167 | lean_file_chunk | fcf68a2e789bd6f819d6e6410108779b931ab70164d846f6a9ff223c646d870c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
module
public import Mathlib.Topology.UniformSpace.Basic
public import Mathlib.Topology.Compactness.Compact
/-!
# Compact sets in un... |
3,209 | lean_file_chunk | c1e989a79c942422d0ed61a091232d2948e83742091b3d48c795685ee55faf37 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.UniformSpace.Defs
/-!
# Construct a `UniformSpace` from a `dist`-like function
In this file we provide a constructor for `U... |
23,987 | lean_file_chunk | 6012aaa02f1d3e87f23ca11f6b6aebb032a50bebecb94b57053d845370668440 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.UniformSpace.CompactConvergence
public import Mathlib.Topology.UniformSpace.Equicontinuity
public import Mathlib.Topology.U... |
4,966 | lean_file_chunk | f5758183e70f076405538db319dd970247f3cf8b8a61fdaa23401ff8924a34f1 | lean_source_file | pkg_mathlib_source_file_chunks | -- `restrict (⋃₀ 𝔖) ∘ F`.
have : IsInducing (restrict (⋃₀ 𝔖) ∘ F) := by
rwa [EquicontinuousOn.inducing_uniformOnFun_iff_pi' 𝔖_compact F_eqcont] at F_ind
-- Thus, we just have to check that the range of this map is compact.
rw [← isCompact_univ_iff, this.isCompact_iff, image_univ]
-- But then we are worki... |
23,985 | lean_file_chunk | def8462c9ef0294ba0c2a218282f225450c65a1605f2412382ae3d5502fa56bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Topology.Algebra.Constructions
public import Mathlib.Topology.Bases
public import Mathlib.Algebra.Order.Group.Nat
public... |
22,854 | lean_file_chunk | 3d8591423e151e4abf1026d04463607847934e3074e53758c9314b4edafb5488 | lean_source_file | pkg_mathlib_source_file_chunks | choose g hgs hgr using fun x : u => x.coe_prop.2
refine ⟨range g, ?_, ?_, ?_⟩
· exact range_subset_iff.2 hgs
· haveI : Fintype u := (fk.inter_of_left _).fintype
exact finite_range g
· filter_upwards [hs, ks] with x xs ⟨y, hy, xy⟩
simp_rw [SetRel.preimage, exists_range_iff]
set z : ↥u := ⟨y, hy, ⟨x, ... |
23,458 | lean_file_chunk | a4f0df142de64a08e8f35c6a4bd80003ec271457a1dc3308c3025150cd227ca4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
module
public import Mathlib.Topology.UniformSpace.AbstractCompletion
/-!
# Hausdorff completions of uniform spaces
The goal is to construct a left... |
22,913 | lean_file_chunk | fc17137d717dcc712e08f3659213b752ac9fbb6f055c0d6ffaf09cf3d7691107 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Yury Kudryashov
-/
module
public import Mathlib.Topology.CompactOpen
public import Mathlib.Topology.Compactness.CompactlyCoherentSpace
public import Mathlib.Topology.Maps.Pr... |
5,825 | lean_file_chunk | b8c56324ae6ebdb1cf26ec2319bbcb5e07f69525105fa195f5e4f5915da2cd88 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Topology.UniformSpace.UniformEmbedding
/-!
# Indexed product of uniform spaces
-/
@[expose] public section
noncomputable section
op... |
2,710 | lean_file_chunk | e0330d40beb61ec346281eace1fa152307ba5eb4bce58459af85e9779e111d0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
module
public import Mathlib.Topology.Algebra.Indicator
public import Mathlib.Topology.ContinuousMap.Algebra
public import Mathlib.Topology.Separation.DisjointCover
... |
23,969 | lean_file_chunk | 02b27eb469da865f0b45a74528281e258054655d234facb449f4e11099d13dcb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Attila Gáspár. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Attila Gáspár
-/
module
public import Mathlib.Topology.Order.Lattice
public import Mathlib.Topology.Sets.VietorisTopology
public import Mathlib.Topology.UniformSpace.UniformEmbedding
... |
10,127 | lean_file_chunk | 3e86701517548898e2b9839cdf3db94d671f74c00933271efc3fb1aae03c9277 | lean_source_file | pkg_mathlib_source_file_chunks | isUniformEmbedding_coe.uniformContinuous_iff.mpr <|
UniformSpace.hausdorff.uniformContinuous_union.comp <|
uniformContinuous_coe.prodMap uniformContinuous_coe
theorem _root_.UniformContinuous.sup_compacts
{f g : α → Compacts β} (hf : UniformContinuous f) (hg : UniformContinuous g) :
UniformContinuous... |
11,490 | lean_file_chunk | 72d5cabe49ae3553ddd1965f62816384875b0ed147ddce4be7b2ca741cd12dea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.UniformSpace.LocallyUniformConvergence
/-!
