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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Continuous.image_eq_of_connectedComponent_eq (h : Continuous f) (a b : α)
(hab : connectedComponent a = connectedComponent b) : f a = f b :=
singleton_eq_singleton_iff.1 <|
h.image_connectedComponent_eq_singleton a ▸
h.image_connectedComponent_eq_singleton b ▸ hab ▸ rfl | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.image_eq_of_connectedComponent_eq | null |
Continuous.connectedComponentsLift (h : Continuous f) : ConnectedComponents α → β := fun x =>
Quotient.liftOn' x f h.image_eq_of_connectedComponent_eq
@[continuity] | def | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsLift | The lift to `connectedComponents α` of a continuous map from `α` to a totally disconnected space |
Continuous.connectedComponentsLift_continuous (h : Continuous f) :
Continuous h.connectedComponentsLift :=
h.quotient_liftOn' <| by convert h.image_eq_of_connectedComponent_eq
@[simp] | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsLift_continuous | null |
Continuous.connectedComponentsLift_apply_coe (h : Continuous f) (x : α) :
h.connectedComponentsLift x = f x :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsLift_apply_coe | null |
Continuous.connectedComponentsLift_comp_coe (h : Continuous f) :
h.connectedComponentsLift ∘ (↑) = f :=
rfl | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsLift_comp_coe | null |
connectedComponents_lift_unique' {β : Sort*} {g₁ g₂ : ConnectedComponents α → β}
(hg : g₁ ∘ ((↑) : α → ConnectedComponents α) = g₂ ∘ (↑)) : g₁ = g₂ :=
ConnectedComponents.surjective_coe.injective_comp_right hg | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | connectedComponents_lift_unique' | null |
Continuous.connectedComponentsLift_unique (h : Continuous f) (g : ConnectedComponents α → β)
(hg : g ∘ (↑) = f) : g = h.connectedComponentsLift :=
connectedComponents_lift_unique' <| hg.trans h.connectedComponentsLift_comp_coe.symm | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsLift_unique | null |
ConnectedComponents.totallyDisconnectedSpace :
TotallyDisconnectedSpace (ConnectedComponents α) := by
rw [totallyDisconnectedSpace_iff_connectedComponent_singleton]
refine ConnectedComponents.surjective_coe.forall.2 fun x => ?_
rw [← ConnectedComponents.isQuotientMap_coe.image_connectedComponent, ←
connectedComponents_preimage_singleton, image_preimage_eq _ ConnectedComponents.surjective_coe]
refine ConnectedComponents.surjective_coe.forall.2 fun y => ?_
rw [connectedComponents_preimage_singleton]
exact isConnected_connectedComponent | instance | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | ConnectedComponents.totallyDisconnectedSpace | null |
Continuous.connectedComponentsMap {β : Type*} [TopologicalSpace β] {f : α → β}
(h : Continuous f) : ConnectedComponents α → ConnectedComponents β :=
Continuous.connectedComponentsLift (ConnectedComponents.continuous_coe.comp h) | def | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsMap | Functoriality of `connectedComponents` |
Continuous.connectedComponentsMap_continuous {β : Type*} [TopologicalSpace β] {f : α → β}
(h : Continuous f) : Continuous h.connectedComponentsMap :=
Continuous.connectedComponentsLift_continuous (ConnectedComponents.continuous_coe.comp h) | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | Continuous.connectedComponentsMap_continuous | null |
IsPreconnected.constant {Y : Type*} [TopologicalSpace Y] [DiscreteTopology Y] {s : Set α}
(hs : IsPreconnected s) {f : α → Y} (hf : ContinuousOn f s) {x y : α} (hx : x ∈ s)
(hy : y ∈ s) : f x = f y :=
(hs.image f hf).subsingleton (mem_image_of_mem f hx) (mem_image_of_mem f hy) | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | IsPreconnected.constant | A preconnected set `s` has the property that every map to a
discrete space that is continuous on `s` is constant on `s` |
PreconnectedSpace.constant {Y : Type*} [TopologicalSpace Y] [DiscreteTopology Y]
(hp : PreconnectedSpace α) {f : α → Y} (hf : Continuous f) {x y : α} : f x = f y :=
