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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isOpenMap_iff_clusterPt_comap :
IsOpenMap f ↔ ∀ x l, ClusterPt (f x) l → ClusterPt x (comap f l) := by
refine ⟨fun hf _ _ ↦ hf.clusterPt_comap, fun h ↦ ?_⟩
simp only [isOpenMap_iff_nhds_le, le_map_iff]
intro x s hs
contrapose! hs
rw [← mem_interior_iff_mem_nhds, mem_interior_iff_not_clusterPt_compl, not_not] at hs ⊢
exact (h _ _ hs).mono <| by simp [subset_preimage_image] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff_clusterPt_comap | null |
isOpenMap_iff_image_interior : IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) :=
⟨IsOpenMap.image_interior_subset, fun hs u hu =>
subset_interior_iff_isOpen.mp <| by simpa only [hu.interior_eq] using hs u⟩
@[deprecated (since := "2025-08-30")] alias isOpenMap_iff_interior := isOpenMap_iff_image_interior | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff_image_interior | null |
isOpenMap_iff_closure_kernImage :
IsOpenMap f ↔ ∀ {s : Set X}, closure (kernImage f s) ⊆ kernImage f (closure s) := by
rw [isOpenMap_iff_image_interior, compl_surjective.forall]
simp [kernImage_eq_compl] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff_closure_kernImage | A map is open if and only if the `Set.kernImage` of every *closed* set is closed. |
protected Topology.IsInducing.isOpenMap (hi : IsInducing f) (ho : IsOpen (range f)) :
IsOpenMap f :=
IsOpenMap.of_nhds_le fun _ => (hi.map_nhds_of_mem _ <| IsOpen.mem_nhds ho <| mem_range_self _).ge | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | Topology.IsInducing.isOpenMap | An inducing map with an open range is an open map. |
protected Dense.preimage {s : Set Y} (hs : Dense s) (hf : IsOpenMap f) :
Dense (f ⁻¹' s) := fun x ↦
hf.preimage_closure_subset_closure_preimage <| hs (f x) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | Dense.preimage | Preimage of a dense set under an open map is dense. |
protected id : IsClosedMap (@id X) := fun s hs => by rwa [image_id] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
protected comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) := by
intro s hs
rw [image_comp]
exact hg _ (hf _ hs) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
protected of_comp_surjective (hf : Surjective f) (hf' : Continuous f)
(hfg : IsClosedMap (g ∘ f)) : IsClosedMap g := by
intro K hK
rw [← image_preimage_eq K hf, ← image_comp]
exact hfg _ (hK.preimage hf') | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp_surjective | null |
closure_image_subset (hf : IsClosedMap f) (s : Set X) :
closure (f '' s) ⊆ f '' closure s :=
closure_minimal (image_mono subset_closure) (hf _ isClosed_closure) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | closure_image_subset | null |
of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
(r_inv : RightInverse f f') : IsClosedMap f := fun s hs => by
rw [image_eq_preimage_of_inverse r_inv l_inv]
exact hs.preimage h | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_inverse | null |
of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
IsClosedMap f := by
intro s hs; rcases eq_empty_or_nonempty s with h2s | h2s
· simp_rw [h2s, image_empty, isClosed_empty]
· exact h s hs h2s | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_nonempty | null |
isClosed_range (hf : IsClosedMap f) : IsClosed (range f) :=
@image_univ _ _ f ▸ hf _ isClosed_univ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_range | null |
isQuotientMap (hcl : IsClosedMap f) (hcont : Continuous f)
(hsurj : Surjective f) : IsQuotientMap f :=
isQuotientMap_iff_isClosed.2 ⟨hsurj, fun s =>
⟨fun hs => hs.preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isQuotientMap | null |
isClosedMap_iff_kernImage :
IsClosedMap f ↔ ∀ {u : Set X}, IsOpen u → IsOpen (kernImage f u) := by
rw [IsClosedMap, compl_surjective.forall]
simp [kernImage_eq_compl] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_kernImage | A map is closed if and only if the `Set.kernImage` of every *open* set is open.
One way to understand this result is that `f : X → Y` is closed if and only if its fibers vary in an
**upper hemicontinuous** way: for any open subset `U ⊆ X`, the set of all `y ∈ Y` such that
`f ⁻¹' {y} ⊆ U` is open in `Y`. |
Topology.IsInducing.isClosedMap (hf : IsInducing f) (h : IsClosed (range f)) :
IsClosedMap f := by
intro s hs
rcases hf.isClosed_iff.1 hs with ⟨t, ht, rfl⟩
rw [image_preimage_eq_inter_range]
exact ht.inter h | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | Topology.IsInducing.isClosedMap | null |
isClosedMap_iff_closure_image :
IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s :=
⟨IsClosedMap.closure_image_subset, fun hs c hc =>
isClosed_of_closure_subset <|
calc
closure (f '' c) ⊆ f '' closure c := hs c
_ = f '' c := by rw [hc.closure_eq]⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_closure_image | null |
isClosedMap_iff_kernImage_interior :
IsClosedMap f ↔ ∀ {s : Set X}, kernImage f (interior s) ⊆ interior (kernImage f s) := by
rw [isClosedMap_iff_closure_image, compl_surjective.forall]
simp [kernImage_eq_compl] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_kernImage_interior | null |
isClosedMap_iff_clusterPt :
IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by
simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt,
and_comm] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_clusterPt | A map `f : X → Y` is closed if and only if for all sets `s`, any cluster point of `f '' s` is
the image by `f` of some cluster point of `s`.
