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 |
|---|---|---|---|---|---|
3,584 | lean_file_chunk | f11770a4d5434ccac21ad691b21beb5264859b541e2b58dc6f78d8e828e80552 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Ben Eltschig. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ben Eltschig
-/
module
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.Topology.Category.TopCat.Limits.Basic
public import Mathlib.Topology.Compactness.DeltaGene... |
1,373 | lean_file_chunk | 5a823dc4312b7130cb14aa76a9710199fd33c995397551f23bc9417dc509b4f0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Order.Category.Frm
public import Mathlib.Topology.Category.CompHaus.Frm
/-!
# The category of locales
This file defines `Locale`, the cate... |
17,909 | lean_file_chunk | 84fd449f1676c14fa1b522c776c3a4dac9a3150837c2ad6fbf0940704b3bf52d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
module
public import Mathlib.CategoryTheory.Monad.Types
public import Mathlib.CategoryTheory.Monad.Limits
public import Mathlib.CategoryTheory.Equivalence
public import Math... |
17,301 | lean_file_chunk | cc44955e4512c6547f577f28086551475ebdaaf3048d0ac57024bc553bc6faf3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
module
public import Mathlib.Topology.Connected.Clopen
/-!
# Totally disconnected and totally separated topological spaces
## Main definitio... |
23,990 | lean_file_chunk | 8b722a22573f911dd8b93030339e2fe87e38769133ecf52d96137b0984c215ff | 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, Yury Kudryashov
-/
module
public import Mathlib.Data.Finite.Sigma
public import Mathlib.Data.Set.Subset
public import Mathlib.Topology.Clopen
public im... |
9,800 | lean_file_chunk | f0ad8fe00d5d875a485031bb25ec035b3c1c266a1e770899d638df79412f7c2d | lean_source_file | pkg_mathlib_source_file_chunks | ⟨fun x => by trivial, fun h1 => h1.symm, fun h1 h2 => h1.trans h2⟩⟩
/-- The quotient of a space by its connected components -/
def ConnectedComponents (α : Type u) [TopologicalSpace α] :=
Quotient (connectedComponentSetoid α)
namespace ConnectedComponents
/-- Coercion from a topological space to the set of connect... |
4,478 | lean_file_chunk | 034accb59b22bcdbf2033b110097f75abcbbedf13f3488f32e52a50ef0daa2e3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.Data.Set.Card.Arithmetic
public import Mathlib.Topology.LocalAtTarget
public import Mathlib.Topology.Separation.Connected
/-!
# Car... |
23,990 | lean_file_chunk | 7ab02241c1f3855316c6f04fbbba0bc09246db26c11a990dbdeff04cef592fb8 | 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, Yury Kudryashov
-/
module
public import Mathlib.Order.SuccPred.Relation
public import Mathlib.Topology.Order.OrderClosed
/-!
# Connected subsets of to... |
12,825 | lean_file_chunk | add42c7334dc0dacb4075f4680f6df0fc5583e530d8032c9c52475eddeda8ff9 | lean_source_file | pkg_mathlib_source_file_chunks | def connectedComponent (x : α) : Set α :=
⋃₀ { s : Set α | IsPreconnected s ∧ x ∈ s }
open Classical in
/-- Given a set `F` in a topological space `α` and a point `x : α`, the connected
component of `x` in `F` is the connected component of `x` in the subtype `F` seen as
a set in `α`. This definition does not make se... |
1,513 | lean_file_chunk | 5e96e043eb9713beb7db55f3af97f6625bf018aacbded1091b0251b3625a5b3a | 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.Algebra.OpenSubgroup
public import Mathlib.Topology.Connected.LocPathConnected
/-! # The path component of the identity in a loc... |
23,968 | lean_file_chunk | 0daeb948c7d7dc252a745a16f87ea78fb7a2964db013629305ac50943ff546d3 | 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
-/
module
public import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib/Topology/Path.lean`, this file defines path components and path-connec... |
566 | lean_file_chunk | ec1b653d22a85d9ac80254b17c9c5b6d15faece14eebf13064494f12cc35e358 | lean_source_file | pkg_mathlib_source_file_chunks | rcases this.exists_path_through_family' p fun _i => True.intro with ⟨γ, t, -, h⟩
exact ⟨γ, t, h⟩
end PathConnectedSpace
/-- The preimage of a singleton in `ZerothHomotopy` is the path component of an element in the
equivalence class. -/
theorem ZerothHomotopy.preimage_singleton_eq_pathComponent (x : X) :
Quotie... |
884 | lean_file_chunk | ef9c240a9fdb59ac662712250ea026f2a58a2724d84947b4cc288c45269ebe93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.Topology.Separation.Hausdorff
public import Mathlib.Topology.Connected.TotallyDisconnected
/-!