IsPreconnected.constant hp.isPreconnected_univ (Continuous.continuousOn hf) trivial trivial | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | PreconnectedSpace.constant | A `PreconnectedSpace` version of `isPreconnected.constant` |
IsPreconnected.constant_of_mapsTo {S : Set α} (hS : IsPreconnected S)
{β} [TopologicalSpace β] {T : Set β} [DiscreteTopology T] {f : α → β} (hc : ContinuousOn f S)
(hTm : MapsTo f S T) {x y : α} (hx : x ∈ S) (hy : y ∈ S) : f x = f y := by
let F : S → T := hTm.restrict f S T
suffices F ⟨x, hx⟩ = F ⟨y, hy⟩ by rwa [← Subtype.coe_inj] at this
exact (isPreconnected_iff_preconnectedSpace.mp hS).constant (hc.mapsToRestrict _) | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | IsPreconnected.constant_of_mapsTo | Refinement of `IsPreconnected.constant` only assuming the map factors through a
discrete subset of the target. |
IsPreconnected.eqOn_const_of_mapsTo {S : Set α} (hS : IsPreconnected S)
{β} [TopologicalSpace β] {T : Set β} [DiscreteTopology T] {f : α → β} (hc : ContinuousOn f S)
(hTm : MapsTo f S T) (hne : T.Nonempty) : ∃ y ∈ T, EqOn f (const α y) S := by
rcases S.eq_empty_or_nonempty with (rfl | ⟨x, hx⟩)
· exact hne.imp fun _ hy => ⟨hy, eqOn_empty _ _⟩
· exact ⟨f x, hTm hx, fun x' hx' => hS.constant_of_mapsTo hc hTm hx' hx⟩ | theorem | Topology | [
"Mathlib.Topology.Connected.Clopen"
] | Mathlib/Topology/Connected/TotallyDisconnected.lean | IsPreconnected.eqOn_const_of_mapsTo | A version of `IsPreconnected.constant_of_mapsTo` that assumes that the codomain is nonempty and
proves that `f` is equal to `const α y` on `S` for some `y ∈ T`. |
instTopologicalSpaceSum [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
TopologicalSpace (X ⊕ Y) :=
coinduced Sum.inl t₁ ⊔ coinduced Sum.inr t₂ | instance | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | instTopologicalSpaceSum | null |
instTopologicalSpaceProd [t₁ : TopologicalSpace X] [t₂ : TopologicalSpace Y] :
TopologicalSpace (X × Y) :=
induced Prod.fst t₁ ⊓ induced Prod.snd t₂ | instance | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | instTopologicalSpaceProd | null |
@[simp]
continuous_prodMk {f : X → Y} {g : X → Z} :
(Continuous fun x => (f x, g x)) ↔ Continuous f ∧ Continuous g :=
continuous_inf_rng.trans <| continuous_induced_rng.and continuous_induced_rng
@[deprecated (since := "2025-03-10")]
alias continuous_prod_mk := continuous_prodMk
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_prodMk | null |
continuous_fst : Continuous (@Prod.fst X Y) :=
(continuous_prodMk.1 continuous_id).1 | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_fst | null |
@[fun_prop]
Continuous.fst {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).1 :=
continuous_fst.comp hf | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.fst | Postcomposing `f` with `Prod.fst` is continuous |
Continuous.fst' {f : X → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.fst :=
hf.comp continuous_fst | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.fst' | Precomposing `f` with `Prod.fst` is continuous |
continuousAt_fst {p : X × Y} : ContinuousAt Prod.fst p :=
continuous_fst.continuousAt | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuousAt_fst | null |
@[fun_prop]
ContinuousAt.fst {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
ContinuousAt (fun x : X => (f x).1) x :=
continuousAt_fst.comp hf | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.fst | Postcomposing `f` with `Prod.fst` is continuous at `x` |
ContinuousAt.fst' {f : X → Z} {x : X} {y : Y} (hf : ContinuousAt f x) :
ContinuousAt (fun x : X × Y => f x.fst) (x, y) :=
ContinuousAt.comp hf continuousAt_fst | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.fst' | Precomposing `f` with `Prod.fst` is continuous at `(x, y)` |
ContinuousAt.fst'' {f : X → Z} {x : X × Y} (hf : ContinuousAt f x.fst) :
ContinuousAt (fun x : X × Y => f x.fst) x :=
hf.comp continuousAt_fst | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.fst'' | Precomposing `f` with `Prod.fst` is continuous at `x : X × Y` |
Filter.Tendsto.fst_nhds {X} {l : Filter X} {f : X → Y × Z} {p : Y × Z}