If you require this for all filters instead of just principal filters, and also that `f` is
continuous, you get the notion of **proper map**. See `isProperMap_iff_clusterPt`. |
isClosedMap_iff_comap_nhdsSet_le :
IsClosedMap f ↔ ∀ {s : Set Y}, comap f (𝓝ˢ s) ≤ 𝓝ˢ (f ⁻¹' s) := by
simp_rw [Filter.le_def, mem_comap'', ← subset_interior_iff_mem_nhdsSet,
← subset_kernImage_iff, isClosedMap_iff_kernImage_interior]
exact ⟨fun H s t hst ↦ hst.trans H, fun H s ↦ H _ subset_rfl⟩
alias ⟨IsClosedMap.comap_nhdsSet_le, _⟩ := isClosedMap_iff_comap_nhdsSet_le | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_comap_nhdsSet_le | null |
isClosedMap_iff_comap_nhds_le :
IsClosedMap f ↔ ∀ {y : Y}, comap f (𝓝 y) ≤ 𝓝ˢ (f ⁻¹' {y}) := by
rw [isClosedMap_iff_comap_nhdsSet_le]
constructor
· exact fun H y ↦ nhdsSet_singleton (x := y) ▸ H
· intro H s
rw [← Set.biUnion_of_singleton s]
simp_rw [preimage_iUnion, nhdsSet_iUnion, comap_iSup, nhdsSet_singleton]
exact iSup₂_mono fun _ _ ↦ H
alias ⟨IsClosedMap.comap_nhds_le, _⟩ := isClosedMap_iff_comap_nhds_le | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap_iff_comap_nhds_le | null |
IsClosedMap.comap_nhds_eq (hf : IsClosedMap f) (hf' : Continuous f) (y : Y) :
comap f (𝓝 y) = 𝓝ˢ (f ⁻¹' {y}) :=
le_antisymm (isClosedMap_iff_comap_nhds_le.mp hf)
(nhdsSet_le.mpr fun x hx ↦ hx ▸ (hf'.tendsto x).le_comap) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.comap_nhds_eq | null |
IsClosedMap.comap_nhdsSet_eq (hf : IsClosedMap f) (hf' : Continuous f) (s : Set Y) :
comap f (𝓝ˢ s) = 𝓝ˢ (f ⁻¹' s) :=
le_antisymm (isClosedMap_iff_comap_nhdsSet_le.mp hf)
(nhdsSet_le.mpr fun x hx ↦ (hf'.tendsto x).le_comap.trans (comap_mono (nhds_le_nhdsSet hx))) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.comap_nhdsSet_eq | null |
IsClosedMap.eventually_nhds_fiber (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y)
(H : ∀ x₀ ∈ f ⁻¹' {y₀}, ∀ᶠ x in 𝓝 x₀, p x) :
∀ᶠ y in 𝓝 y₀, ∀ x ∈ f ⁻¹' {y}, p x := by
rw [← eventually_nhdsSet_iff_forall] at H
replace H := H.filter_mono hf.comap_nhds_le
rwa [eventually_comap] at H | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.eventually_nhds_fiber | Assume `f` is a closed map. If some property `p` holds around every point in the fiber of `f`
at `y₀`, then for any `y` close enough to `y₀` we have that `p` holds on the fiber at `y`. |
IsClosedMap.frequently_nhds_fiber (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y)
(H : ∃ᶠ y in 𝓝 y₀, ∃ x ∈ f ⁻¹' {y}, p x) :
∃ x₀ ∈ f ⁻¹' {y₀}, ∃ᶠ x in 𝓝 x₀, p x := by
/-
Note: this result could also be seen as a reformulation of `isClosedMap_iff_clusterPt`.
One would then be able to deduce the `eventually` statement,
and then go back to `isClosedMap_iff_comap_nhdsSet_le`.
Ultimately, this makes no difference.
