fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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proj_clift : proj (T.clift (e, γ) i) = γ i := by
simp [clift, liftCM, proj_lift] | theorem | Topology | [
"Mathlib.Data.Bundle",
"Mathlib.Data.Set.Image",
"Mathlib.Topology.CompactOpen",
"Mathlib.Topology.OpenPartialHomeomorph",
"Mathlib.Topology.Order.Basic"
] | Mathlib/Topology/FiberBundle/Trivialization.lean | proj_clift | null |
IsGδ (s : Set X) : Prop :=
∃ T : Set (Set X), (∀ t ∈ T, IsOpen t) ∧ T.Countable ∧ s = ⋂₀ T | def | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ | A Gδ set is a countable intersection of open sets. |
IsOpen.isGδ {s : Set X} (h : IsOpen s) : IsGδ s :=
⟨{s}, by simp [h], countable_singleton _, (Set.sInter_singleton _).symm⟩
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsOpen.isGδ | An open set is a Gδ set. |
protected IsGδ.empty : IsGδ (∅ : Set X) :=
isOpen_empty.isGδ
@[simp] | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.empty | null |
protected IsGδ.univ : IsGδ (univ : Set X) :=
isOpen_univ.isGδ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.univ | null |
IsGδ.biInter_of_isOpen {I : Set ι} (hI : I.Countable) {f : ι → Set X}
(hf : ∀ i ∈ I, IsOpen (f i)) : IsGδ (⋂ i ∈ I, f i) :=
⟨f '' I, by rwa [forall_mem_image], hI.image _, by rw [sInter_image]⟩ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.biInter_of_isOpen | null |
IsGδ.iInter_of_isOpen [Countable ι'] {f : ι' → Set X} (hf : ∀ i, IsOpen (f i)) :
IsGδ (⋂ i, f i) :=
⟨range f, by rwa [forall_mem_range], countable_range _, by rw [sInter_range]⟩ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.iInter_of_isOpen | null |
isGδ_iff_eq_iInter_nat {s : Set X} :
IsGδ s ↔ ∃ (f : ℕ → Set X), (∀ n, IsOpen (f n)) ∧ s = ⋂ n, f n := by
refine ⟨?_, ?_⟩
· rintro ⟨T, hT, T_count, rfl⟩
rcases Set.eq_empty_or_nonempty T with rfl | hT
· exact ⟨fun _n ↦ univ, fun _n ↦ isOpen_univ, by simp⟩
· obtain ⟨f, hf⟩ : ∃ (f : ℕ → Set X), T = range f := Countable.exists_eq_range T_count hT
exact ⟨f, by simp_all, by simp [hf]⟩
· rintro ⟨f, hf, rfl⟩
exact .iInter_of_isOpen hf
alias ⟨IsGδ.eq_iInter_nat, _⟩ := isGδ_iff_eq_iInter_nat | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | isGδ_iff_eq_iInter_nat | null |
protected IsGδ.iInter [Countable ι'] {s : ι' → Set X} (hs : ∀ i, IsGδ (s i)) :
IsGδ (⋂ i, s i) := by
choose T hTo hTc hTs using hs
obtain rfl : s = fun i => ⋂₀ T i := funext hTs
refine ⟨⋃ i, T i, ?_, countable_iUnion hTc, (sInter_iUnion _).symm⟩
simpa [@forall_swap ι'] using hTo | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.iInter | The intersection of an encodable family of Gδ sets is a Gδ set. |
IsGδ.biInter {s : Set ι} (hs : s.Countable) {t : ∀ i ∈ s, Set X}
(ht : ∀ (i) (hi : i ∈ s), IsGδ (t i hi)) : IsGδ (⋂ i ∈ s, t i ‹_›) := by
rw [biInter_eq_iInter]
haveI := hs.to_subtype
exact .iInter fun x => ht x x.2 | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.biInter | null |
IsGδ.sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by
simpa only [sInter_eq_biInter] using IsGδ.biInter hS h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.sInter | A countable intersection of Gδ sets is a Gδ set. |
IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by
rw [inter_eq_iInter]
exact .iInter (Bool.forall_bool.2 ⟨ht, hs⟩) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.inter | null |
IsGδ.union {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∪ t) := by
rcases hs with ⟨S, Sopen, Scount, rfl⟩
rcases ht with ⟨T, Topen, Tcount, rfl⟩
rw [sInter_union_sInter]
refine .biInter_of_isOpen (Scount.prod Tcount) ?_
rintro ⟨a, b⟩ ⟨ha, hb⟩
exact (Sopen a ha).union (Topen b hb) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.union | The union of two Gδ sets is a Gδ set. |