(h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).1) l (𝓝 <| p.1) :=
continuousAt_fst.tendsto.comp h
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Tendsto.fst_nhds | null |
continuous_snd : Continuous (@Prod.snd X Y) :=
(continuous_prodMk.1 continuous_id).2 | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_snd | null |
@[fun_prop]
Continuous.snd {f : X → Y × Z} (hf : Continuous f) : Continuous fun x : X => (f x).2 :=
continuous_snd.comp hf | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.snd | Postcomposing `f` with `Prod.snd` is continuous |
Continuous.snd' {f : Y → Z} (hf : Continuous f) : Continuous fun x : X × Y => f x.snd :=
hf.comp continuous_snd | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.snd' | Precomposing `f` with `Prod.snd` is continuous |
continuousAt_snd {p : X × Y} : ContinuousAt Prod.snd p :=
continuous_snd.continuousAt | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuousAt_snd | null |
@[fun_prop]
ContinuousAt.snd {f : X → Y × Z} {x : X} (hf : ContinuousAt f x) :
ContinuousAt (fun x : X => (f x).2) x :=
continuousAt_snd.comp hf | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.snd | Postcomposing `f` with `Prod.snd` is continuous at `x` |
ContinuousAt.snd' {f : Y → Z} {x : X} {y : Y} (hf : ContinuousAt f y) :
ContinuousAt (fun x : X × Y => f x.snd) (x, y) :=
ContinuousAt.comp hf continuousAt_snd | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.snd' | Precomposing `f` with `Prod.snd` is continuous at `(x, y)` |
ContinuousAt.snd'' {f : Y → Z} {x : X × Y} (hf : ContinuousAt f x.snd) :
ContinuousAt (fun x : X × Y => f x.snd) x :=
hf.comp continuousAt_snd | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.snd'' | Precomposing `f` with `Prod.snd` is continuous at `x : X × Y` |
Filter.Tendsto.snd_nhds {X} {l : Filter X} {f : X → Y × Z} {p : Y × Z}
(h : Tendsto f l (𝓝 p)) : Tendsto (fun a ↦ (f a).2) l (𝓝 <| p.2) :=
continuousAt_snd.tendsto.comp h
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Tendsto.snd_nhds | null |
Continuous.prodMk {f : Z → X} {g : Z → Y} (hf : Continuous f) (hg : Continuous g) :
Continuous fun x => (f x, g x) :=
continuous_prodMk.2 ⟨hf, hg⟩
@[deprecated (since := "2025-03-10")]
alias Continuous.prod_mk := Continuous.prodMk
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.prodMk | null |
Continuous.prodMk_right (x : X) : Continuous fun y : Y => (x, y) := by fun_prop
@[deprecated (since := "2025-03-10")]
alias Continuous.Prod.mk := Continuous.prodMk_right
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.prodMk_right | null |
Continuous.prodMk_left (y : Y) : Continuous fun x : X => (x, y) := by fun_prop
@[deprecated (since := "2025-03-10")]
alias Continuous.Prod.mk_left := Continuous.prodMk_left | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.prodMk_left | null |
IsClosed.setOf_mapsTo {α : Type*} {f : X → α → Z} {s : Set α} {t : Set Z} (ht : IsClosed t)
(hf : ∀ a ∈ s, Continuous (f · a)) : IsClosed {x | MapsTo (f x) s t} := by
simpa only [MapsTo, setOf_forall] using isClosed_biInter fun y hy ↦ ht.preimage (hf y hy) | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsClosed.setOf_mapsTo | If `f x y` is continuous in `x` for all `y ∈ s`,
then the set of `x` such that `f x` maps `s` to `t` is closed. |
Continuous.comp₂ {g : X × Y → Z} (hg : Continuous g) {e : W → X} (he : Continuous e)
{f : W → Y} (hf : Continuous f) : Continuous fun w => g (e w, f w) :=
hg.comp <| he.prodMk hf | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.comp₂ | null |
Continuous.comp₃ {g : X × Y × Z → ε} (hg : Continuous g) {e : W → X} (he : Continuous e)
{f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) :
Continuous fun w => g (e w, f w, k w) :=
hg.comp₂ he <| hf.prodMk hk | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.comp₃ | null |
Continuous.comp₄ {g : X × Y × Z × ζ → ε} (hg : Continuous g) {e : W → X} (he : Continuous e)
{f : W → Y} (hf : Continuous f) {k : W → Z} (hk : Continuous k) {l : W → ζ}
(hl : Continuous l) : Continuous fun w => g (e w, f w, k w, l w) :=
hg.comp₃ he hf <| hk.prodMk hl