-/
revert H
contrapose
simpa only [not_frequently, not_exists, not_and] using hf.eventually_nhds_fiber y₀ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.frequently_nhds_fiber | Assume `f` is a closed map. If there are points `y` arbitrarily close to `y₀` such that `p`
holds for at least some `x ∈ f ⁻¹' {y}`, then one can find `x₀ ∈ f ⁻¹' {y₀}` such that there
are points `x` arbitrarily close to `x₀` which satisfy `p`. |
IsClosedMap.closure_image_eq_of_continuous
(f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) :
closure (f '' s) = f '' closure s :=
subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.closure_image_eq_of_continuous | null |
IsClosedMap.lift'_closure_map_eq
(f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) :
(map f F).lift' closure = map f (F.lift' closure) := by
rw [map_lift'_eq2 (monotone_closure Y), map_lift'_eq (monotone_closure X)]
congr 1
ext s : 1
exact f_closed.closure_image_eq_of_continuous f_cont s | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.lift'_closure_map_eq | null |
IsClosedMap.mapClusterPt_iff_lift'_closure
{F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} :
MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by
rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont,
← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsClosedMap.mapClusterPt_iff_lift'_closure | null |
IsOpenEmbedding.isEmbedding (hf : IsOpenEmbedding f) : IsEmbedding f := hf.toIsEmbedding | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.isEmbedding | null |
IsOpenEmbedding.isInducing (hf : IsOpenEmbedding f) : IsInducing f :=
hf.isEmbedding.isInducing | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.isInducing | null |
IsOpenEmbedding.isOpenMap (hf : IsOpenEmbedding f) : IsOpenMap f :=
hf.isEmbedding.isInducing.isOpenMap hf.isOpen_range | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.isOpenMap | null |
IsOpenEmbedding.map_nhds_eq (hf : IsOpenEmbedding f) (x : X) :
map f (𝓝 x) = 𝓝 (f x) :=
hf.isEmbedding.map_nhds_of_mem _ <| hf.isOpen_range.mem_nhds <| mem_range_self _ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.map_nhds_eq | null |
IsOpenEmbedding.isOpen_iff_image_isOpen (hf : IsOpenEmbedding f) {s : Set X} :
IsOpen s ↔ IsOpen (f '' s) where
mp := hf.isOpenMap s
mpr h := by convert ← h.preimage hf.isEmbedding.continuous; apply preimage_image_eq _ hf.injective | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.isOpen_iff_image_isOpen | null |
IsOpenEmbedding.tendsto_nhds_iff [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y}
(hg : IsOpenEmbedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) :=
hg.isEmbedding.tendsto_nhds_iff | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.tendsto_nhds_iff | null |
IsOpenEmbedding.tendsto_nhds_iff' (hf : IsOpenEmbedding f) {l : Filter Z} {x : X} :
Tendsto (g ∘ f) (𝓝 x) l ↔ Tendsto g (𝓝 (f x)) l := by
rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.tendsto_nhds_iff' | null |
IsOpenEmbedding.continuousAt_iff [TopologicalSpace Z] (hf : IsOpenEmbedding f) {x : X} :
ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) :=
hf.tendsto_nhds_iff' | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.continuousAt_iff | null |
IsOpenEmbedding.continuous (hf : IsOpenEmbedding f) : Continuous f :=
hf.isEmbedding.continuous | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.continuous | null |
IsOpenEmbedding.isOpen_iff_preimage_isOpen (hf : IsOpenEmbedding f) {s : Set Y}
(hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) := by
rw [hf.isOpen_iff_image_isOpen, image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.isOpen_iff_preimage_isOpen | null |
IsOpenEmbedding.of_isEmbedding_isOpenMap (h₁ : IsEmbedding f) (h₂ : IsOpenMap f) :
IsOpenEmbedding f :=
⟨h₁, h₂.isOpen_range⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.of_isEmbedding_isOpenMap | null |
IsEmbedding.isOpenEmbedding_of_surjective (hf : IsEmbedding f) (hsurj : f.Surjective) :
IsOpenEmbedding f :=
⟨hf, hsurj.range_eq ▸ isOpen_univ⟩
alias IsOpenEmbedding.of_isEmbedding := IsEmbedding.isOpenEmbedding_of_surjective | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsEmbedding.isOpenEmbedding_of_surjective | A surjective embedding is an `IsOpenEmbedding`. |
isOpenEmbedding_iff_isEmbedding_isOpenMap : IsOpenEmbedding f ↔ IsEmbedding f ∧ IsOpenMap f :=
⟨fun h => ⟨h.1, h.isOpenMap⟩, fun h => .of_isEmbedding_isOpenMap h.1 h.2⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenEmbedding_iff_isEmbedding_isOpenMap | null |
IsOpenEmbedding.of_continuous_injective_isOpenMap
(h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : IsOpenEmbedding f := by
simp only [isOpenEmbedding_iff_isEmbedding_isOpenMap, isEmbedding_iff, isInducing_iff_nhds, *,
