statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
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tendstoLocallyUniformlyOn_iUnion {ι' : Sort*} {S : ι' → Set α} (hS : ∀ i, IsOpen (S i))
(h : ∀ i, TendstoLocallyUniformlyOn F f p (S i)) :
TendstoLocallyUniformlyOn F f p (⋃ i, S i) | (isOpen_iUnion hS).tendstoLocallyUniformlyOn_iff_forall_tendsto.2 fun _x hx =>
let ⟨i, hi⟩ := mem_iUnion.1 hx
(hS i).tendstoLocallyUniformlyOn_iff_forall_tendsto.1 (h i) _ hi | theorem | tendstoLocallyUniformlyOn_iUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsOpen",
"Set",
"TendstoLocallyUniformlyOn",
"isOpen_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_biUnion {s : Set γ} {S : γ → Set α} (hS : ∀ i ∈ s, IsOpen (S i))
(h : ∀ i ∈ s, TendstoLocallyUniformlyOn F f p (S i)) :
TendstoLocallyUniformlyOn F f p (⋃ i ∈ s, S i) | tendstoLocallyUniformlyOn_iUnion (fun i => isOpen_iUnion (hS i))
fun i ↦ tendstoLocallyUniformlyOn_iUnion (hS i) (h i) | theorem | tendstoLocallyUniformlyOn_biUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsOpen",
"Set",
"TendstoLocallyUniformlyOn",
"isOpen_iUnion",
"tendstoLocallyUniformlyOn_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_sUnion (S : Set (Set α)) (hS : ∀ s ∈ S, IsOpen s)
(h : ∀ s ∈ S, TendstoLocallyUniformlyOn F f p s) : TendstoLocallyUniformlyOn F f p (⋃₀ S) | by
rw [sUnion_eq_biUnion]
exact tendstoLocallyUniformlyOn_biUnion hS h | theorem | tendstoLocallyUniformlyOn_sUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsOpen",
"Set",
"TendstoLocallyUniformlyOn",
"tendstoLocallyUniformlyOn_biUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.union (hs₁ : IsOpen s) (hs₂ : IsOpen s')
(h₁ : TendstoLocallyUniformlyOn F f p s) (h₂ : TendstoLocallyUniformlyOn F f p s') :
TendstoLocallyUniformlyOn F f p (s ∪ s') | by
rw [← sUnion_pair]
refine tendstoLocallyUniformlyOn_sUnion _ ?_ ?_ <;> simp [*] | theorem | TendstoLocallyUniformlyOn.union | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsOpen",
"TendstoLocallyUniformlyOn",
"tendstoLocallyUniformlyOn_sUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.tendstoLocallyUniformlyOn
(h : TendstoLocallyUniformly F f p) : TendstoLocallyUniformlyOn F f p s | (tendstoLocallyUniformlyOn_univ.mpr h).mono (subset_univ _) | theorem | TendstoLocallyUniformly.tendstoLocallyUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"TendstoLocallyUniformlyOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace [CompactSpace α] :
TendstoLocallyUniformly F f p ↔ TendstoUniformly F f p | by
refine ⟨fun h V hV => ?_, TendstoUniformly.tendstoLocallyUniformly⟩
choose U hU using h V hV
obtain ⟨t, ht⟩ := isCompact_univ.elim_nhds_subcover' (fun k _ => U k) fun k _ => (hU k).1
replace hU := fun x : t => (hU x).2
rw [← eventually_all] at hU
refine hU.mono fun i hi x => ?_
specialize ht (mem_univ ... | theorem | tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"CompactSpace",
"SetCoe.exists",
"TendstoLocallyUniformly",
"TendstoUniformly"
] | On a compact space, locally uniform convergence is just uniform convergence. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact (hs : IsCompact s) :
TendstoLocallyUniformlyOn F f p s ↔ TendstoUniformlyOn F f p s | by
haveI : CompactSpace s := isCompact_iff_compactSpace.mp hs
refine ⟨fun h => ?_, TendstoUniformlyOn.tendstoLocallyUniformlyOn⟩
rwa [tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe,
tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace, ←
tendstoUniformlyOn_iff_tendstoUniformly_comp... | theorem | tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"CompactSpace",
"IsCompact",
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOn",
"tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe",
"tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace",
"tendstoUniformlyOn_iff_tendstoUniformly_comp_coe"