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.comp₄ | null |
Continuous.prodMap {f : Z → X} {g : W → Y} (hf : Continuous f) (hg : Continuous g) :
Continuous (Prod.map f g) :=
hf.fst'.prodMk hg.snd' | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.prodMap | null |
continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
{tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
(h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by
haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _))
have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _))
have h_continuous_id := @Continuous.prodMap _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_inf_dom_left₂ | A version of `continuous_inf_dom_left` for binary functions |
continuous_inf_dom_right₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
{tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
(h : by haveI := ta2; haveI := tb2; exact Continuous fun p : X × Y => f p.1 p.2) : by
haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _))
have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _))
have h_continuous_id := @Continuous.prodMap _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_inf_dom_right₂ | A version of `continuous_inf_dom_right` for binary functions |
continuous_sInf_dom₂ {X Y Z} {f : X → Y → Z} {tas : Set (TopologicalSpace X)}
{tbs : Set (TopologicalSpace Y)} {tX : TopologicalSpace X} {tY : TopologicalSpace Y}
{tc : TopologicalSpace Z} (hX : tX ∈ tas) (hY : tY ∈ tbs)
(hf : Continuous fun p : X × Y => f p.1 p.2) : by
haveI := sInf tas; haveI := sInf tbs
exact @Continuous _ _ _ tc fun p : X × Y => f p.1 p.2 := by
have hX := continuous_sInf_dom hX continuous_id
have hY := continuous_sInf_dom hY continuous_id
have h_continuous_id := @Continuous.prodMap _ _ _ _ tX tY (sInf tas) (sInf tbs) _ _ hX hY
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_sInf_dom₂ | A version of `continuous_sInf_dom` for binary functions |
Filter.Eventually.prod_inl_nhds {p : X → Prop} {x : X} (h : ∀ᶠ x in 𝓝 x, p x) (y : Y) :
∀ᶠ x in 𝓝 (x, y), p (x : X × Y).1 :=
continuousAt_fst h | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Eventually.prod_inl_nhds | null |
Filter.Eventually.prod_inr_nhds {p : Y → Prop} {y : Y} (h : ∀ᶠ x in 𝓝 y, p x) (x : X) :
∀ᶠ x in 𝓝 (x, y), p (x : X × Y).2 :=
continuousAt_snd h | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Eventually.prod_inr_nhds | null |
Filter.Eventually.prodMk_nhds {px : X → Prop} {x} (hx : ∀ᶠ x in 𝓝 x, px x) {py : Y → Prop}
{y} (hy : ∀ᶠ y in 𝓝 y, py y) : ∀ᶠ p in 𝓝 (x, y), px (p : X × Y).1 ∧ py p.2 :=
(hx.prod_inl_nhds y).and (hy.prod_inr_nhds x)
@[deprecated (since := "2025-03-10")]
alias Filter.Eventually.prod_mk_nhds := Filter.Eventually.prodMk_nhds | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Eventually.prodMk_nhds | null |
continuous_swap : Continuous (Prod.swap : X × Y → Y × X) :=
continuous_snd.prodMk continuous_fst | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_swap | null |
isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs ↦ by
rw [image_swap_eq_preimage_swap]
exact hs.preimage continuous_swap | lemma | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isClosedMap_swap | null |
Continuous.uncurry_left {f : X → Y → Z} (x : X) (h : Continuous (uncurry f)) :
Continuous (f x) :=
h.comp (.prodMk_right _) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.uncurry_left | null |
Continuous.uncurry_right {f : X → Y → Z} (y : Y) (h : Continuous (uncurry f)) :
Continuous fun a => f a y :=
h.comp (.prodMk_left _) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.uncurry_right | null |
continuous_curry {g : X × Y → Z} (x : X) (h : Continuous g) : Continuous (curry g x) :=
Continuous.uncurry_left x h | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | continuous_curry | null |
IsOpen.prod {s : Set X} {t : Set Y} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ˢ t) :=
(hs.preimage continuous_fst).inter (ht.preimage continuous_snd) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | IsOpen.prod | null |