and_true]
exact fun x =>
le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 x) _ h₂ ▸ comap_mono (h₃.nhds_le _)) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsOpenEmbedding.of_continuous_injective_isOpenMap | null |
isOpenEmbedding_iff_continuous_injective_isOpenMap :
IsOpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f :=
⟨fun h => ⟨h.continuous, h.injective, h.isOpenMap⟩, fun h =>
.of_continuous_injective_isOpenMap h.1 h.2.1 h.2.2⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenEmbedding_iff_continuous_injective_isOpenMap | null |
protected id : IsOpenEmbedding (@id X) := ⟨.id, IsOpenMap.id.isOpen_range⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
protected comp (hg : IsOpenEmbedding g)
(hf : IsOpenEmbedding f) : IsOpenEmbedding (g ∘ f) :=
⟨hg.1.comp hf.1, (hg.isOpenMap.comp hf.isOpenMap).isOpen_range⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
isOpenMap_iff (hg : IsOpenEmbedding g) :
IsOpenMap f ↔ IsOpenMap (g ∘ f) := by
simp_rw [isOpenMap_iff_nhds_le, ← map_map, comp, ← hg.map_nhds_eq, map_le_map_iff hg.injective] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff | null |
of_comp_iff (f : X → Y) (hg : IsOpenEmbedding g) :
IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding f := by
simp only [isOpenEmbedding_iff_continuous_injective_isOpenMap, ← hg.isOpenMap_iff, ←
hg.1.continuous_iff, hg.injective.of_comp_iff] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp_iff | null |
of_comp (f : X → Y) (hg : IsOpenEmbedding g) (h : IsOpenEmbedding (g ∘ f)) :
IsOpenEmbedding f := (IsOpenEmbedding.of_comp_iff f hg).1 h | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp | null |
of_isEmpty [IsEmpty X] (f : X → Y) : IsOpenEmbedding f :=
of_isEmbedding_isOpenMap (.of_subsingleton f) (.of_isEmpty f) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_isEmpty | null |
image_mem_nhds {f : X → Y} (hf : IsOpenEmbedding f) {s : Set X} {x : X} :
f '' s ∈ 𝓝 (f x) ↔ s ∈ 𝓝 x := by
rw [← hf.map_nhds_eq, mem_map, preimage_image_eq _ hf.injective] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | image_mem_nhds | null |
isEmbedding (hf : IsClosedEmbedding f) : IsEmbedding f := hf.toIsEmbedding | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isEmbedding | null |
isInducing (hf : IsClosedEmbedding f) : IsInducing f := hf.isEmbedding.isInducing | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isInducing | null |
continuous (hf : IsClosedEmbedding f) : Continuous f := hf.isEmbedding.continuous | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous | null |
tendsto_nhds_iff {g : ι → X} {l : Filter ι} {x : X} (hf : IsClosedEmbedding f) :
Tendsto g l (𝓝 x) ↔ Tendsto (f ∘ g) l (𝓝 (f x)) := hf.isEmbedding.tendsto_nhds_iff | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | tendsto_nhds_iff | null |
isClosedMap (hf : IsClosedEmbedding f) : IsClosedMap f :=
hf.isEmbedding.isInducing.isClosedMap hf.isClosed_range | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedMap | null |
isClosed_iff_image_isClosed (hf : IsClosedEmbedding f) {s : Set X} :
IsClosed s ↔ IsClosed (f '' s) :=
⟨hf.isClosedMap s, fun h => by
rw [← preimage_image_eq s hf.injective]
exact h.preimage hf.continuous⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_iff_image_isClosed | null |
isClosed_iff_preimage_isClosed (hf : IsClosedEmbedding f) {s : Set Y}
(hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) := by
rw [hf.isClosed_iff_image_isClosed, image_preimage_eq_of_subset hs] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_iff_preimage_isClosed | null |
of_isEmbedding_isClosedMap (h₁ : IsEmbedding f) (h₂ : IsClosedMap f) :
IsClosedEmbedding f :=
⟨h₁, image_univ (f := f) ▸ h₂ univ isClosed_univ⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_isEmbedding_isClosedMap | null |
of_continuous_injective_isClosedMap (h₁ : Continuous f) (h₂ : Injective f)
(h₃ : IsClosedMap f) : IsClosedEmbedding f := by
refine .of_isEmbedding_isClosedMap ⟨⟨?_⟩, h₂⟩ h₃
refine h₁.le_induced.antisymm fun s hs => ?_
refine ⟨(f '' sᶜ)ᶜ, (h₃ _ hs.isClosed_compl).isOpen_compl, ?_⟩
rw [preimage_compl, preimage_image_eq _ h₂, compl_compl] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_continuous_injective_isClosedMap | null |
isClosedEmbedding_iff_continuous_injective_isClosedMap {f : X → Y} :
IsClosedEmbedding f ↔ Continuous f ∧ Injective f ∧ IsClosedMap f where
mp h := ⟨h.continuous, h.injective, h.isClosedMap⟩
mpr h := .of_continuous_injective_isClosedMap h.1 h.2.1 h.2.2 | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosedEmbedding_iff_continuous_injective_isClosedMap | null |
protected id : IsClosedEmbedding (@id X) := ⟨.id, IsClosedMap.id.isClosed_range⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