] | For a compact set `s`, locally uniform convergence on `s` is just uniform convergence on `s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
TendstoLocallyUniformlyOn.comp [TopologicalSpace γ] {t : Set γ}
(h : TendstoLocallyUniformlyOn F f p s) (g : γ → α) (hg : MapsTo g t s)
(cg : ContinuousOn g t) : TendstoLocallyUniformlyOn (fun n => F n ∘ g) (f ∘ g) p t | by
intro u hu x hx
rcases h u hu (g x) (hg hx) with ⟨a, ha, H⟩
have : g ⁻¹' a ∈ 𝓝[t] x :=
(cg x hx).preimage_mem_nhdsWithin' (nhdsWithin_mono (g x) hg.image_subset ha)
exact ⟨g ⁻¹' a, this, H.mono fun n hn y hy => hn _ hy⟩ | theorem | TendstoLocallyUniformlyOn.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"ContinuousOn",
"Set",
"TendstoLocallyUniformlyOn",
"TopologicalSpace",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.comp [TopologicalSpace γ] (h : TendstoLocallyUniformly F f p)
(g : γ → α) (cg : Continuous g) : TendstoLocallyUniformly (fun n => F n ∘ g) (f ∘ g) p | by
rw [← tendstoLocallyUniformlyOn_univ] at h ⊢
rw [← continuousOn_univ] at cg
exact h.comp _ (mapsTo_univ _ _) cg | theorem | TendstoLocallyUniformly.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Continuous",
"TendstoLocallyUniformly",
"TopologicalSpace",
"continuousOn_univ",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.comp_tendstoLocallyUniformlyOn {t : Set β}
(hg : UniformContinuousOn g t) (hf : TendstoLocallyUniformlyOn F f p s)
(hfs : MapsTo f s t) (hFs : ∀ᶠ n in p, MapsTo (F n) s t) :
TendstoLocallyUniformlyOn (g ∘ F ·) (g ∘ f) p s | by
rw [tendstoLocallyUniformlyOn_iff_forall_tendsto] at hf ⊢
refine fun x hx ↦ Tendsto.comp hg (tendsto_inf.mpr ⟨hf x hx, tendsto_principal.mpr ?_⟩)
filter_upwards [hFs.prod_mk eventually_mem_nhdsWithin] with y hy using ⟨hfs hy.2, hy.1 hy.2⟩ | theorem | UniformContinuousOn.comp_tendstoLocallyUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Set",
"TendstoLocallyUniformlyOn",
"UniformContinuousOn",
"eventually_mem_nhdsWithin",
"tendstoLocallyUniformlyOn_iff_forall_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.comp_tendstoLocallyUniformly {t : Set β}
(hg : UniformContinuousOn g t) (hf : TendstoLocallyUniformly F f p)
(hfs : ∀ x, f x ∈ t) (hFs : ∀ᶠ n in p, ∀ x, F n x ∈ t) :
TendstoLocallyUniformly (g ∘ F ·) (g ∘ f) p | by
rw [← tendstoLocallyUniformlyOn_univ] at *
apply hg.comp_tendstoLocallyUniformlyOn hf <;> simpa [MapsTo] | theorem | UniformContinuousOn.comp_tendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Set",
"TendstoLocallyUniformly",
"UniformContinuousOn",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.comp_tendstoLocallyUniformlyOn (hg : UniformContinuous g)
(hf : TendstoLocallyUniformlyOn F f p s) :
TendstoLocallyUniformlyOn (g ∘ F ·) (g ∘ f) p s | hg.uniformContinuousOn.comp_tendstoLocallyUniformlyOn hf (mapsTo_univ _ _) <| .of_forall fun _ ↦
mapsTo_univ _ _ | theorem | UniformContinuous.comp_tendstoLocallyUniformlyOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.comp_tendstoLocallyUniformly (hg : UniformContinuous g)
(hf : TendstoLocallyUniformly F f p) :
TendstoLocallyUniformly (g ∘ F ·) (g ∘ f) p | (hg.uniformContinuousOn (s := univ)).comp_tendstoLocallyUniformly hf (by simp) (by simp) | theorem | UniformContinuous.comp_tendstoLocallyUniformly | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.prodMk [UniformSpace γ] {G : ι → α → γ} {g : α → γ}
(hF : TendstoLocallyUniformlyOn F f p s) (hG : TendstoLocallyUniformlyOn G g p s) :
TendstoLocallyUniformlyOn (fun n x ↦ (F n x, G n x)) (fun x ↦ (f x, g x)) p s | by
rw [tendstoLocallyUniformlyOn_iff_forall_tendsto] at *
intro x hx
rw [uniformity_prod_eq_comap_prod, tendsto_comap_iff]
exact (hF x hx).prodMk (hG x hx) | theorem | TendstoLocallyUniformlyOn.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"UniformSpace",
"tendstoLocallyUniformlyOn_iff_forall_tendsto",
"uniformity_prod_eq_comap_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.piProd [UniformSpace γ] {G : ι → α → γ} {g : α → γ}