# Separation and (dis)connectedness ... |
14,449 | lean_file_chunk | 28225d6f6ecc36f2e7f0b7bea35ec5c17f4fcb282f7b296dfc596f620426793e | 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, Ben Eltschig
-/
module
public import Mathlib.Topology.Connected.PathConnected
public import Mathlib.Topology.AlexandrovDiscrete
/-!
# Locally path-connected spaces
T... |
7,897 | lean_file_chunk | 2ce28c7e7241474fd24bfa0b9d0fb35a8ceb9acacc4e24097ebd1ce924223829 | 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.Connected.Basic
public import Mathlib.Topology.Connected.Clopen
/-!
# Locally connected topological spaces
A topological ... |
1,598 | lean_file_chunk | 49c89796a798358320fe7b53686721c23dd8e5b072f72b09df65a03c276ae261 | 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.Complex.Basic
public import Mathlib.Data.Real.StarOrdered
public import Mathlib.Topology.ContinuousMap.StarOrdered
/-! # Instanc... |
4,980 | lean_file_chunk | ac07d1917c704a2ab2596e9400c83063b2cb901f1b886381d5d4632ef43307ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Analysis.SpecialFunctions.Bernstein
public import Mathlib.Topology.Algebra.Algebra
public import Mathlib.Topology.ContinuousMap.Compact
/-!... |
23,977 | lean_file_chunk | ce7a77b1677dc0a53681fbe6be3648f4e0aee8361aaf20515dd64c4df94ba370 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Heather Macbeth
-/
module
public import Mathlib.Algebra.Algebra.Subalgebra.Tower
public import Mathlib.Analysis.RCLike.Basic
public import Mathlib.Topology.Algebra.Star.Re... |
10,148 | lean_file_chunk | e736d3f2640eca9e197536a848941e58b10c9ed36b725d5bade274eb3f7fea9d | lean_source_file | pkg_mathlib_source_file_chunks | AlgHom.ext fun x => this (by trivial)
rw [← polynomialFunctions.topologicalClosure s]
exact Subalgebra.topologicalClosure_minimal
(polynomialFunctions.le_equalizer s φ ψ h) (isClosed_eq hφ hψ)
/-- Continuous star algebra homomorphisms from `C(s, 𝕜)` into a star `𝕜`-algebra `A` which agree
at `X : 𝕜[X]` (int... |
18,373 | lean_file_chunk | 1c7a0706dfc249469f115d2b5ad6f76c58d0b2c634bb7c199eb0f215fff671d3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Topology.ContinuousMap.Bounded.Star
public import Mathlib.Topology.ContinuousMap.Star
public import Mathlib.Topology.UniformSpace.Compact
pu... |
20,032 | lean_file_chunk | 7499eb4d259b451015cf7d53fe7ffac4cb90c82267d57acc8de6d22ea5f89db3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Algebra.Algebra
public import Mathlib.Topology.ContinuousMap.Compact
public import Mathlib.Topology.UrysohnsLemma
public import M... |
6,560 | lean_file_chunk | 68fc85dce58c36280116c6a760141d9346dfb6ae6ed9fa02a22516ab777715a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Nicolò Cavalleri
-/
module
public import Mathlib.Topology.Algebra.Star
public import Mathlib.Algebra.Star.StarAlgHom
public import Mathlib.Topology.ContinuousMap.Algebra
... |
6,146 | lean_file_chunk | b1ab8831067a75ce37586cf2cbcf16b236776af5d3749956c2695680677b60cb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
module
public import Mathlib.Topology.CompactOpen
public import Mathlib.Topology.Order.ProjIcc
/-!
# Continuous bundled maps on intervals
In this file we prove a... |
3,366 | lean_file_chunk | a3da22bde2825776d88d8511d6deac4fcdb6ee7f16e0ddebfe4d75d92ef1ac6b | 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.CompactOpen
/-!
# Equivalence between `C(X, Σ i, Y i)` and `Σ i, C(X, Y i)`
If `X` is a connected topological space, then f... |
6,670 | lean_file_chunk | c8ad65f6f19e44916e744265cbb786167435b39ae587c2ef6af222aaa7a1da46 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.ContinuousMap.Basic
/-!