comp (hg : IsClosedEmbedding g) (hf : IsClosedEmbedding f) :
IsClosedEmbedding (g ∘ f) :=
⟨hg.isEmbedding.comp hf.isEmbedding, (hg.isClosedMap.comp hf.isClosedMap).isClosed_range⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
of_comp_iff (hg : IsClosedEmbedding g) : IsClosedEmbedding (g ∘ f) ↔ IsClosedEmbedding f := by
simp_rw [isClosedEmbedding_iff, hg.isEmbedding.of_comp_iff, Set.range_comp,
← hg.isClosed_iff_image_isClosed] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp_iff | null |
closure_image_eq (hf : IsClosedEmbedding f) (s : Set X) :
closure (f '' s) = f '' closure s :=
hf.isClosedMap.closure_image_eq_of_continuous hf.continuous s | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | closure_image_eq | null |
protected id : IsOpenQuotientMap (id : X → X) := ⟨surjective_id, continuous_id, .id⟩ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | id | null |
isQuotientMap (h : IsOpenQuotientMap f) : IsQuotientMap f :=
h.isOpenMap.isQuotientMap h.continuous h.surjective | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | isQuotientMap | An open quotient map is a quotient map. |
iff_isOpenMap_isQuotientMap : IsOpenQuotientMap f ↔ IsOpenMap f ∧ IsQuotientMap f :=
⟨fun h ↦ ⟨h.isOpenMap, h.isQuotientMap⟩, fun ⟨ho, hq⟩ ↦ ⟨hq.surjective, hq.continuous, ho⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | iff_isOpenMap_isQuotientMap | null |
of_isOpenMap_isQuotientMap (ho : IsOpenMap f) (hq : IsQuotientMap f) :
IsOpenQuotientMap f :=
iff_isOpenMap_isQuotientMap.2 ⟨ho, hq⟩ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | of_isOpenMap_isQuotientMap | null |
comp {g : Y → Z} (hg : IsOpenQuotientMap g) (hf : IsOpenQuotientMap f) :
IsOpenQuotientMap (g ∘ f) :=
⟨.comp hg.1 hf.1, .comp hg.2 hf.2, .comp hg.3 hf.3⟩ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | comp | null |
map_nhds_eq (h : IsOpenQuotientMap f) (x : X) : map f (𝓝 x) = 𝓝 (f x) :=
le_antisymm h.continuous.continuousAt <| h.isOpenMap.nhds_le _ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | map_nhds_eq | null |
continuous_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} :
Continuous (g ∘ f) ↔ Continuous g :=
h.isQuotientMap.continuous_iff.symm | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | continuous_comp_iff | null |
continuousAt_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} :
ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
simp only [ContinuousAt, ← h.map_nhds_eq, tendsto_map'_iff, comp_def] | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | continuousAt_comp_iff | null |
dense_preimage_iff (h : IsOpenQuotientMap f) {s : Set Y} : Dense (f ⁻¹' s) ↔ Dense s :=
⟨fun hs ↦ h.surjective.denseRange.dense_of_mapsTo h.continuous hs (mapsTo_preimage _ _),
fun hs ↦ hs.preimage h.isOpenMap⟩ | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | dense_preimage_iff | null |
Topology.IsInducing.isOpenQuotientMap_of_surjective (ind : IsInducing f)
(surj : Function.Surjective f) : IsOpenQuotientMap f where
surjective := surj
continuous := ind.continuous
isOpenMap U U_open := by
obtain ⟨V, hV, rfl⟩ := ind.isOpen_iff.mp U_open
rwa [V.image_preimage_eq surj] | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | Topology.IsInducing.isOpenQuotientMap_of_surjective | null |
Topology.IsInducing.isQuotientMap_of_surjective (ind : IsInducing f)
(surj : Function.Surjective f) : IsQuotientMap f :=
(ind.isOpenQuotientMap_of_surjective surj).isQuotientMap | theorem | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | Topology.IsInducing.isQuotientMap_of_surjective | null |
coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing
(h : g ∘ p = q ∘ f)
(hf : IsInducing f) (hp : Function.Surjective p)
(hq : IsOpenQuotientMap q) (hg : Function.Injective g)
(H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) :
‹TopologicalSpace A›.coinduced p = ‹TopologicalSpace D›.induced g := by
ext U
change IsOpen (p ⁻¹' U) ↔ ∃ V, _
simp_rw [hf.isOpen_iff,
(Set.image_surjective.mpr hq.surjective).exists,
← hq.isQuotientMap.isOpen_preimage]
constructor
· rintro ⟨V, hV, e⟩
refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩
ext x
obtain ⟨x, rfl⟩ := hp x
constructor
· rintro ⟨y, hy, e'⟩
obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩
rw [← hg ((congr_fun h y).trans e')]
exact e.le hy
· intro H
exact ⟨f x, e.ge H, congr_fun h.symm x⟩
· rintro ⟨V, hV, rfl⟩
refine ⟨_, hV, ?_⟩
simp_rw [← Set.preimage_comp, h] | lemma | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing | Given the following diagram with `f` inducing, `p` surjective,
`q` an open quotient map, and `g` injective. Suppose the image of `A` in `B` is stable
under the equivalence mod `q`, then the coinduced topology on `C` (from `A`)
coincides with the induced topology (from `D`).