(hF : TendstoLocallyUniformlyOn F f p s) (hG : TendstoLocallyUniformlyOn G g p s) :
TendstoLocallyUniformlyOn (fun n ↦ Function.prod (F n) (G n)) (Function.prod f g) p s | hF.prodMk hG | theorem | TendstoLocallyUniformlyOn.piProd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Function.prod",
"TendstoLocallyUniformlyOn",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.prodMk [UniformSpace γ] {G : ι → α → γ} {g : α → γ}
(hF : TendstoLocallyUniformly F f p) (hG : TendstoLocallyUniformly G g p) :
TendstoLocallyUniformly (fun n x ↦ (F n x, G n x)) (fun x ↦ (f x, g x)) p | by
rw [← tendstoLocallyUniformlyOn_univ] at *
exact hF.prodMk hG | theorem | TendstoLocallyUniformly.prodMk | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"UniformSpace",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.piProd [UniformSpace γ] {G : ι → α → γ} {g : α → γ}
(hF : TendstoLocallyUniformly F f p) (hG : TendstoLocallyUniformly G g p) :
TendstoLocallyUniformly (fun n ↦ Function.prod (F n) (G n)) (Function.prod f g) p | hF.prodMk hG | theorem | TendstoLocallyUniformly.piProd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Function.prod",
"TendstoLocallyUniformly",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_of_forall_exists_nhds
(h : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, TendstoUniformlyOn F f p t) :
TendstoLocallyUniformlyOn F f p s | by
refine tendstoLocallyUniformlyOn_iff_forall_tendsto.mpr fun x hx ↦ ?_
obtain ⟨t, ht, htr⟩ := h x hx
rw [tendstoUniformlyOn_iff_tendsto] at htr
exact htr.mono_left <| prod_mono_right _ <| le_principal_iff.mpr ht | lemma | tendstoLocallyUniformlyOn_of_forall_exists_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOn",
"tendstoUniformlyOn_iff_tendsto"
] | If every `x ∈ s` has a neighbourhood within `s` on which `F i` tends uniformly to `f`, then
`F i` tends locally uniformly on `s` to `f`.
Note this is **not** a tautology, since our definition of `TendstoLocallyUniformlyOn` is slightly
more general (although the conditions are equivalent if `β` is locally compact and `... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformly_of_forall_exists_nhds
(h : ∀ x, ∃ t ∈ 𝓝 x, TendstoUniformlyOn F f p t) :
TendstoLocallyUniformly F f p | tendstoLocallyUniformlyOn_univ.mp
<| tendstoLocallyUniformlyOn_of_forall_exists_nhds (by simpa using h) | lemma | tendstoLocallyUniformly_of_forall_exists_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"TendstoUniformlyOn",
"tendstoLocallyUniformlyOn_of_forall_exists_nhds"
] | If every `x` has a neighbourhood on which `F i` tends uniformly to `f`, then `F i` tends
locally uniformly to `f`. (Special case of `tendstoLocallyUniformlyOn_of_forall_exists_nhds`
where `s = univ`.) | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendstoLocallyUniformlyOn_TFAE [LocallyCompactSpace α] (G : ι → α → β) (g : α → β)
(p : Filter ι) (hs : IsOpen s) :
List.TFAE [
TendstoLocallyUniformlyOn G g p s,
∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn G g p K,
∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v] | by
tfae_have 1 → 2
| h, K, hK1, hK2 =>
(tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK2).mp (h.mono hK1)
tfae_have 2 → 3
| h, x, hx => by
obtain ⟨K, ⟨hK1, hK2⟩, hK3⟩ := (compact_basis_nhds x).mem_iff.mp (hs.mem_nhds hx)
exact ⟨K, nhdsWithin_le_nhds hK1, h K hK3 hK2⟩
tfae_have 3 → 1... | theorem | tendstoLocallyUniformlyOn_TFAE | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Filter",
"IsCompact",
"IsOpen",
"List.TFAE",
"LocallyCompactSpace",
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOn",
"compact_basis_nhds",
"nhdsWithin_le_nhds",
"tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_iff_forall_isCompact [LocallyCompactSpace α] (hs : IsOpen s) :