IsGδ.sUnion {S : Set (Set X)} (hS : S.Finite) (h : ∀ s ∈ S, IsGδ s) : IsGδ (⋃₀ S) := by
induction S, hS using Set.Finite.induction_on with
| empty => simp
| insert _ _ ih =>
simp only [forall_mem_insert, sUnion_insert] at *
exact h.1.union (ih h.2) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.sUnion | The union of finitely many Gδ sets is a Gδ set, `Set.sUnion` version. |
IsGδ.biUnion {s : Set ι} (hs : s.Finite) {f : ι → Set X} (h : ∀ i ∈ s, IsGδ (f i)) :
IsGδ (⋃ i ∈ s, f i) := by
rw [← sUnion_image]
exact .sUnion (hs.image _) (forall_mem_image.2 h) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.biUnion | The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. |
IsGδ.iUnion [Finite ι'] {f : ι' → Set X} (h : ∀ i, IsGδ (f i)) : IsGδ (⋃ i, f i) :=
.sUnion (finite_range _) <| forall_mem_range.2 h | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsGδ.iUnion | The union of finitely many Gδ sets is a Gδ set, bounded indexed union version. |
residual (X : Type*) [TopologicalSpace X] : Filter X :=
Filter.countableGenerate { t | IsOpen t ∧ Dense t } | def | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | residual | A set `s` is called *residual* if it includes a countable intersection of dense open sets. |
countableInterFilter_residual : CountableInterFilter (residual X) := by
rw [residual]; infer_instance | instance | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | countableInterFilter_residual | null |
residual_of_dense_open {s : Set X} (ho : IsOpen s) (hd : Dense s) : s ∈ residual X :=
CountableGenerateSets.basic ⟨ho, hd⟩ | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | residual_of_dense_open | Dense open sets are residual. |
residual_of_dense_Gδ {s : Set X} (ho : IsGδ s) (hd : Dense s) : s ∈ residual X := by
rcases ho with ⟨T, To, Tct, rfl⟩
exact
(countable_sInter_mem Tct).mpr fun t tT =>
residual_of_dense_open (To t tT) (hd.mono (sInter_subset_of_mem tT)) | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | residual_of_dense_Gδ | Dense Gδ sets are residual. |
mem_residual_iff {s : Set X} :
s ∈ residual X ↔
∃ S : Set (Set X), (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s :=
mem_countableGenerate_iff.trans <| by simp_rw [subset_def, mem_setOf, forall_and, and_assoc] | theorem | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | mem_residual_iff | A set is residual iff it includes a countable intersection of dense open sets. |
IsNowhereDense (s : Set X) := interior (closure s) = ∅ | def | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsNowhereDense | A set is called **nowhere dense** iff its closure has empty interior. |
@[simp]
isNowhereDense_empty : IsNowhereDense (∅ : Set X) := by
rw [IsNowhereDense, closure_empty, interior_empty] | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | isNowhereDense_empty | The empty set is nowhere dense. |
IsClosed.isNowhereDense_iff {s : Set X} (hs : IsClosed s) :
IsNowhereDense s ↔ interior s = ∅ := by
rw [IsNowhereDense, IsClosed.closure_eq hs] | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsClosed.isNowhereDense_iff | A closed set is nowhere dense iff its interior is empty. |
protected IsNowhereDense.closure {s : Set X} (hs : IsNowhereDense s) :
IsNowhereDense (closure s) := by
rwa [IsNowhereDense, closure_closure] | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsNowhereDense.closure | If a set `s` is nowhere dense, so is its closure. |
IsNowhereDense.subset_of_closed_isNowhereDense {s : Set X} (hs : IsNowhereDense s) :
∃ t : Set X, s ⊆ t ∧ IsNowhereDense t ∧ IsClosed t :=
⟨closure s, subset_closure, ⟨hs.closure, isClosed_closure⟩⟩ | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsNowhereDense.subset_of_closed_isNowhereDense | A nowhere dense set `s` is contained in a closed nowhere dense set (namely, its closure). |