```
A -f→ B
∣ ∣
p q
↓ ↓
C -g→ D
```
A typical application is when `K ≤ H` are subgroups of `G`, then the quotient topology on `H/K`
is also the subspace topology from `G/K`. |
isEmbedding_of_isOpenQuotientMap_of_isInducing
(h : g ∘ p = q ∘ f)
(hf : IsInducing f) (hp : IsQuotientMap p)
(hq : IsOpenQuotientMap q) (hg : Function.Injective g)
(H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) :
IsEmbedding g :=
⟨⟨hp.eq_coinduced.trans (coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing
f g p q h hf hp.surjective hq hg H)⟩, hg⟩ | lemma | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | isEmbedding_of_isOpenQuotientMap_of_isInducing | null |
isQuotientMap_of_isOpenQuotientMap_of_isInducing
(h : g ∘ p = q ∘ f)
(hf : IsInducing f) (hp : Surjective p)
(hq : IsOpenQuotientMap q) (hg : IsEmbedding g)
(H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) :
IsQuotientMap p :=
⟨hp, hg.eq_induced.trans ((coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing
f g p q h hf hp hq hg.injective H)).symm⟩ | lemma | Topology | [
"Mathlib.Topology.Maps.Basic"
] | Mathlib/Topology/Maps/OpenQuotient.lean | isQuotientMap_of_isOpenQuotientMap_of_isInducing | null |
LipschitzAdd [AddMonoid β] : Prop where
lipschitz_add : ∃ C, LipschitzWith C fun p : β × β => p.1 + p.2 | class | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | LipschitzAdd | Class `LipschitzAdd M` says that the addition `(+) : X × X → X` is Lipschitz jointly in
the two arguments. |
@[to_additive]
LipschitzMul [Monoid β] : Prop where
lipschitz_mul : ∃ C, LipschitzWith C fun p : β × β => p.1 * p.2
variable [Monoid β] | class | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | LipschitzMul | Class `LipschitzMul M` says that the multiplication `(*) : X × X → X` is Lipschitz jointly
in the two arguments. |
@[to_additive /-- The Lipschitz constant of an `AddMonoid` `β` satisfying `LipschitzAdd` -/]
LipschitzMul.C [_i : LipschitzMul β] : ℝ≥0 := Classical.choose _i.lipschitz_mul
variable {β}
@[to_additive] | def | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | LipschitzMul.C | The Lipschitz constant of a monoid `β` satisfying `LipschitzMul` |
lipschitzWith_lipschitz_const_mul_edist [_i : LipschitzMul β] :
LipschitzWith (LipschitzMul.C β) fun p : β × β => p.1 * p.2 :=
Classical.choose_spec _i.lipschitz_mul
variable [LipschitzMul β]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | lipschitzWith_lipschitz_const_mul_edist | null |
lipschitz_with_lipschitz_const_mul :
∀ p q : β × β, dist (p.1 * p.2) (q.1 * q.2) ≤ LipschitzMul.C β * dist p q := by
rw [← lipschitzWith_iff_dist_le_mul]
exact lipschitzWith_lipschitz_const_mul_edist
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | lipschitz_with_lipschitz_const_mul | null |
@[to_additive]
Submonoid.lipschitzMul (s : Submonoid β) : LipschitzMul s where
lipschitz_mul := ⟨LipschitzMul.C β, by
rintro ⟨x₁, x₂⟩ ⟨y₁, y₂⟩
convert lipschitzWith_lipschitz_const_mul_edist ⟨(x₁ : β), x₂⟩ ⟨y₁, y₂⟩ using 1⟩
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | Submonoid.lipschitzMul | null |
MulOpposite.lipschitzMul : LipschitzMul βᵐᵒᵖ where
lipschitz_mul := ⟨LipschitzMul.C β, fun ⟨x₁, x₂⟩ ⟨y₁, y₂⟩ =>
(lipschitzWith_lipschitz_const_mul_edist ⟨x₂.unop, x₁.unop⟩ ⟨y₂.unop, y₁.unop⟩).trans_eq
(congr_arg _ <| max_comm _ _)⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | MulOpposite.lipschitzMul | null |
Real.hasLipschitzAdd : LipschitzAdd ℝ where
lipschitz_add := ⟨2, LipschitzWith.of_dist_le_mul fun p q => by
simp only [Real.dist_eq, Prod.dist_eq, NNReal.coe_ofNat,
add_sub_add_comm, two_mul]
refine le_trans (abs_add_le (p.1 - q.1) (p.2 - q.2)) ?_
exact add_le_add (le_max_left _ _) (le_max_right _ _)⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | Real.hasLipschitzAdd | null |
NNReal.hasLipschitzAdd : LipschitzAdd ℝ≥0 where
lipschitz_add := ⟨LipschitzAdd.C ℝ, by
rintro ⟨x₁, x₂⟩ ⟨y₁, y₂⟩
exact lipschitzWith_lipschitz_const_add_edist ⟨(x₁ : ℝ), x₂⟩ ⟨y₁, y₂⟩⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | NNReal.hasLipschitzAdd | null |
IsBoundedSMul : Prop where
dist_smul_pair' : ∀ x : α, ∀ y₁ y₂ : β, dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂
dist_pair_smul' : ∀ x₁ x₂ : α, ∀ y : β, dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0
@[deprecated (since := "2025-03-10")] alias BoundedSMul := IsBoundedSMul
variable {α β}
variable [IsBoundedSMul α β] | class | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | IsBoundedSMul | Mixin typeclass on a scalar action of a metric space `α` on a metric space `β` both with
distinguished points `0`, requiring compatibility of the action in the sense that
`dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂` and
`dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0`.