TendstoLocallyUniformlyOn F f p s ↔ ∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn F f p K | (tendstoLocallyUniformlyOn_TFAE F f p hs).out 0 1 | theorem | tendstoLocallyUniformlyOn_iff_forall_isCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsCompact",
"IsOpen",
"LocallyCompactSpace",
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOn",
"tendstoLocallyUniformlyOn_TFAE"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformly_iff_forall_isCompact [LocallyCompactSpace α] :
TendstoLocallyUniformly F f p ↔ ∀ K : Set α, IsCompact K → TendstoUniformlyOn F f p K | by
simp only [← tendstoLocallyUniformlyOn_univ,
tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_univ, Set.subset_univ, forall_true_left] | lemma | tendstoLocallyUniformly_iff_forall_isCompact | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"IsCompact",
"LocallyCompactSpace",
"Set",
"Set.subset_univ",
"TendstoLocallyUniformly",
"TendstoUniformlyOn",
"forall_true_left",
"isOpen_univ",
"tendstoLocallyUniformlyOn_iff_forall_isCompact",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformlyOn_iff_filter :
TendstoLocallyUniformlyOn F f p s ↔ ∀ x ∈ s, TendstoUniformlyOnFilter F f p (𝓝[s] x) | by
simp only [TendstoUniformlyOnFilter, eventually_prod_iff]
constructor
· rintro h x hx u hu
obtain ⟨s, hs1, hs2⟩ := h u hu x hx
exact ⟨_, hs2, _, eventually_of_mem hs1 fun x => id, fun hi y hy => hi y hy⟩
· rintro h u hu x hx
obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu
exact ⟨{a | pb a}, hpb, ... | theorem | tendstoLocallyUniformlyOn_iff_filter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"TendstoUniformlyOnFilter"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendstoLocallyUniformly_iff_filter :
TendstoLocallyUniformly F f p ↔ ∀ x, TendstoUniformlyOnFilter F f p (𝓝 x) | by
simpa [← tendstoLocallyUniformlyOn_univ, ← nhdsWithin_univ] using
@tendstoLocallyUniformlyOn_iff_filter _ _ _ _ _ F f univ p | theorem | tendstoLocallyUniformly_iff_filter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"TendstoUniformlyOnFilter",
"nhdsWithin_univ",
"tendstoLocallyUniformlyOn_iff_filter",
"tendstoLocallyUniformlyOn_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.tendsto_at (hf : TendstoLocallyUniformlyOn F f p s) {a : α}
(ha : a ∈ s) : Tendsto (fun i => F i a) p (𝓝 (f a)) | by
refine ((tendstoLocallyUniformlyOn_iff_filter.mp hf) a ha).tendsto_at ?_
simpa only [Filter.principal_singleton] using pure_le_nhdsWithin ha | theorem | TendstoLocallyUniformlyOn.tendsto_at | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Filter.principal_singleton",
"TendstoLocallyUniformlyOn",
"pure_le_nhdsWithin"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.unique [p.NeBot] [T2Space β] {g : α → β}
(hf : TendstoLocallyUniformlyOn F f p s) (hg : TendstoLocallyUniformlyOn F g p s) :
s.EqOn f g | fun _a ha => tendsto_nhds_unique (hf.tendsto_at ha) (hg.tendsto_at ha) | theorem | TendstoLocallyUniformlyOn.unique | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"T2Space",
"TendstoLocallyUniformlyOn",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.congr_inseparable {G : ι → α → β}
(hf : TendstoLocallyUniformlyOn F f p s)
(hg : ∀ᶠ n in p, ∀ x ∈ s, Inseparable (F n x) (G n x)) : TendstoLocallyUniformlyOn G f p s | by
have hg : ∀ᶠ x in p ×ˢ 𝓟 s, Inseparable (F x.1 x.2) (G x.1 x.2) := by
simpa using eventually_prod_principal_iff.2 hg
rw [tendstoLocallyUniformlyOn_iff_forall_tendsto] at hf ⊢
refine forall₂_imp (fun x hx hf => ?_) hf
rw [uniformity_hasBasis_open.tendsto_right_iff] at hf ⊢
exact fun i hi => (hf i hi).m... | theorem | TendstoLocallyUniformlyOn.congr_inseparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Inseparable",
"TendstoLocallyUniformlyOn",
"forall₂_imp",
"tendstoLocallyUniformlyOn_iff_forall_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.congr {G : ι → α → β} (hf : TendstoLocallyUniformlyOn F f p s)