isClosed_isNowhereDense_iff_compl {s : Set X} :
IsClosed s ∧ IsNowhereDense s ↔ IsOpen sᶜ ∧ Dense sᶜ := by
rw [and_congr_right IsClosed.isNowhereDense_iff,
isOpen_compl_iff, interior_eq_empty_iff_dense_compl] | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | isClosed_isNowhereDense_iff_compl | A set `s` is closed and nowhere dense iff its complement `sᶜ` is open and dense. |
IsMeagre (s : Set X) := sᶜ ∈ residual X | def | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsMeagre | A set is called **meagre** iff its complement is a residual (or comeagre) set. |
IsMeagre.empty : IsMeagre (∅ : Set X) := by
rw [IsMeagre, compl_empty]
exact Filter.univ_mem | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsMeagre.empty | The empty set is meagre. |
IsMeagre.mono {s t : Set X} (hs : IsMeagre s) (hts : t ⊆ s) : IsMeagre t :=
Filter.mem_of_superset hs (compl_subset_compl.mpr hts) | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsMeagre.mono | Subsets of meagre sets are meagre. |
IsMeagre.inter {s t : Set X} (hs : IsMeagre s) : IsMeagre (s ∩ t) :=
hs.mono inter_subset_left | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsMeagre.inter | An intersection with a meagre set is meagre. |
IsMeagre.union {s t : Set X} (hs : IsMeagre s) (ht : IsMeagre t) : IsMeagre (s ∪ t) := by
rw [IsMeagre, compl_union]
exact inter_mem hs ht | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | IsMeagre.union | A union of two meagre sets is meagre. |
isMeagre_iUnion [Countable ι] {f : ι → Set X} (hs : ∀ i, IsMeagre (f i)) :
IsMeagre (⋃ i, f i) := by
rw [IsMeagre, compl_iUnion]
exact countable_iInter_mem.mpr hs | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | isMeagre_iUnion | A countable union of meagre sets is meagre. |
isMeagre_iff_countable_union_isNowhereDense {s : Set X} :
IsMeagre s ↔ ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S := by
rw [IsMeagre, mem_residual_iff, compl_bijective.surjective.image_surjective.exists]
simp_rw [← and_assoc, ← forall_and, forall_mem_image, ← isClosed_isNowhereDense_iff_compl,
sInter_image, ← compl_iUnion₂, compl_subset_compl, ← sUnion_eq_biUnion, and_assoc]
refine ⟨fun ⟨S, hS, hc, hsub⟩ ↦ ⟨S, fun s hs ↦ (hS hs).2, ?_, hsub⟩, ?_⟩
· rw [← compl_compl_image S]; exact hc.image _
· intro ⟨S, hS, hc, hsub⟩
use closure '' S
rw [forall_mem_image]
exact ⟨fun s hs ↦ ⟨isClosed_closure, (hS s hs).closure⟩,
(hc.image _).image _, hsub.trans (sUnion_mono_subsets fun s ↦ subset_closure)⟩ | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | isMeagre_iff_countable_union_isNowhereDense | A set is meagre iff it is contained in a countable union of nowhere dense sets. |
nonempty_of_not_isMeagre {s : Set X} (hs : ¬IsMeagre s) : s.Nonempty := by
contrapose! hs
simpa [hs] using IsMeagre.empty | lemma | Topology | [
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Closure"
] | Mathlib/Topology/GDelta/Basic.lean | nonempty_of_not_isMeagre | A set of second category (i.e. non-meagre) is nonempty. |
IsGδ.setOf_continuousAt [PseudoMetrizableSpace Y] (f : X → Y) :
IsGδ { x | ContinuousAt f x } := by
let _ := pseudoMetrizableSpacePseudoMetric Y
obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toHasBasis,
forall_prop_of_true, setOf_forall]
refine .iInter fun k ↦ IsOpen.isGδ <| isOpen_iff_mem_nhds.2 fun x ↦ ?_
rintro ⟨s, ⟨hsx, hso⟩, hsU⟩
filter_upwards [IsOpen.mem_nhds hso hsx] with _ hy using ⟨s, ⟨hy, hso⟩, hsU⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.HausdorffDistance",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Separation.GDelta"
] | Mathlib/Topology/GDelta/MetrizableSpace.lean | IsGδ.setOf_continuousAt | null |