If `[NormedDivisionRing α] [SeminormedAddCommGroup β] [Module α β]` are assumed, then prefer writing
`[NormSMulClass α β]` instead of using `[IsBoundedSMul α β]`, since while equivalent, typeclass
search can only infer the latter from the former and not vice versa. |
dist_smul_pair (x : α) (y₁ y₂ : β) : dist (x • y₁) (x • y₂) ≤ dist x 0 * dist y₁ y₂ :=
IsBoundedSMul.dist_smul_pair' x y₁ y₂ | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | dist_smul_pair | null |
dist_pair_smul (x₁ x₂ : α) (y : β) : dist (x₁ • y) (x₂ • y) ≤ dist x₁ x₂ * dist y 0 :=
IsBoundedSMul.dist_pair_smul' x₁ x₂ y | theorem | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | dist_pair_smul | null |
IsBoundedSMul.op [SMul αᵐᵒᵖ β] [IsCentralScalar α β] : IsBoundedSMul αᵐᵒᵖ β where
dist_smul_pair' :=
MulOpposite.rec' fun x y₁ y₂ => by simpa only [op_smul_eq_smul] using dist_smul_pair x y₁ y₂
dist_pair_smul' :=
MulOpposite.rec' fun x₁ =>
MulOpposite.rec' fun x₂ y => by simpa only [op_smul_eq_smul] using dist_pair_smul x₁ x₂ y | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | IsBoundedSMul.op | The typeclass `IsBoundedSMul` on a metric-space scalar action implies continuity of the
action. -/
instance (priority := 100) IsBoundedSMul.continuousSMul : ContinuousSMul α β where
continuous_smul := by
rw [Metric.continuous_iff]
rintro ⟨a, b⟩ ε ε0
obtain ⟨δ, δ0, hδε⟩ : ∃ δ > 0, δ * (δ + dist b 0) + dist a 0 * δ < ε := by
have : Continuous fun δ ↦ δ * (δ + dist b 0) + dist a 0 * δ := by fun_prop
refine ((this.tendsto' _ _ ?_).eventually (gt_mem_nhds ε0)).exists_gt
simp
refine ⟨δ, δ0, fun (a', b') hab' => ?_⟩
obtain ⟨ha, hb⟩ := max_lt_iff.1 hab'
calc dist (a' • b') (a • b)
≤ dist (a' • b') (a • b') + dist (a • b') (a • b) := dist_triangle ..
_ ≤ dist a' a * dist b' 0 + dist a 0 * dist b' b :=
add_le_add (dist_pair_smul _ _ _) (dist_smul_pair _ _ _)
_ ≤ δ * (δ + dist b 0) + dist a 0 * δ := by
have : dist b' 0 ≤ δ + dist b 0 := (dist_triangle _ _ _).trans <| add_le_add_right hb.le _
gcongr
_ < ε := hδε
instance (priority := 100) IsBoundedSMul.toUniformContinuousConstSMul :
UniformContinuousConstSMul α β :=
⟨fun c => ((lipschitzWith_iff_dist_le_mul (K := nndist c 0)).2 fun _ _ =>
dist_smul_pair c _ _).uniformContinuous⟩
-- this instance could be deduced from `NormedSpace.isBoundedSMul`, but we prove it separately
-- here so that it is available earlier in the hierarchy
instance Real.isBoundedSMul : IsBoundedSMul ℝ ℝ where
dist_smul_pair' x y₁ y₂ := by simpa [Real.dist_eq, mul_sub] using (abs_mul x (y₁ - y₂)).le
dist_pair_smul' x₁ x₂ y := by simpa [Real.dist_eq, sub_mul] using (abs_mul (x₁ - x₂) y).le
instance NNReal.isBoundedSMul : IsBoundedSMul ℝ≥0 ℝ≥0 where
dist_smul_pair' x y₁ y₂ := by convert dist_smul_pair (x : ℝ) (y₁ : ℝ) y₂ using 1
dist_pair_smul' x₁ x₂ y := by convert dist_pair_smul (x₁ : ℝ) x₂ (y : ℝ) using 1
/-- If a scalar is central, then its right action is bounded when its left action is. |
Pi.instIsBoundedSMul {α : Type*} {β : ι → Type*} [PseudoMetricSpace α]
[∀ i, PseudoMetricSpace (β i)] [Zero α] [∀ i, Zero (β i)] [∀ i, SMul α (β i)]
[∀ i, IsBoundedSMul α (β i)] : IsBoundedSMul α (∀ i, β i) where
dist_smul_pair' x y₁ y₂ :=
(dist_pi_le_iff <| by positivity).2 fun _ ↦
(dist_smul_pair _ _ _).trans <| mul_le_mul_of_nonneg_left (dist_le_pi_dist _ _ _) dist_nonneg
dist_pair_smul' x₁ x₂ y :=
(dist_pi_le_iff <| by positivity).2 fun _ ↦
(dist_pair_smul _ _ _).trans <| mul_le_mul_of_nonneg_left (dist_le_pi_dist _ 0 _) dist_nonneg | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | Pi.instIsBoundedSMul | null |
Pi.instIsBoundedSMul' {α β : ι → Type*} [∀ i, PseudoMetricSpace (α i)]
[∀ i, PseudoMetricSpace (β i)] [∀ i, Zero (α i)] [∀ i, Zero (β i)] [∀ i, SMul (α i) (β i)]