nhds_prod_eq {x : X} {y : Y} : 𝓝 (x, y) = 𝓝 x ×ˢ 𝓝 y := by
rw [prod_eq_inf, instTopologicalSpaceProd, nhds_inf (t₁ := TopologicalSpace.induced Prod.fst _)
(t₂ := TopologicalSpace.induced Prod.snd _), nhds_induced, nhds_induced] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | nhds_prod_eq | null |
nhdsWithin_prod_eq (x : X) (y : Y) (s : Set X) (t : Set Y) :
𝓝[s ×ˢ t] (x, y) = 𝓝[s] x ×ˢ 𝓝[t] y := by
simp only [nhdsWithin, nhds_prod_eq, ← prod_inf_prod, prod_principal_principal] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | nhdsWithin_prod_eq | null |
Prod.instNeBotNhdsWithinIio [Preorder X] [Preorder Y] {x : X × Y}
[hx₁ : (𝓝[<] x.1).NeBot] [hx₂ : (𝓝[<] x.2).NeBot] : (𝓝[<] x).NeBot := by
refine (hx₁.prod hx₂).mono ?_
rw [← nhdsWithin_prod_eq]
exact nhdsWithin_mono _ fun _ ⟨h₁, h₂⟩ ↦ Prod.lt_iff.2 <| .inl ⟨h₁, h₂.le⟩ | instance | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Prod.instNeBotNhdsWithinIio | null |
Prod.instNeBotNhdsWithinIoi [Preorder X] [Preorder Y] {x : X × Y}
[hx₁ : (𝓝[>] x.1).NeBot] [hx₂ : (𝓝[>] x.2).NeBot] : (𝓝[>] x).NeBot := by
refine (hx₁.prod hx₂).mono ?_
rw [← nhdsWithin_prod_eq]
exact nhdsWithin_mono _ fun _ ⟨h₁, h₂⟩ ↦ Prod.lt_iff.2 <| .inl ⟨h₁, h₂.le⟩ | instance | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Prod.instNeBotNhdsWithinIoi | null |
mem_nhds_prod_iff {x : X} {y : Y} {s : Set (X × Y)} :
s ∈ 𝓝 (x, y) ↔ ∃ u ∈ 𝓝 x, ∃ v ∈ 𝓝 y, u ×ˢ v ⊆ s := by rw [nhds_prod_eq, mem_prod_iff] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | mem_nhds_prod_iff | null |
mem_nhdsWithin_prod_iff {x : X} {y : Y} {s : Set (X × Y)} {tx : Set X} {ty : Set Y} :
s ∈ 𝓝[tx ×ˢ ty] (x, y) ↔ ∃ u ∈ 𝓝[tx] x, ∃ v ∈ 𝓝[ty] y, u ×ˢ v ⊆ s := by
rw [nhdsWithin_prod_eq, mem_prod_iff] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | mem_nhdsWithin_prod_iff | null |
Filter.HasBasis.prod_nhds {ιX ιY : Type*} {px : ιX → Prop} {py : ιY → Prop}
{sx : ιX → Set X} {sy : ιY → Set Y} {x : X} {y : Y} (hx : (𝓝 x).HasBasis px sx)
(hy : (𝓝 y).HasBasis py sy) :
(𝓝 (x, y)).HasBasis (fun i : ιX × ιY => px i.1 ∧ py i.2) fun i => sx i.1 ×ˢ sy i.2 := by
rw [nhds_prod_eq]
exact hx.prod hy | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.HasBasis.prod_nhds | null |
Filter.HasBasis.prod_nhds' {ιX ιY : Type*} {pX : ιX → Prop} {pY : ιY → Prop}
{sx : ιX → Set X} {sy : ιY → Set Y} {p : X × Y} (hx : (𝓝 p.1).HasBasis pX sx)
(hy : (𝓝 p.2).HasBasis pY sy) :
(𝓝 p).HasBasis (fun i : ιX × ιY => pX i.1 ∧ pY i.2) fun i => sx i.1 ×ˢ sy i.2 :=
hx.prod_nhds hy | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.HasBasis.prod_nhds' | null |
mem_nhds_prod_iff' {x : X} {y : Y} {s : Set (X × Y)} :
s ∈ 𝓝 (x, y) ↔ ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧ u ×ˢ v ⊆ s :=
((nhds_basis_opens x).prod_nhds (nhds_basis_opens y)).mem_iff.trans <| by
simp only [Prod.exists, and_comm, and_assoc, and_left_comm] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | mem_nhds_prod_iff' | null |
Prod.tendsto_iff {X} (seq : X → Y × Z) {f : Filter X} (p : Y × Z) :
Tendsto seq f (𝓝 p) ↔
Tendsto (fun n => (seq n).fst) f (𝓝 p.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 p.snd) := by
rw [nhds_prod_eq, Filter.tendsto_prod_iff'] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Prod.tendsto_iff | null |
prod_mem_nhds_iff {s : Set X} {t : Set Y} {x : X} {y : Y} :
s ×ˢ t ∈ 𝓝 (x, y) ↔ s ∈ 𝓝 x ∧ t ∈ 𝓝 y := by rw [nhds_prod_eq, prod_mem_prod_iff] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prod_mem_nhds_iff | null |
prod_mem_nhds {s : Set X} {t : Set Y} {x : X} {y : Y} (hx : s ∈ 𝓝 x) (hy : t ∈ 𝓝 y) :
s ×ˢ t ∈ 𝓝 (x, y) :=
prod_mem_nhds_iff.2 ⟨hx, hy⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prod_mem_nhds | null |
isOpen_setOf_disjoint_nhds_nhds : IsOpen { p : X × X | Disjoint (𝓝 p.1) (𝓝 p.2) } := by
simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq]
intro x y h
obtain ⟨U, hU, V, hV, hd⟩ := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h
exact mem_nhds_prod_iff'.mpr ⟨U, V, hU.2, hU.1, hV.2, hV.1, fun ⟨x', y'⟩ ⟨hx', hy'⟩ =>
disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')⟩ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_setOf_disjoint_nhds_nhds | null |