ContinuousEval (F : Type*) (X Y : outParam Type*) [FunLike F X Y]
[TopologicalSpace F] [TopologicalSpace X] [TopologicalSpace Y] : Prop where
/-- Evaluation of a bundled morphism at a point is continuous in both variables. -/
continuous_eval : Continuous fun fx : F × X ↦ fx.1 fx.2
export ContinuousEval (continuous_eval)
variable {F X Y Z : Type*} [FunLike F X Y]
[TopologicalSpace F] [TopologicalSpace X] [TopologicalSpace Y] [ContinuousEval F X Y]
[TopologicalSpace Z] {f : Z → F} {g : Z → X} {s : Set Z} {z : Z}
@[continuity, fun_prop] | class | Topology | [
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/Hom/ContinuousEval.lean | ContinuousEval | A typeclass saying that `F` is a bundled morphism class (in the sense of `FunLike`)
with a topology such that `fun (f, x) : F × X ↦ f x` is a continuous function. |
protected Continuous.eval (hf : Continuous f) (hg : Continuous g) :
Continuous fun z ↦ f z (g z) :=
continuous_eval.comp (hf.prodMk hg) | theorem | Topology | [
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/Hom/ContinuousEval.lean | Continuous.eval | null |
ContinuousEval.of_continuous_forget {F' : Type*} [FunLike F' X Y] [TopologicalSpace F']
{f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g := by intro; rfl) :
ContinuousEval F' X Y where
continuous_eval := by simpa only [← hf] using hc.fst'.eval continuous_snd | theorem | Topology | [
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/Hom/ContinuousEval.lean | ContinuousEval.of_continuous_forget | If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEval`,
then `F'` satisfies this predicate too.
The word "forget" in the name is motivated by the term "forgetful functor". |
protected Filter.Tendsto.eval {α : Type*} {l : Filter α} {f : α → F} {f₀ : F}
{g : α → X} {x₀ : X} (hf : Tendsto f l (𝓝 f₀)) (hg : Tendsto g l (𝓝 x₀)) :
Tendsto (fun a ↦ f a (g a)) l (𝓝 (f₀ x₀)) :=
(ContinuousEval.continuous_eval.tendsto _).comp (hf.prodMk_nhds hg)
protected nonrec theorem ContinuousAt.eval (hf : ContinuousAt f z) (hg : ContinuousAt g z) :
ContinuousAt (fun z ↦ f z (g z)) z :=
hf.eval hg
protected nonrec theorem ContinuousWithinAt.eval (hf : ContinuousWithinAt f s z)
(hg : ContinuousWithinAt g s z) : ContinuousWithinAt (fun z ↦ f z (g z)) s z :=
hf.eval hg | theorem | Topology | [
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/Hom/ContinuousEval.lean | Filter.Tendsto.eval | null |
protected ContinuousOn.eval (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun z ↦ f z (g z)) s :=
fun z hz ↦ (hf z hz).eval (hg z hz) | theorem | Topology | [
"Mathlib.Topology.Hom.ContinuousEvalConst",
"Mathlib.Topology.ContinuousMap.Defs"
] | Mathlib/Topology/Hom/ContinuousEval.lean | ContinuousOn.eval | null |
ContinuousEvalConst (F : Type*) (α X : outParam Type*) [FunLike F α X]
[TopologicalSpace F] [TopologicalSpace X] : Prop where
continuous_eval_const (x : α) : Continuous fun f : F ↦ f x
export ContinuousEvalConst (continuous_eval_const) | class | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | ContinuousEvalConst | A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`)
with a topology on `F` such that evaluation at a point is continuous in `f : F`. |
ContinuousEvalConst.of_continuous_forget {F' : Type*} [FunLike F' α X] [TopologicalSpace F']
{f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g := by intro; rfl) :
ContinuousEvalConst F' α X where
continuous_eval_const x := by simpa only [← hf] using (continuous_eval_const x).comp hc
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | ContinuousEvalConst.of_continuous_forget | If a type `F'` of bundled morphisms admits a continuous projection
to a type satisfying `ContinuousEvalConst`,
then `F'` satisfies this predicate too.