# Uniform approximation
In this file, we give lemmas ensuring that a functio... |
23,981 | lean_file_chunk | 13b00ecdf5a1eb4cd342501e413752b7ab910c017fed156003d7c1b10462db30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
module
public import Mathlib.Data.Rel
public import Mathlib.Order.Filter.SmallSets
public import Mathlib.Topology.UniformSpace.Defs
p... |
22,817 | lean_file_chunk | 801c7ff7ec0a48eb563be3da10032fcd3fb5362218003264a0451c1dd613a3b2 | lean_source_file | pkg_mathlib_source_file_chunks | exact tendsto_iInf' ⟨u, h₁⟩ hf
theorem uniformContinuous_sInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : Set (UniformSpace β)} :
UniformContinuous[u₁, sInf u₂] f ↔ ∀ u ∈ u₂, UniformContinuous[u₁, u] f := by
delta UniformContinuous
rw [sInf_eq_iInf', iInf_uniformity, tendsto_iInf, SetCoe.forall]
theorem unifo... |
23,966 | lean_file_chunk | 185458c0fb73f22b3a3632240e392bec1acfd3aeeda0af22d000bc380b311107 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges un... |
8,759 | lean_file_chunk | 886a032ac18ab68118dcd2f848ff03c2631c7debcc077bf21616d7e84f9b7867 | lean_source_file | pkg_mathlib_source_file_chunks | theorem TendstoUniformlyOn.uniformCauchySeqOn (hF : TendstoUniformlyOn F f p s) :
UniformCauchySeqOn F p s :=
uniformCauchySeqOn_iff_uniformCauchySeqOnFilter.mpr
hF.tendstoUniformlyOnFilter.uniformCauchySeqOnFilter
/-- A uniformly Cauchy sequence converges uniformly to its limit -/
theorem UniformCauchySeqOn... |
3,608 | lean_file_chunk | 2a54b891b73c2988cc12867d4b778c8d89174abd7ec814a72097c0cdfc7d1f50 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Topology.Path
public import Mathlib.Topology.UniformSpace.CompactConvergence
public import Mathlib.Topology.UniformSpace.HeineCantor
p... |
23,957 | lean_file_chunk | bdfc921c6e13c6aa5968965f0cb2886dccf57c4e78bcfca9af8ae0f3da91b14c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
module
public import Mathlib.Data.Rel.Cover
public import Mathlib.Topology.Order
/-!
# Uniform spaces
Uniform spaces are a generali... |
16,350 | lean_file_chunk | b92bff7764150355f1636eb49f29da765bd3b15924ea498f4da4a2cd50fb1c7f | lean_source_file | pkg_mathlib_source_file_chunks | let ⟨_t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs
let ⟨t', ht', _, ht'₂⟩ := symm_of_uniformity ht₁
⟨t', ht', SetRel.symm _, Subset.trans (monotone_id.relComp monotone_id ht'₂) ht₂⟩
theorem uniformity_le_symm : 𝓤 α ≤ map Prod.swap (𝓤 α) := by
rw [map_swap_eq_comap_swap]; exact tendsto_swap_uniformity.le_comap
... |
5,079 | lean_file_chunk | ae888ff8a088b444892ff0dcce835db385d1a0e60c9c6d2a8aaaff6651b02c7e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Aaron Liu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Liu
-/
module
public import Mathlib.Topology.Separation.CompletelyRegular
import Mathlib.Topology.UniformSpace.OfCompactT2
/-!
# Uniformizable Spaces
A topological space is unifor... |
23,994 | lean_file_chunk | 8a56a5816042bcdbe074edfc229d156e25fa2afa736249eefbe90f1b88f36e81 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
module
public import Mathlib.Topology.UniformSpace.Cauchy
public import Mathlib.Topology.UniformSpace.Separation
public import Mat... |
6,565 | lean_file_chunk | 128c01e28259785f2be2268cd3d37e002093f5db4ae430efa62fa39379aa83b7 | lean_source_file | pkg_mathlib_source_file_chunks | include h_f in
theorem uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuous ψ := fun d hd =>
let ⟨s, hs, hs_comp⟩ := comp3_mem_uniformity hd
have h_pnt : ∀ {a m}, m ∈ 𝓝 a → ∃ c ∈ f '' (e ⁻¹' m), (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s :=
fun {a m} hm =>
have nb : NeBot (map f (comap e (𝓝 a))) :=
... |
15,755 | lean_file_chunk | 8664e1999b24f25692362ffa50326c7707f72b30c2724a0d928246e049961706 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Topology.Separation.Regular
public import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff properti... |
4,593 | lean_file_chunk | 95b529a0de6ed8f10e7f60d3c893e276802e288c84238160c3b143839eacfdab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Topology.Instances.Rat
public import Mathlib.Topology.UniformSpace.AbsoluteValue
public import Mathlib.Topology.UniformSpace.Completion
... |
15,179 | lean_file_chunk | dda719a2ef4a22e1f6c41a604306ba0d145c0b1a4492a28b65ebc0e7855fdec3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton,
Anatole Dedecker
-/
module
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Topology.... |
8,688 | lean_file_chunk | fdd95060b4c5b8bbcb684ceb95ec9925c055c20c9e67095ee8b21c7e7310941c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module
public import Mathlib.Analysis.Normed.Order.Lattice
public import Mathlib.Topology.ContinuousMap.Ordered
public import Mathlib.Topology.UniformSpace.CompactConv... |
7,119 | lean_file_chunk | a3da86618feb56cde08ae0509853f8da6ff07bbb30cd699815e8e6a61a854bd2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Topology.Algebra.Support
public import Mathlib.Topology.UniformSpace.Compact
public import Mathlib.Topology.UniformSpac... |
23,949 | lean_file_chunk | 129c2641cba334486daf3133f826dd9cd29394156877c2f0602f5a411069ceab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and ... |
23,950 | lean_file_chunk | c17e8320e7826f0a5ace8e2db8cca0f1c8d5e3a81e23c81893cfb898acabeb7c | lean_source_file | pkg_mathlib_source_file_chunks | theorem equicontinuousAt_iff_continuousAt {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ ContinuousAt (ofFun ∘ Function.swap F : X → ι →ᵤ α) x₀ := by
rw [ContinuousAt, (UniformFun.hasBasis_nhds ι α _).tendsto_right_iff]
rfl
/-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff the function... |
5,421 | lean_file_chunk | 0520a67d693aa59e241878ce13179141fcaa0854f96c7c646fa6f53544868409 | lean_source_file | pkg_mathlib_source_file_chunks | the ambient topology. So it turns out to be easier to re-do the proof by hand.