Filter.Eventually.prod_nhds {p : X → Prop} {q : Y → Prop} {x : X} {y : Y}
(hx : ∀ᶠ x in 𝓝 x, p x) (hy : ∀ᶠ y in 𝓝 y, q y) : ∀ᶠ z : X × Y in 𝓝 (x, y), p z.1 ∧ q z.2 :=
prod_mem_nhds hx hy | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Eventually.prod_nhds | null |
Filter.EventuallyEq.prodMap_nhds {α β : Type*} {f₁ f₂ : X → α} {g₁ g₂ : Y → β}
{x : X} {y : Y} (hf : f₁ =ᶠ[𝓝 x] f₂) (hg : g₁ =ᶠ[𝓝 y] g₂) :
Prod.map f₁ g₁ =ᶠ[𝓝 (x, y)] Prod.map f₂ g₂ := by
rw [nhds_prod_eq]
exact hf.prodMap hg | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.EventuallyEq.prodMap_nhds | null |
Filter.EventuallyLE.prodMap_nhds {α β : Type*} [LE α] [LE β] {f₁ f₂ : X → α} {g₁ g₂ : Y → β}
{x : X} {y : Y} (hf : f₁ ≤ᶠ[𝓝 x] f₂) (hg : g₁ ≤ᶠ[𝓝 y] g₂) :
Prod.map f₁ g₁ ≤ᶠ[𝓝 (x, y)] Prod.map f₂ g₂ := by
rw [nhds_prod_eq]
exact hf.prodMap hg | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.EventuallyLE.prodMap_nhds | null |
nhds_swap (x : X) (y : Y) : 𝓝 (x, y) = (𝓝 (y, x)).map Prod.swap := by
rw [nhds_prod_eq, Filter.prod_comm, nhds_prod_eq] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | nhds_swap | null |
Filter.Tendsto.prodMk_nhds {γ} {x : X} {y : Y} {f : Filter γ} {mx : γ → X} {my : γ → Y}
(hx : Tendsto mx f (𝓝 x)) (hy : Tendsto my f (𝓝 y)) :
Tendsto (fun c => (mx c, my c)) f (𝓝 (x, y)) := by
rw [nhds_prod_eq]
exact hx.prodMk hy
@[deprecated (since := "2025-03-10")]
alias Filter.Tendsto.prod_mk_nhds := Filter.Tendsto.prodMk_nhds | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Tendsto.prodMk_nhds | null |
Filter.Tendsto.prodMap_nhds {x : X} {y : Y} {z : Z} {w : W} {f : X → Y} {g : Z → W}
(hf : Tendsto f (𝓝 x) (𝓝 y)) (hg : Tendsto g (𝓝 z) (𝓝 w)) :
Tendsto (Prod.map f g) (𝓝 (x, z)) (𝓝 (y, w)) := by
rw [nhds_prod_eq, nhds_prod_eq]
exact hf.prodMap hg | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Tendsto.prodMap_nhds | null |
Filter.Eventually.curry_nhds {p : X × Y → Prop} {x : X} {y : Y}
(h : ∀ᶠ x in 𝓝 (x, y), p x) : ∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') := by
rw [nhds_prod_eq] at h
exact h.curry
@[fun_prop] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Filter.Eventually.curry_nhds | null |
ContinuousAt.prodMk {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt f x)
(hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
hf.prodMk_nhds hg
@[deprecated (since := "2025-03-10")]
alias ContinuousAt.prod := ContinuousAt.prodMk | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.prodMk | null |
ContinuousAt.prodMap {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.fst)
(hg : ContinuousAt g p.snd) : ContinuousAt (Prod.map f g) p :=
hf.fst''.prodMk hg.snd'' | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.prodMap | null |
ContinuousAt.prodMap' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x)
(hg : ContinuousAt g y) : ContinuousAt (Prod.map f g) (x, y) :=
hf.prodMap hg | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.prodMap' | A version of `ContinuousAt.prodMap` that avoids `Prod.fst`/`Prod.snd`
by assuming that the point is `(x, y)`. |
ContinuousAt.comp₂ {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X}
(hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) :
ContinuousAt (fun x ↦ f (g x, h x)) x :=
ContinuousAt.comp hf (hg.prodMk hh) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.comp₂ | null |
ContinuousAt.comp₂_of_eq {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X} {y : Y × Z}
(hf : ContinuousAt f y) (hg : ContinuousAt g x) (hh : ContinuousAt h x) (e : (g x, h x) = y) :
ContinuousAt (fun x ↦ f (g x, h x)) x := by
rw [← e] at hf
exact hf.comp₂ hg hh | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | ContinuousAt.comp₂_of_eq | null |
Continuous.curry_left {f : X × Y → Z} (hf : Continuous f) {y : Y} :
Continuous fun x ↦ f (x, y) :=
hf.comp (.prodMk_left _)
alias Continuous.along_fst := Continuous.curry_left | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.curry_left | Continuous functions on products are continuous in their first argument |