# Cocompact continuous maps
The type of *cocompact continuous maps* are those which tend to the cocompa... |
23,985 | lean_file_chunk | 99a47160b9cd0693e801ed759dcd5e1645e7b5d229e27dce2ea3e6a138389502 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yoh Tanimoto
-/
module
public import Mathlib.Algebra.Order.Module.PositiveLinearMap
public import Mathlib.Topology.Algebra.Order.Support
public import Mathlib.Topology.ContinuousMap.Zer... |
7,917 | lean_file_chunk | 5a00279f6c746bf78bb5ce2224d3ba73ec8b934da76e388f4ca913dd361b70a3 | lean_source_file | pkg_mathlib_source_file_chunks | rw [tsupport, tsupport, ← closure_union]
apply closure_mono
intro x hx
contrapose hx
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, Set.mem_union, Function.mem_support,
ne_eq, not_or, Decidable.not_not, ContinuousMap.coe_sub, Pi.sub_apply] at hx ⊢
simp [hx.1, hx.2]
· ext x
s... |
16,495 | lean_file_chunk | 3b5f430acc9572225d3f8da4a70713d5bacac93763140aacd6a250703942aafc | 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
-/
module
public import Mathlib.Data.Set.UnionLift
public import Mathlib.Topology.ContinuousMap.Defs
public import Mathlib.Topology.Homeomorph.Defs
public import Ma... |
4,813 | lean_file_chunk | 197ac4a7beb6400d533a36b187de6284080d2e69644bee56cbc0cc109487917a | 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, Yury Kudryashov
-/
module
public import Mathlib.Data.FunLike.Basic
public import Mathlib.Tactic.Continuity
public import Mathlib.Tactic.Lift
public import Mathlib.... |
5,284 | lean_file_chunk | d5182ccac75a064c1e6be9525d571e66b7b3a85f9c85600e91d202a0ece16cb9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 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.CompactOpen
/-!
# Second countable topology on `C(X, Y)`
In this file we prove that `C(X, Y)` with compact-open topology ha... |
3,195 | lean_file_chunk | 1a596ad844bdbd2cc75b95169c254c71ca02f5f4924fb72fd61659c822e6bd9f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam
-/
module
public import Mathlib.Topology.Order.Lattice
public import Mathlib.Topology.Order.ProjIcc
public import Mathlib.Topology.ContinuousMap.Defs
/-!
# ... |
2,286 | lean_file_chunk | 3c8417e9aa30fa3a308f8f1cb2ea50a72f5284bf2db42aa83e245443d70545bf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
module
public import Mathlib.Topology.Order
public import Mathlib.Topology.Sets.Opens
public import Mathlib.Topology.ContinuousMap.Basic
/-!
# Any T0 sp... |
1,692 | lean_file_chunk | 3c24b4bb079875d2d7302791f9672854882527b96ea8e08f331a477bdedda51c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Nicolò Cavalleri
-/
module
public import Mathlib.Algebra.Ring.Periodic
public import Mathlib.Topology.ContinuousMap.Algebra
/-!
# Sums of translates of a continuous funct... |
2,615 | lean_file_chunk | 4d7b498bbf40c2516d7c1a5c2b756456fd8d0b535f5b0d2891411f2ddb6aab8f | 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
-/
module
public import Mathlib.Topology.LocallyConstant.Algebra
public import Mathlib.Topology.ContinuousMap.Basic
public import Mathlib.Topology.ContinuousMap.Algebra
/-!
... |
19,274 | lean_file_chunk | be5013907308870093df5434a2ce4920fe40d9993976289957f561d1071d6c36 | 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.Topology.ContinuousMap.Algebra
public import Mathlib.Topology.ContinuousMap.Compact
/-!
# Continuous maps sending zero to zero
This is t... |
7,721 | lean_file_chunk | db66298074ce3fdd1401a9db81405b498e7dbeee9191f078d45e16478d045cea | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Topology.Algebra.Polynomial
public import Mathlib.Topology.ContinuousMap.Star
public import Mathlib.Topology.UnitInterval
public import Math... |
5,374 | lean_file_chunk | 97c55c5c3390df5d4e5e1a9ec11b0f0f8bbb3ba00093126d029299681ce090ac | 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.Algebra.Order.Star.Basic
public import Mathlib.Topology.ContinuousMap.ContinuousMapZero
public import Mathlib.Topology.ContinuousMap.Order... |
4,540 | lean_file_chunk | d7eff1823fb2d3b311872daf2df7a61e6468199233d00831e764a2ed6a637e68 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yoh Tanimoto
-/
module
public import Mathlib.Topology.ContinuousMap.Bounded.Normed
/-!