-/
/-- If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise on `S ∪ {x₀} : Set X`* along some nontrivial
filter, and if the family `𝓕` is equicontinuous at `x₀ : X` within `S`, then the limit is
continuous at `x₀` within `S`. -/
theorem Fil... |
23,950 | lean_file_chunk | 285c1ce6cb28e733a2fa656b77f191e13e23a1a9dff48f694752789d1f42718c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
module
public import Mathlib.Topology.Coherent
public import Mathlib.Topology.UniformSpace.Equiv
public import Mathlib.Topology.UniformSpace.Pi
public import Mat... |
23,991 | lean_file_chunk | ff1bf11d9b6244f0b048579918e362903c0b60d6e309ee1563058fddb95f35d0 | lean_source_file | pkg_mathlib_source_file_chunks | UniformSpace.comap_iInf, UniformFun.iInf_eq]
refine iInf_congr fun i => ?_
rw [← UniformSpace.comap_comap, UniformFun.comap_eq]
rfl
-- Like in the previous lemma, the diagram actually commutes by definition
/-- The set of continuous functions is closed in the uniform convergence topology.
This is a ... |
13,197 | lean_file_chunk | af597b9f1f0f330f167ee3eeee58158925914872996ce0b874966a881dd14e7a | lean_source_file | pkg_mathlib_source_file_chunks | theorem t2Space_of_covering [T2Space β] (h : ⋃₀ 𝔖 = univ) : T2Space (α →ᵤ[𝔖] β) where
t2 f g hfg := by
obtain ⟨x, hx⟩ := not_forall.mp (mt funext hfg)
obtain ⟨s, hs, hxs⟩ : ∃ s ∈ 𝔖, x ∈ s := mem_sUnion.mp (h.symm ▸ True.intro)
exact separated_by_continuous (uniformContinuous_eval_of_mem β 𝔖 hxs hs).co... |
14,789 | lean_file_chunk | 8817d3ea8ba79d77829fd650975d2b6beed2f3472a8b7fbb4d744b0aeb2f28be | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Topology.UniformSpace.UniformEmbedding
public import Mathlib.Topology.UniformSpace.Equiv
/-!
# Abstract theory of Hausdorff completions... |
17,879 | lean_file_chunk | 532ef4020d8f04caa3f3ec6ef3e1632fd7a43c26e0d93693f4504ae6dc5d20ea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.UniformSpace.UniformConvergence
/-!
# Locally uniform convergence
We define a sequence of functions `Fₙ` to *converge l... |
3,395 | lean_file_chunk | f94cfdd0b5745e1fb3e8794a549a3f98de9d9455c4e15679bdb6a7b8c73baf63 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Antoine Chambert-Loir, María Inés de Frutos Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Antoine Chambert-Loir, María Inés de Frutos Fernández
-/
module
public import Mathlib.Topology.UniformSpace.Basi... |
1,508 | lean_file_chunk | 144e8e1dd311696a466fa4f26e4614a4a23dd2a3b204c6753da771afcd02c360 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Topology.UniformSpace.UniformEmbedding
/-!
# Theory of complete separated uniform spaces.
This file is for elementary lemmas tha... |
1,554 | lean_file_chunk | ceb1cddd95a7c1c42b94ced03b73771452727217831ee2682ca2b31e7af80dc6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
module
public import Mathlib.Algebra.Order.AbsoluteValue.Basic
public import Mathlib.Algebra.Order.Field.Basic
public import Mathlib.Topology.UniformSpace.OfFun
/-!... |
1,856 | lean_file_chunk | e08b5bf9b41b4887fbb549f2e83e52dd159c58aca11dc37e6158663d2e72036d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
module
public import Mathlib.Topology.ContinuousMap.Basic
public import Mathlib.Topology.MetricSpace.Cauchy
/-!
# The reals are complete
This file ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.