Continuous.curry_right {f : X × Y → Z} (hf : Continuous f) {x : X} :
Continuous fun y ↦ f (x, y) :=
hf.comp (.prodMk_right _)
alias Continuous.along_snd := Continuous.curry_right | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | Continuous.curry_right | Continuous functions on products are continuous in their second argument |
prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : Set (Set Y)}
(hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
@instTopologicalSpaceProd X Y (generateFrom s) (generateFrom t) =
generateFrom (image2 (· ×ˢ ·) s t) :=
let G := generateFrom (image2 (· ×ˢ ·) s t)
le_antisymm
(le_generateFrom fun _ ⟨_, hu, _, hv, g_eq⟩ =>
g_eq.symm ▸
@IsOpen.prod _ _ (generateFrom s) (generateFrom t) _ _ (GenerateOpen.basic _ hu)
(GenerateOpen.basic _ hv))
(le_inf
(coinduced_le_iff_le_induced.mp <|
le_generateFrom fun u hu =>
have : ⋃ v ∈ t, u ×ˢ v = Prod.fst ⁻¹' u := by
simp_rw [← prod_iUnion, ← sUnion_eq_biUnion, ht, prod_univ]
show G.IsOpen (Prod.fst ⁻¹' u) by
rw [← this]
exact
isOpen_iUnion fun v =>
isOpen_iUnion fun hv => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩)
(coinduced_le_iff_le_induced.mp <|
le_generateFrom fun v hv =>
have : ⋃ u ∈ s, u ×ˢ v = Prod.snd ⁻¹' v := by
simp_rw [← iUnion_prod_const, ← sUnion_eq_biUnion, hs, univ_prod]
show G.IsOpen (Prod.snd ⁻¹' v) by
rw [← this]
exact
isOpen_iUnion fun u =>
isOpen_iUnion fun hu => GenerateOpen.basic _ ⟨_, hu, _, hv, rfl⟩)) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prod_generateFrom_generateFrom_eq | null |
prod_eq_generateFrom :
instTopologicalSpaceProd =
generateFrom { g | ∃ (s : Set X) (t : Set Y), IsOpen s ∧ IsOpen t ∧ g = s ×ˢ t } :=
le_antisymm (le_generateFrom fun _ ⟨_, _, hs, ht, g_eq⟩ => g_eq.symm ▸ hs.prod ht)
(le_inf
(forall_mem_image.2 fun t ht =>
GenerateOpen.basic _ ⟨t, univ, by simpa [Set.prod_eq] using ht⟩)
(forall_mem_image.2 fun t ht =>
GenerateOpen.basic _ ⟨univ, t, by simpa [Set.prod_eq] using ht⟩)) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prod_eq_generateFrom | null |
isOpen_prod_iff {s : Set (X × Y)} :
IsOpen s ↔ ∀ a b, (a, b) ∈ s →
∃ u v, IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s :=
isOpen_iff_mem_nhds.trans <| by simp_rw [Prod.forall, mem_nhds_prod_iff', and_left_comm] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_prod_iff | null |
prod_induced_induced {X Z} (f : X → Y) (g : Z → W) :
@instTopologicalSpaceProd X Z (induced f ‹_›) (induced g ‹_›) =
induced (fun p => (f p.1, g p.2)) instTopologicalSpaceProd := by
delta instTopologicalSpaceProd
simp_rw [induced_inf, induced_compose]
rfl | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | prod_induced_induced | A product of induced topologies is induced by the product map |
exists_nhds_square {s : Set (X × X)} {x : X} (hx : s ∈ 𝓝 (x, x)) :
∃ U : Set X, IsOpen U ∧ x ∈ U ∧ U ×ˢ U ⊆ s := by
simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and_assoc, and_left_comm] using hx | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | exists_nhds_square | Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
that is a subset of `s`. |
map_fst_nhdsWithin (x : X × Y) : map Prod.fst (𝓝[Prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 := by
refine le_antisymm (continuousAt_fst.mono_left inf_le_left) fun s hs => ?_
rcases x with ⟨x, y⟩
rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
rcases hs with ⟨u, hu, v, hv, H⟩
simp only [prod_subset_iff, mem_singleton_iff, mem_setOf_eq, mem_preimage] at H
exact mem_of_superset hu fun z hz => H _ hz _ (mem_of_mem_nhds hv) rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | map_fst_nhdsWithin | `Prod.fst` maps neighborhood of `x : X × Y` within the section `Prod.snd ⁻¹' {x.2}`
to `𝓝 x.1`. |
map_fst_nhds (x : X × Y) : map Prod.fst (𝓝 x) = 𝓝 x.1 :=
le_antisymm continuousAt_fst <| (map_fst_nhdsWithin x).symm.trans_le (map_mono inf_le_left) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | map_fst_nhds | null |
isOpenMap_fst : IsOpenMap (@Prod.fst X Y) :=
isOpenMap_iff_nhds_le.2 fun x => (map_fst_nhds x).ge | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_fst | The first projection in a product of topological spaces sends open sets to open sets. |