The word "forget" in the name is motivated by the term "forgetful functor". |
protected Continuous.eval_const (hf : Continuous f) (x : α) : Continuous (f · x) :=
(continuous_eval_const x).comp hf | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | Continuous.eval_const | null |
continuous_coeFun : Continuous (DFunLike.coe : F → α → X) :=
continuous_pi continuous_eval_const | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | continuous_coeFun | null |
protected Continuous.coeFun (hf : Continuous f) : Continuous fun z ↦ ⇑(f z) :=
continuous_pi hf.eval_const | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | Continuous.coeFun | null |
protected Filter.Tendsto.eval_const {ι : Type*} {l : Filter ι} {f : ι → F} {g : F}
(hf : Tendsto f l (𝓝 g)) (a : α) : Tendsto (f · a) l (𝓝 (g a)) :=
((continuous_id.eval_const a).tendsto _).comp hf | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | Filter.Tendsto.eval_const | null |
protected Filter.Tendsto.coeFun {ι : Type*} {l : Filter ι} {f : ι → F} {g : F}
(hf : Tendsto f l (𝓝 g)) : Tendsto (fun i ↦ ⇑(f i)) l (𝓝 ⇑g) :=
(continuous_id.coeFun.tendsto _).comp hf
protected nonrec theorem ContinuousAt.eval_const (hf : ContinuousAt f z) (x : α) :
ContinuousAt (f · x) z :=
hf.eval_const x
protected nonrec theorem ContinuousAt.coeFun (hf : ContinuousAt f z) :
ContinuousAt (fun z ↦ ⇑(f z)) z :=
hf.coeFun
protected nonrec theorem ContinuousWithinAt.eval_const (hf : ContinuousWithinAt f s z) (x : α) :
ContinuousWithinAt (f · x) s z :=
hf.eval_const x
protected nonrec theorem ContinuousWithinAt.coeFun (hf : ContinuousWithinAt f s z) :
ContinuousWithinAt (fun z ↦ ⇑(f z)) s z :=
hf.coeFun | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | Filter.Tendsto.coeFun | null |
protected ContinuousOn.eval_const (hf : ContinuousOn f s) (x : α) :
ContinuousOn (f · x) s :=
fun z hz ↦ (hf z hz).eval_const x | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | ContinuousOn.eval_const | null |
protected ContinuousOn.coeFun (hf : ContinuousOn f s) (x : α) : ContinuousOn (f · x) s :=
fun z hz ↦ (hf z hz).eval_const x | theorem | Topology | [
"Mathlib.Topology.Constructions"
] | Mathlib/Topology/Hom/ContinuousEvalConst.lean | ContinuousOn.coeFun | null |
ContinuousOpenMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] extends
ContinuousMap α β where
map_open' : IsOpenMap toFun
@[inherit_doc] infixr:25 " →CO " => ContinuousOpenMap | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | ContinuousOpenMap | The type of continuous open maps from `α` to `β`, aka Priestley homomorphisms. |
ContinuousOpenMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
[TopologicalSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where
map_open (f : F) : IsOpenMap f | class | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | ContinuousOpenMapClass | `ContinuousOpenMapClass F α β` states that `F` is a type of continuous open maps.