# Compactly supported bounded continuous functions
The two-sided ideal of compactly supported bo... |
23,956 | lean_file_chunk | 947ac7e57683047354c15f8e41bc4b434c80f3dfd25141f48d2129d3fe7bcc8f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Nicolò Cavalleri
-/
module
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.Algebra.Algebra.Subalgebra.Basic
public import Mathlib.Tactic.FieldSimp
public im... |
8,630 | lean_file_chunk | 429ea2f228cc57461c141ec8e55cb5ea0e4bf5f3fc49497eb7773c62cebf26f8 | lean_source_file | pkg_mathlib_source_file_chunks | def evalCLM (x : α) : C(α, M) →L[R] M where
toFun f := f x
map_add' _ _ := add_apply _ _ x
map_smul' _ _ := smul_apply _ _ x
end ContinuousMap
end ModuleStructure
section AlgebraStructure
/-!
### Algebra structure
In this section we show that continuous functions valued in a topological algebra `A` over a ri... |
1,231 | lean_file_chunk | bc40d3cb5361ebd1889c5c131f9828764eaf6f0f16b55c4f70f357210ba265b1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Nicolò Cavalleri
-/
module
public import Mathlib.Algebra.Algebra.Pi
public import Mathlib.Algebra.Order.Group.Lattice
public import Mathlib.Topology.ContinuousMap.Algebra
... |
4,273 | lean_file_chunk | 26e569e3b2795e5357a2e1159b3f8c6594fa6eb9fc166474ab3d160d5b6338cb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Ring.Units
public import Mathlib.Algebra.Algebra.Spectrum.Basic
public import Mathlib.Topology.ContinuousMap.Algebra
/-!
... |
22,739 | lean_file_chunk | 6d6e920237a44eb3f155ae7f0707835f616331a1cf9be8fb7eaada149076ed3c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.ContinuousMap.Bounded.Star
public import Mathlib.Topology.ContinuousMap.CocompactMap
/-!
# Continuous functions vanishing at inf... |
1,056 | lean_file_chunk | 029122763cca2542967aca5c9f13a11f8581a37451888225e28308886692ba6f | 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.ContinuousMap.Algebra
public import Mathlib.Topology.Algebra.Module.LocallyConvex
/-!
# The space of continuous maps is a lo... |
23,918 | lean_file_chunk | 07c7354e15967a9e28f2875ac56758b7c3616f89d0f296abde0c104987d4ade8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected
public import M... |
3,193 | lean_file_chunk | 7cc132d9a5b5bbe340c54d1c6828bacc5ee6298d9b4b67efc58beb17899cd717 | lean_source_file | pkg_mathlib_source_file_chunks | /-- A continuous map `f` from a path connected, locally path-connected space `A` to another
space `X` lifts uniquely through a covering map `p : E → X` (such that `f a₀` is lifted to `e₀`)
if `f⁎ π₁(A, a₀) ⊆ p⁎ π₁(E, e₀)`. Proposition 1.33 of [hatcher02], known as
the lifting criterion. -/
theorem existsUnique_co... |
8,347 | lean_file_chunk | 36810d948b4d993b85699c4a2b5cd4f42eaea9fffa064b1280f4953ec6825e12 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
module
public import Mathlib.Topology.Constructions
public import Mathlib.Topology.Homotopy.Path
/-!
# Product of homotopies
In this file, we introduce def... |
23,992 | lean_file_chunk | d4b24d9a7ebc722c020f0d85ae8377bce76c08b7f541ba1d90a6518591c958ef | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
module
public import Mathlib.Topology.Order.ProjIcc
public import Mathlib.Topology.ContinuousMap.Ordered
public import Mathlib.Topology.CompactOpen
public import Mathl... |
509 | lean_file_chunk | 3c10583332e6353a4a71653f6e5bfb201199ef96acb8f007fd8937242792bc75 | lean_source_file | pkg_mathlib_source_file_chunks | h₀.map2 HomotopyRel.trans h₁
theorem equivalence : Equivalence fun f g : C(X, Y) => HomotopicRel f g S :=
⟨refl, by apply symm, by apply trans⟩
theorem comp_continuousMap ⦃f₀ f₁ : C(X, Y)⦄ (h : f₀.HomotopicRel f₁ S) (g : C(Y, Z)) :
(g.comp f₀).HomotopicRel (g.comp f₁) S := h.map (·.compContinuousMap g)
end Hom... |
10,387 | lean_file_chunk | 9b5e7dff40700eacadb3700f4aef573a3545cdaacc3821e42bd7b4836ecd45d6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Filippo A. E. Nuccio Mortarino Majno di Capriglio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Junyan Xu
-/
module
public import Mathlib.Topology.CompactOpen
public import Mathlib.Topology.Homotopy.Basic
public import Mat... |
16,006 | lean_file_chunk | 0ad7cd4f696c3455a218cbf5d869f4463aad0334877fae640fb76b7fd9811089 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
module
public import Mathlib.Topology.Homotopy.Basic
public import Mathlib.Topology.Connected.PathConnected
public import Mathlib.Analysis.Convex.Basic
/-!