[∀ i, IsBoundedSMul (α i) (β i)] : IsBoundedSMul (∀ i, α i) (∀ i, β i) where
dist_smul_pair' x y₁ y₂ :=
(dist_pi_le_iff <| by positivity).2 fun _ ↦
(dist_smul_pair _ _ _).trans <|
mul_le_mul (dist_le_pi_dist _ 0 _) (dist_le_pi_dist _ _ _) dist_nonneg dist_nonneg
dist_pair_smul' x₁ x₂ y :=
(dist_pi_le_iff <| by positivity).2 fun _ ↦
(dist_pair_smul _ _ _).trans <|
mul_le_mul (dist_le_pi_dist _ _ _) (dist_le_pi_dist _ 0 _) dist_nonneg dist_nonneg | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | Pi.instIsBoundedSMul' | null |
Prod.instIsBoundedSMul {α β γ : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
[PseudoMetricSpace γ] [Zero α] [Zero β] [Zero γ] [SMul α β] [SMul α γ] [IsBoundedSMul α β]
[IsBoundedSMul α γ] : IsBoundedSMul α (β × γ) where
dist_smul_pair' _x _y₁ _y₂ :=
max_le ((dist_smul_pair _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_left _ _) dist_nonneg)
((dist_smul_pair _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_right _ _) dist_nonneg)
dist_pair_smul' _x₁ _x₂ _y :=
max_le ((dist_pair_smul _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_left _ _) dist_nonneg)
((dist_pair_smul _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_right _ _) dist_nonneg) | instance | Topology | [
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.SeparationQuotient.Basic",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/Algebra.lean | Prod.instIsBoundedSMul | null |
AntilipschitzWith [PseudoEMetricSpace α] [PseudoEMetricSpace β] (K : ℝ≥0) (f : α → β) :=
∀ x y, edist x y ≤ K * edist (f x) (f y) | def | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | AntilipschitzWith | We say that `f : α → β` is `AntilipschitzWith K` if for any two points `x`, `y` we have
`edist x y ≤ K * edist (f x) (f y)`. |
protected AntilipschitzWith.edist_lt_top [PseudoEMetricSpace α] [PseudoMetricSpace β]
{K : ℝ≥0} {f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y < ⊤ :=
(h x y).trans_lt <| ENNReal.mul_lt_top ENNReal.coe_lt_top (edist_lt_top _ _) | lemma | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | AntilipschitzWith.edist_lt_top | null |
AntilipschitzWith.edist_ne_top [PseudoEMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{f : α → β} (h : AntilipschitzWith K f) (x y : α) : edist x y ≠ ⊤ :=
(h.edist_lt_top x y).ne | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | AntilipschitzWith.edist_ne_top | null |
antilipschitzWith_iff_le_mul_nndist :
AntilipschitzWith K f ↔ ∀ x y, nndist x y ≤ K * nndist (f x) (f y) := by
simp only [AntilipschitzWith, edist_nndist]
norm_cast
alias ⟨AntilipschitzWith.le_mul_nndist, AntilipschitzWith.of_le_mul_nndist⟩ :=
antilipschitzWith_iff_le_mul_nndist | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | antilipschitzWith_iff_le_mul_nndist | null |
antilipschitzWith_iff_le_mul_dist :
AntilipschitzWith K f ↔ ∀ x y, dist x y ≤ K * dist (f x) (f y) := by
simp only [antilipschitzWith_iff_le_mul_nndist, dist_nndist]
norm_cast
alias ⟨AntilipschitzWith.le_mul_dist, AntilipschitzWith.of_le_mul_dist⟩ :=
antilipschitzWith_iff_le_mul_dist | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | antilipschitzWith_iff_le_mul_dist | null |
mul_le_nndist (hf : AntilipschitzWith K f) (x y : α) :
K⁻¹ * nndist x y ≤ nndist (f x) (f y) := by
simpa only [div_eq_inv_mul] using NNReal.div_le_of_le_mul' (hf.le_mul_nndist x y) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | mul_le_nndist | null |
mul_le_dist (hf : AntilipschitzWith K f) (x y : α) :
(K⁻¹ * dist x y : ℝ) ≤ dist (f x) (f y) := mod_cast hf.mul_le_nndist x y | theorem | Topology | [
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.MetricSpace.Bounded"
] | Mathlib/Topology/MetricSpace/Antilipschitz.lean | mul_le_dist | null |
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