You should extend this class when you extend `ContinuousOpenMap`. |
instFunLike : FunLike (α →CO β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | instFunLike | null |
toFun_eq_coe {f : α →CO β} : f.toFun = (f : α → β) :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | toFun_eq_coe | null |
@[simp]
coe_toContinuousMap (f : α →CO β) : (f.toContinuousMap : α → β) = f := rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | coe_toContinuousMap | `simp`-normal form of `toFun_eq_coe`. |
ext {f g : α →CO β} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext f g h | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | ext | null |
protected copy (f : α →CO β) (f' : α → β) (h : f' = f) : α →CO β :=
⟨f.toContinuousMap.copy f' <| h, h.symm.subst f.map_open'⟩
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | copy | Copy of a `ContinuousOpenMap` with a new `ContinuousMap` equal to the old one. Useful to fix
definitional equalities. |
coe_copy (f : α →CO β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | coe_copy | null |
copy_eq (f : α →CO β) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
DFunLike.ext' h
variable (α) | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | copy_eq | null |
protected id : α →CO α :=
⟨ContinuousMap.id _, IsOpenMap.id⟩ | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | id | `id` as a `ContinuousOpenMap`. |
@[simp, norm_cast]
coe_id : ⇑(ContinuousOpenMap.id α) = id :=
rfl
variable {α}
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | coe_id | null |
id_apply (a : α) : ContinuousOpenMap.id α a = a :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | id_apply | null |
comp (f : β →CO γ) (g : α →CO β) : ContinuousOpenMap α γ :=
⟨f.toContinuousMap.comp g.toContinuousMap, f.map_open'.comp g.map_open'⟩
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | comp | Composition of `ContinuousOpenMap`s as a `ContinuousOpenMap`. |
coe_comp (f : β →CO γ) (g : α →CO β) : (f.comp g : α → γ) = f ∘ g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | coe_comp | null |
comp_apply (f : β →CO γ) (g : α →CO β) (a : α) : (f.comp g) a = f (g a) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | comp_apply | null |
comp_assoc (f : γ →CO δ) (g : β →CO γ) (h : α →CO β) :
(f.comp g).comp h = f.comp (g.comp h) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | comp_assoc | null |
comp_id (f : α →CO β) : f.comp (ContinuousOpenMap.id α) = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | comp_id | null |
id_comp (f : α →CO β) : (ContinuousOpenMap.id β).comp f = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | id_comp | null |
cancel_right {g₁ g₂ : β →CO γ} {f : α →CO β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=
⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, fun h => congr_arg₂ _ h rfl⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | cancel_right | null |
cancel_left {g : β →CO γ} {f₁ f₂ : α →CO β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Basic"
] | Mathlib/Topology/Hom/Open.lean | cancel_left | null |
Homeomorph (X : Type*) (Y : Type*) [TopologicalSpace X] [TopologicalSpace Y]
extends X ≃ Y where
/-- The forward map of a homeomorphism is a continuous function. -/
continuous_toFun : Continuous toFun := by continuity
/-- The inverse map of a homeomorphism is a continuous function. -/
continuous_invFun : Continuous invFun := by continuity
@[inherit_doc]
infixl:25 " ≃ₜ " => Homeomorph | structure | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | Homeomorph | Homeomorphism between `X` and `Y`, also called topological isomorphism |
toEquiv_injective : Function.Injective (toEquiv : X ≃ₜ Y → X ≃ Y)
| ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | toEquiv_injective | null |
@[simp] homeomorph_mk_coe (a : X ≃ Y) (b c) : (Homeomorph.mk a b c : X → Y) = a :=
rfl | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | homeomorph_mk_coe | null |
protected empty [IsEmpty X] [IsEmpty Y] : X ≃ₜ Y where
__ := Equiv.equivOfIsEmpty X Y | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | empty | The unique homeomorphism between two empty types. |
@[symm]
protected symm (h : X ≃ₜ Y) : Y ≃ₜ X where
continuous_toFun := h.continuous_invFun
continuous_invFun := h.continuous_toFun
toEquiv := h.toEquiv.symm
@[simp] theorem symm_symm (h : X ≃ₜ Y) : h.symm.symm = h := rfl | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm | Inverse of a homeomorphism. |
symm_bijective : Function.Bijective (Homeomorph.symm : (X ≃ₜ Y) → Y ≃ₜ X) :=
Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_bijective | null |