# Homotopy... |
4,003 | lean_file_chunk | 8ac873808c9b7375004e06a48ca13be62b19b182bf86879612219caf0abf6c30 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
module
public import Mathlib.Topology.Homotopy.Path
public import Mathlib.Topology.Homotopy.Equiv
/-!
# Contractible spaces
In this file, we define `Contra... |
5,441 | lean_file_chunk | b33f78d057e253a805e74ce53478c8ae7c18cd102a8dba4f3c889c913acb1d12 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
module
public import Mathlib.Topology.Homotopy.Basic
/-!
# Homotopy equivalences between topological spaces
In this file, we define homotopy equivalences between to... |
8,366 | lean_file_chunk | 150b05f7c243da254de33047aca6e6d0cd762fd562d29abc8841da37bf92ff78 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Topology.Homotopy.Contractible
public import Mathlib.Topology.Homotopy.Basic
public import Mathlib.Topology.Connected.LocPathConnected
publ... |
1,166 | lean_file_chunk | 51ad0601c134ef629bdc54eedf1cf4e50032a2d75150e51a44637fe5fa8f1803 | 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.Topology.Homotopy.Path
public import Mathlib.Analysis.Convex.PathConnected
/-!
# Affine homotopy between two continuous maps
I... |
23,982 | lean_file_chunk | 97949c209454d372aa2d854f58f222928e00b3e45f54bd732f1392fab0501f73 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
module
public import Mathlib.Algebra.Group.Ext
public import Mathlib.Algebra.Group.TransferInstance
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.Fun... |
1,293 | lean_file_chunk | 5db23600057d975f485f51eff4d3236d44907fddeafe18c2a0305f049efc1012 | lean_source_file | pkg_mathlib_source_file_chunks | fast_instance% @EckmannHilton.commGroup (HomotopyGroup N X x) _ 1
(isUnital_auxGroup <| Classical.choose h) _
(by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ⟨d⟩
apply congr_arg Quotient.mk'
simp only [fromLoop_trans_toLoop, transAt_distrib <| Classical.choose_spec h, coe_toEquiv,
loopHomeo_apply, coe_symm_toE... |
15,426 | lean_file_chunk | 08fe16a06bc10062b997154c81c4eb9b19d51a90434c23abe67221f371c4cadb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
module
public import Mathlib.Topology.Bases
public import Mathlib.Topology.DenseEmbedding
public import Mathlib.Topology.Connected.TotallyDisconnected
/-! # Stone-Čech co... |
9,171 | lean_file_chunk | fe9f100c4db05be5c4e4020271f69bb0895f2b895f8ec6f4b26ec4ec6eef9492 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Analysis.Normed.Group.Basic
public import Mathlib.Topology.MetricSpace.ProperSpace.Real
public import Mathlib.Analysis.Normed.Ring.Lemmas
/... |
11,642 | lean_file_chunk | 2697df1c0fb872b40f0363aaf6c78a70dd8e06a8d46dd9b990224398d0b3f0f4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 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.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets a... |
6,360 | lean_file_chunk | bcb1e54d516ce2047cbd509fa4c66fb4f50ed3b247ba280f75bf0dfa3a557f24 | 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.Group.TypeTags.Basic
public import Mathlib.Topology.Bornology.Basic
/-!
# Bornology structure on products and subtypes
In th... |
8,709 | lean_file_chunk | ff87d09c1e82641d169da9bbbc258974cc2fe7a2f83368d96774ad5b276edd77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
module
public import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
public import Mathlib.Algebra.Ring.Action.Pointwise.Set
public import Mat... |
5,692 | lean_file_chunk | 61a87d5c5ac2575c01c945a8634df70a299d9898d6a5ba4b6e112748bb6fb828 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Topology.Bornology.Basic
/-!
# Locally bounded maps
This file defines locally bounded maps between bornologies.
We use the `DFunLike` des... |
1,021 | lean_file_chunk | 2515065aa260b666b7c809caf8f4c2ff45f47c324ffac28fc4b35bd89b312017 | 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.MetricSpace.Bounded
public import Mathlib.Topology.Order.Bornology
/-!