map_snd_nhdsWithin (x : X × Y) : map Prod.snd (𝓝[Prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 := by
refine le_antisymm (continuousAt_snd.mono_left inf_le_left) fun s hs => ?_
rcases x with ⟨x, y⟩
rw [mem_map, nhdsWithin, mem_inf_principal, mem_nhds_prod_iff] at hs
rcases hs with ⟨u, hu, v, hv, H⟩
simp only [prod_subset_iff, mem_singleton_iff, mem_setOf_eq, mem_preimage] at H
exact mem_of_superset hv fun z hz => H _ (mem_of_mem_nhds hu) _ hz rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | map_snd_nhdsWithin | `Prod.snd` maps neighborhood of `x : X × Y` within the section `Prod.fst ⁻¹' {x.1}`
to `𝓝 x.2`. |
map_snd_nhds (x : X × Y) : map Prod.snd (𝓝 x) = 𝓝 x.2 :=
le_antisymm continuousAt_snd <| (map_snd_nhdsWithin x).symm.trans_le (map_mono inf_le_left) | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | map_snd_nhds | null |
isOpenMap_snd : IsOpenMap (@Prod.snd X Y) :=
isOpenMap_iff_nhds_le.2 fun x => (map_snd_nhds x).ge | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpenMap_snd | The second projection in a product of topological spaces sends open sets to open sets. |
isOpen_prod_iff' {s : Set X} {t : Set Y} :
IsOpen (s ×ˢ t) ↔ IsOpen s ∧ IsOpen t ∨ s = ∅ ∨ t = ∅ := by
rcases (s ×ˢ t).eq_empty_or_nonempty with h | h
· simp [h, prod_eq_empty_iff.1 h]
· have st : s.Nonempty ∧ t.Nonempty := prod_nonempty_iff.1 h
constructor
· intro (H : IsOpen (s ×ˢ t))
refine Or.inl ⟨?_, ?_⟩
· simpa only [fst_image_prod _ st.2] using isOpenMap_fst _ H
· simpa only [snd_image_prod st.1 t] using isOpenMap_snd _ H
· intro H
simp only [st.1.ne_empty, st.2.ne_empty, or_false] at H
exact H.1.prod H.2 | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isOpen_prod_iff' | A product set is open in a product space if and only if each factor is open, or one of them is
empty |
isQuotientMap_fst [Nonempty Y] : IsQuotientMap (Prod.fst : X × Y → X) :=
isOpenMap_fst.isQuotientMap continuous_fst Prod.fst_surjective | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isQuotientMap_fst | null |
isQuotientMap_snd [Nonempty X] : IsQuotientMap (Prod.snd : X × Y → Y) :=
isOpenMap_snd.isQuotientMap continuous_snd Prod.snd_surjective | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | isQuotientMap_snd | null |
closure_prod_eq {s : Set X} {t : Set Y} : closure (s ×ˢ t) = closure s ×ˢ closure t :=
ext fun ⟨a, b⟩ => by
simp_rw [mem_prod, mem_closure_iff_nhdsWithin_neBot, nhdsWithin_prod_eq, prod_neBot] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | closure_prod_eq | null |
interior_prod_eq (s : Set X) (t : Set Y) : interior (s ×ˢ t) = interior s ×ˢ interior t :=
ext fun ⟨a, b⟩ => by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | interior_prod_eq | null |
frontier_prod_eq (s : Set X) (t : Set Y) :
frontier (s ×ˢ t) = closure s ×ˢ frontier t ∪ frontier s ×ˢ closure t := by
simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | frontier_prod_eq | null |
frontier_prod_univ_eq (s : Set X) :
frontier (s ×ˢ (univ : Set Y)) = frontier s ×ˢ univ := by
simp [frontier_prod_eq]
@[simp] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | frontier_prod_univ_eq | null |
frontier_univ_prod_eq (s : Set Y) :
frontier ((univ : Set X) ×ˢ s) = univ ×ˢ frontier s := by
simp [frontier_prod_eq] | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | frontier_univ_prod_eq | null |
map_mem_closure₂ {f : X → Y → Z} {x : X} {y : Y} {s : Set X} {t : Set Y} {u : Set Z}
(hf : Continuous (uncurry f)) (hx : x ∈ closure s) (hy : y ∈ closure t)
(h : ∀ a ∈ s, ∀ b ∈ t, f a b ∈ u) : f x y ∈ closure u :=
have H₁ : (x, y) ∈ closure (s ×ˢ t) := by simpa only [closure_prod_eq] using mk_mem_prod hx hy
have H₂ : MapsTo (uncurry f) (s ×ˢ t) u := forall_prod_set.2 h
H₂.closure hf H₁ | theorem | Topology | [
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Maps.Basic",
"Mathlib.Topology.Separation.SeparatedNhds"
] | Mathlib/Topology/Constructions/SumProd.lean | map_mem_closure₂ | null |
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