Simps.symm_apply (h : X ≃ₜ Y) : Y → X :=
h.symm
initialize_simps_projections Homeomorph (toFun → apply, invFun → symm_apply)
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | Simps.symm_apply | See Note [custom simps projection] |
coe_toEquiv (h : X ≃ₜ Y) : ⇑h.toEquiv = h :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | coe_toEquiv | null |
coe_symm_toEquiv (h : X ≃ₜ Y) : ⇑h.toEquiv.symm = h.symm :=
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | coe_symm_toEquiv | null |
ext {h h' : X ≃ₜ Y} (H : ∀ x, h x = h' x) : h = h' :=
DFunLike.ext _ _ H | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | ext | null |
@[simps! -fullyApplied apply]
protected refl (X : Type*) [TopologicalSpace X] : X ≃ₜ X where
continuous_toFun := continuous_id
continuous_invFun := continuous_id
toEquiv := Equiv.refl X | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | refl | Identity map as a homeomorphism. |
@[trans]
protected trans (h₁ : X ≃ₜ Y) (h₂ : Y ≃ₜ Z) : X ≃ₜ Z where
continuous_toFun := h₂.continuous_toFun.comp h₁.continuous_toFun
continuous_invFun := h₁.continuous_invFun.comp h₂.continuous_invFun
toEquiv := Equiv.trans h₁.toEquiv h₂.toEquiv
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | trans | Composition of two homeomorphisms. |
trans_apply (h₁ : X ≃ₜ Y) (h₂ : Y ≃ₜ Z) (x : X) : h₁.trans h₂ x = h₂ (h₁ x) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | trans_apply | null |
symm_trans_apply (f : X ≃ₜ Y) (g : Y ≃ₜ Z) (z : Z) :
(f.trans g).symm z = f.symm (g.symm z) := rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_trans_apply | null |
homeomorph_mk_coe_symm (a : X ≃ Y) (b c) :
((Homeomorph.mk a b c).symm : Y → X) = a.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | homeomorph_mk_coe_symm | null |
refl_symm : (Homeomorph.refl X).symm = Homeomorph.refl X :=
rfl
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | refl_symm | null |
protected continuous (h : X ≃ₜ Y) : Continuous h :=
h.continuous_toFun
@[continuity] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | continuous | null |
protected continuous_symm (h : X ≃ₜ Y) : Continuous h.symm :=
h.continuous_invFun
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | continuous_symm | null |
apply_symm_apply (h : X ≃ₜ Y) (y : Y) : h (h.symm y) = y :=
h.toEquiv.apply_symm_apply y
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | apply_symm_apply | null |
symm_apply_apply (h : X ≃ₜ Y) (x : X) : h.symm (h x) = x :=
h.toEquiv.symm_apply_apply x
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_apply_apply | null |
self_trans_symm (h : X ≃ₜ Y) : h.trans h.symm = Homeomorph.refl X := by
ext
apply symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | self_trans_symm | null |
symm_trans_self (h : X ≃ₜ Y) : h.symm.trans h = Homeomorph.refl Y := by
ext
apply apply_symm_apply | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_trans_self | null |
protected bijective (h : X ≃ₜ Y) : Function.Bijective h :=
h.toEquiv.bijective | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | bijective | null |
protected injective (h : X ≃ₜ Y) : Function.Injective h :=
h.toEquiv.injective | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | injective | null |
protected surjective (h : X ≃ₜ Y) : Function.Surjective h :=
h.toEquiv.surjective | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | surjective | null |
changeInv (f : X ≃ₜ Y) (g : Y → X) (hg : Function.RightInverse g f) : X ≃ₜ Y :=
haveI : g = f.symm := (f.left_inv.eq_rightInverse hg).symm
{ toFun := f
invFun := g
left_inv := by convert f.left_inv
right_inv := by convert f.right_inv using 1
continuous_toFun := f.continuous
continuous_invFun := by convert f.symm.continuous }
@[simp] | def | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | changeInv | Change the homeomorphism `f` to make the inverse function definitionally equal to `g`. |
symm_comp_self (h : X ≃ₜ Y) : h.symm ∘ h = id :=
funext h.symm_apply_apply
@[simp] | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | symm_comp_self | null |
self_comp_symm (h : X ≃ₜ Y) : h ∘ h.symm = id :=
funext h.apply_symm_apply | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | self_comp_symm | null |
range_coe (h : X ≃ₜ Y) : range h = univ := by simp | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | range_coe | null |
image_symm (h : X ≃ₜ Y) : image h.symm = preimage h :=
funext h.symm.toEquiv.image_eq_preimage | theorem | Topology | [
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Homeomorph/Defs.lean | image_symm | null |
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