# The reals are equipped with their ord... |
23,985 | lean_file_chunk | fcd4a9fa426a4f43c90233d1a2f50d6b8f2af3bbbbc5bbee27470e813f328fd9 | 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.Homeomorph.Defs
public import Mathlib.Topology.Maps.Basic
public import Mathlib.Topology.Separa... |
20,657 | lean_file_chunk | 63c8ee6dc11f3cdf76029c162ff31154202e2e56cf4079ba8379c54c8cbe5578 | lean_source_file | pkg_mathlib_source_file_chunks | lemma requires `f` to be jointly continuous, whereas here we only require continuity in each
variable separately. -/
theorem map_mem_closure₂' {f : X → Y → Z} {x : X} {y : Y} {s : Set X} {t : Set Y} {u : Set Z}
(hf₁ : ∀ x, Continuous (f x)) (hf₂ : ∀ y, Continuous (f · y))
(hx : x ∈ closure s) (hy : y ∈ closure ... |
23,954 | lean_file_chunk | 1795b9e1c48efa9a6c866c57dbef1301449e40778618fd001f841a1670ac97c5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. 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
-/
module
public import Mathlib.Logic.Equiv.Fin.Basic
public import Mathlib.Topology.Connected.LocallyConnec... |
780 | lean_file_chunk | c75469debcb7b99e5756e69fdaae37836c1aa2a511c0141786152a8347955def | lean_source_file | pkg_mathlib_source_file_chunks | ConnectedComponents X ≃ₜ ConnectedComponents Y :=
IsHomeomorph.homeomorph hf.continuous.connectedComponentsMap <| by
have hbij := hf.connectedComponentsMap_bijective hf'
exact ⟨hf.continuous.connectedComponentsMap_continuous,
hf.connectedComponentsMap.isOpenMap_of_injective hbij.injective, hbij⟩
variab... |
1,225 | lean_file_chunk | 5c25bc0e83318b5efb85b9007abe6352d1a8ebe23236467e96a1daa3f4316306 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
module
public import Mathlib.Topology.Homeomorph.Defs
/-!
# Transfer topological structure across `Equiv`s
We show how to transport a topological space structu... |
18,554 | lean_file_chunk | 20a9fdb276e9225759d4dd1c28f49788e21d09441f838162db35a1cfde95bbfd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Reid Barton. 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
-/
module
public import Mathlib.Topology.ContinuousMap.Defs
public import Mathlib.Topology.Maps.Basic
/-!
#... |
9,528 | lean_file_chunk | 9f96b6ec254e4f40153da7e20dfaeb6a36da8bbae87c82f20524bf3d8e7f0714 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Order.Ideal
public import Mathlib.Topology.Sets.Compacts
public import Mathlib.Topology.Sets.OpenCover
public import Mathlib.Topology.Spectral... |
1,403 | lean_file_chunk | 96281db39cdf0f009c083c7986cc19ce14e30d90c81ec30862df9c4a872cb43b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Fangming Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fangming Li, Panagiotis Angelinos
-/
module
public import Mathlib.Topology.Sober
public import Mathlib.Topology.Spectral.Prespectral
/-!
# Spectral spaces
A topological space is spectr... |
7,483 | lean_file_chunk | 372fc5c8deeb1f5e278feb3570fa0ce3e1db56448fca28d4207ce993a36f78c6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Jiedong Jiang, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jiedong Jiang, Christian Merten
-/
module
public import Mathlib.Topology.Spectral.Basic
public import Mathlib.Topology.JacobsonSpace
public import Mat... |
6,458 | lean_file_chunk | 84ce91662c955ba636cec0fa6c4fdf9f8a97b45a3f1319abd37475ab4d5ce3da | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Tactic.StacksAttribute
public import Mathlib.Topology.ContinuousMap.Basic
public import Mathlib.Topology.Maps.Proper.Basic
/-!
# Spectral m... |
7,921 | lean_file_chunk | 0c86b676eb268607645734ca4e54a1639f29ceaa03559570b6c86439340db9b1 | 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.Tactic.GCongr
public import Mathlib.Topology.Compactness.Paracompact
public import Mathlib.Topology.EMetricSpace.Basic
public import M... |
3,046 | lean_file_chunk | cad08a742aa89909579dec77618bb9d3b581030c481aedc4eb44ffe9c49fb858 | 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.EMetricSpace.Basic
public import Mathlib.Topology.UniformSpac... |
17,171 | lean_file_chunk | 53e15c66811dbefc25cfbf243e4e96245294b924e0cb216dfc820a768787cf60 | 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.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.Algebra.... |
23,944 | lean_file_chunk | 21aa60a1bc98c98a3c1af5bae566fc5a8236834fe4973a7c6c08364458ac0af2 | 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.Data.ENNReal.Inv
public import Mathlib.Topology.UniformSpace.Basic
pub... |
10,860 | lean_file_chunk | 4c9f5423ae9b03d18bfe6e42b888c8723622c2fba4a9f39d6652f8e8c985cf3c | lean_source_file | pkg_mathlib_source_file_chunks | mem_of_superset (eball_mem_nhds x ε0) eball_subset_closedEBall
theorem eball_prod_same [PseudoEMetricSpace β] (x : α) (y : β) (r : ℝ≥0∞) :
eball x r ×ˢ eball y r = eball (x, y) r :=
ext fun z => by simp [Prod.edist_eq]
theorem closedEBall_prod_same [PseudoEMetricSpace β] (x : α) (y : β) (r : ℝ≥0∞) :
closedE... |
7,873 | lean_file_chunk | 2b2f7af2604b079e5eb9c6e2256cf1da39fb43af8c2c1ab6e0950d48c7fe8f8a | 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.EMetricSpace.Pi
/-!
# Diameters of sets in extended metric s... |
23,993 | lean_file_chunk | 3b1307cb32298093eac35e3d5bd7afc8ab60fd00bc5b4150f8d9578b95938899 | 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, Winston Yin
-/
module
public import Mathlib.Algebra.Group.End
public import Mathlib.Tactic.Finiteness
public... |
2,062 | lean_file_chunk | e0184dba55d6f94278620e777775c0453c3b24d6453e9111bcce711ed17705ca | lean_source_file | pkg_mathlib_source_file_chunks | (hb : ∀ b, LipschitzWith K fun x => f (x, b)) : Continuous f :=
continuous_prod_of_dense_continuous_lipschitzWith f K dense_univ (fun _ _ ↦ ha _) hb
theorem continuousOn_prod_of_subset_closure_continuousOn_lipschitzOnWith' [TopologicalSpace α]
[PseudoEMetricSpace β] [PseudoEMetricSpace γ] (f : α × β → γ) {s : Se... |
23,959 | lean_file_chunk | 6ee97faa243ed722dd5292b474b838331a46d733260e3db8136a5cbe1f44a30c | 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.Order.Interval.Set.ProjIcc
public import Mathlib.Tactic.Finiteness
public import Mathlib.Topology.UniformSpace.UniformConvergenceT... |
23,980 | lean_file_chunk | 5663646fc1c5e0a8d82c68c5702602dac810459f4f2d5f2a010c6965ef73c3d7 | lean_source_file | pkg_mathlib_source_file_chunks | exacts [(Set.subsingleton_Icc_of_ge (hφ hy hx h)).anti Set.inter_subset_right,
(Set.subsingleton_Icc_of_ge h).anti Set.inter_subset_right]
theorem comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) := by
apply le_antisymm (comp... |
9,865 | lean_file_chunk | e145dc6bb32e2758c0866ddb290ee213a057b2eae52eacd084db2aa54a4974a4 | lean_source_file | pkg_mathlib_source_file_chunks | · simp only [variationOnFromTo, hab, ↓reduceIte, ENNReal.abs_toReal]
exact ENNReal.toReal_mono hf (eVariationOn.mono _ inter_subset_left)
· simp only [variationOnFromTo, hab, ↓reduceIte, abs_neg, ENNReal.abs_toReal]
exact ENNReal.toReal_mono hf (eVariationOn.mono _ inter_subset_left)
protected theorem eq_of_... |
22,795 | lean_file_chunk | 8dfec310d521f43aa6483e38c52a1f95558d3d927127665065c8ca4ad805417b | 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, David Ledvinka
-/
module
public import Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp
public import Mathlib.Order.CompletePartialOrder
/-!
# Pair Reduction
The goal... |
6,730 | lean_file_chunk | abaa5e836d8da4db0b105b6a30fe19843fa28a6fc42b4155f14ae93e49ba0091 | 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.Topology.MetricSpace.Pseudo.Basic
public import Mathlib.Topology.MetricSpace.Pseudo.Lemmas
public import Mathlib.Topology.MetricSp... |
23,992 | lean_file_chunk | 33f37f29fe2e8d6992184ee1235c65d7753fddd9079baa37fc4deb5e546ddb62 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open t... |
9,884 | lean_file_chunk | d4c4a6e94a7a5ac949531cd8ca26e54e2a72e15247f0e76a40d63f9a7d571938 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The closure of a set equals the intersection of its closed thickenings of positive radii. -/
theorem closure_eq_iInter_cthickening (E : Set α) :
closure E = ⋂ (δ : ℝ) (_ : 0 < δ), cthickening δ E := by
rw [← cthickening_zero]
exact cthickening_eq_iInter_cthickening E
/-- The closure of a set equals the int... |
23,922 | lean_file_chunk | 19c084cd2dc729ff679b0ce47d461f30455e42d73b01f2bf06bab99ddb4f5e38 | 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.Analysis.Calculus.ContDiff.RCLike
public import Mathlib.MeasureTheory.Measure.Hausdorff
import Mathlib.Analysis.Convex.Intrinsic
/-!
... |
1,196 | lean_file_chunk | 12100c3f8e3f94910d5bb46e11dcd25710c608d3455adecb182a8a191c6d6336 | lean_source_file | pkg_mathlib_source_file_chunks | theorem ContDiffOn.dense_compl_image_of_dimH_lt_finrank [FiniteDimensional ℝ F] {f : E → F}
{s t : Set E} (h : ContDiffOn ℝ 1 f s) (hc : Convex ℝ s) (ht : t ⊆ s)
(htF : dimH t < finrank ℝ F) : Dense (f '' t)ᶜ :=
dense_compl_of_dimH_lt_finrank <| (h.dimH_image_le hc ht).trans_lt htF
/-- A particular case of S... |
23,927 | lean_file_chunk | 1883ea4ca5e4fd06e2804c2a45fc6d0cea2a8da8e16dcfc4b7acd5c01186bcd5 | 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.Logic.Encodable.Pi
public import Mathlib.SetTheory.Cardinal.Basic
public import Mathlib.Topology.MetricSpace.Closeds
public import... |
23,958 | lean_file_chunk | 2cb9a8a0eeeee24639d1bffa592b1ff057acb0b7211c0f138337abbb5d14795c | lean_source_file | pkg_mathlib_source_file_chunks | _ = dist x y + dist y z := by rw [x.toGHSpace_rep, y.toGHSpace_rep, z.toGHSpace_rep]
end GHSpace --section
end GromovHausdorff
/-- In particular, nonempty compacts of a metric space map to `GHSpace`.
We register this in the `TopologicalSpace` namespace to take advantage
of the notation `p.toGHSpace`. -/
def Topolog... |
3,843 | lean_file_chunk | e177cf8e437b173b846055ff288b4935f9dba9a1b5b1bb8391cc26db168927c4 | lean_source_file | pkg_mathlib_source_file_chunks | refine Metric.complete_of_convergent_controlled_sequences d this fun u hu => ?_
-- `X n` is a representative of `u n`
let X n := (u n).Rep
-- glue them together successively in an optimal way, getting a sequence of metric spaces `Y n`
let Y := auxGluing X
-- this equality is true by definition but Lean unfold... |
18,457 | lean_file_chunk | f1eb455d321d5afb16261ea829394a7966cc52d03f373c499a14c8ecabc3ff99 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
module
public import Mathlib.Topology.MetricSpace.Antilipschitz
public import Mathlib.Topology.MetricSpace.Isometry
public import Mathlib.Topology.MetricSpace.Lipschit... |
17,167 | lean_file_chunk | 99a41b51325ce132fea12764e0718e5059c271574fbb5cafff4844283d9a210c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
module
public import Mathlib.Data.ENNReal.Lemmas
public import Mathlib.Topology.MetricSpace.Thickening
public import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# ... |
18,293 | lean_file_chunk | e13edf0c344a59713efe1eafd43dcad217c3c974ec72ad9b2133c83df6e753fe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
module
public import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately du... |
13,415 | lean_file_chunk | b39e9eb76708b596eac55984687de131de32403b45589c576a62dba2fa73bf13 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
module
public import Mathlib.Topology.MetricSpace.Holder
/-!
# Hölder norm
This file defines the Hölder (semi-)norm for Hölder functions alongside some basic properties.... |
923 | lean_file_chunk | 162390a5f2ca3fcddf63b465a548c4ef30e31cb5cb8ea454e5fdec9675bd12cf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
module
public import Mathlib.Topology.MetricSpace.Basic
/-!
# Transfer metric space structures across `Equiv`s
In this file, we transfer a distance and (pseudo... |
4,127 | lean_file_chunk | 54cbb9877e45dc3ac75ea2e18d59063521e0dec76fff2e81496e7c1fc1279a9f | 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.Topology.MetricSpace.Antilipschitz
public import Mathlib.Topology.MetricSpace.Lipschitz
/-! # Bilipschitz equivalence
A common pattern i... |
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