(hg : ∀ n, s.EqOn (F n) (G n)) : TendstoLocallyUniformlyOn G f p s | hf.congr_inseparable (.of_forall fun n _ hx => .of_eq (hg n hx)) | theorem | TendstoLocallyUniformlyOn.congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.congr_inseparable_right {g : α → β}
(hf : TendstoLocallyUniformlyOn F f p s)
(hg : ∀ x ∈ s, Inseparable (f x) (g x)) : TendstoLocallyUniformlyOn F g p s | by
have hg : ∀ᶠ x in p ×ˢ 𝓟 s, Inseparable (f x.2) (g x.2) := by
rw [eventually_prod_principal_iff]
exact .of_forall fun _ => hg
rw [tendstoLocallyUniformlyOn_iff_forall_tendsto] at hf ⊢
refine forall₂_imp (fun x hx hf => ?_) hf
rw [uniformity_hasBasis_open.tendsto_right_iff] at hf ⊢
exact fun i hi =... | theorem | TendstoLocallyUniformlyOn.congr_inseparable_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Inseparable",
"TendstoLocallyUniformlyOn",
"forall₂_imp",
"tendstoLocallyUniformlyOn_iff_forall_tendsto"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformlyOn.congr_right {g : α → β} (hf : TendstoLocallyUniformlyOn F f p s)
(hg : s.EqOn f g) : TendstoLocallyUniformlyOn F g p s | hf.congr_inseparable_right fun _ hx => .of_eq (hg hx) | theorem | TendstoLocallyUniformlyOn.congr_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformlyOn",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.congr_inseparable {G : ι → α → β}
(hf : TendstoLocallyUniformly F f p)
(hg : ∀ᶠ n in p, ∀ x, Inseparable (F n x) (G n x)) : TendstoLocallyUniformly G f p | tendstoLocallyUniformlyOn_univ.1
(hf.tendstoLocallyUniformlyOn.congr_inseparable (by simpa using hg)) | theorem | TendstoLocallyUniformly.congr_inseparable | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Inseparable",
"TendstoLocallyUniformly"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.congr {G : ι → α → β} (hf : TendstoLocallyUniformly F f p)
(hg : ∀ n x, F n x = G n x) : TendstoLocallyUniformly G f p | hf.congr_inseparable (.of_forall fun n x => .of_eq (hg n x)) | theorem | TendstoLocallyUniformly.congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.congr_inseparable_right {g : α → β}
(hf : TendstoLocallyUniformly F f p)
(hg : ∀ x, Inseparable (f x) (g x)) : TendstoLocallyUniformly F g p | tendstoLocallyUniformlyOn_univ.1
(hf.tendstoLocallyUniformlyOn.congr_inseparable_right (by simpa using hg)) | theorem | TendstoLocallyUniformly.congr_inseparable_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"Inseparable",
"TendstoLocallyUniformly"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
TendstoLocallyUniformly.congr_right {g : α → β} (hf : TendstoLocallyUniformly F f p)
(hg : ∀ x, f x = g x) : TendstoLocallyUniformly F g p | hf.congr_inseparable_right fun x => .of_eq (hg x) | theorem | TendstoLocallyUniformly.congr_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/LocallyUniformConvergence.lean | [] | [
"TendstoLocallyUniformly",
"of_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instUniformSpace : UniformSpace (Matrix m n 𝕜) | inferInstanceAs <| UniformSpace (m → n → 𝕜) | instance | Matrix.instUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Matrix.lean | [] | [
"Matrix",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instIsUniformAddGroup [AddGroup 𝕜] [IsUniformAddGroup 𝕜] :
IsUniformAddGroup (Matrix m n 𝕜) | inferInstanceAs <| IsUniformAddGroup (m → n → 𝕜) | instance | Matrix.instIsUniformAddGroup | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Matrix.lean | [] | [
"AddGroup",
"IsUniformAddGroup",
"Matrix"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity :
𝓤 (Matrix m n 𝕜) = ⨅ (i : m) (j : n), (𝓤 𝕜).comap fun a => (a.1 i j, a.2 i j) | by
erw [Pi.uniformity]
simp_rw [Pi.uniformity, Filter.comap_iInf, Filter.comap_comap]
rfl | theorem | Matrix.uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Matrix.lean | [] | [
"Filter.comap_comap",
"Filter.comap_iInf",
"Matrix",
"Pi.uniformity",
"comap",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous {β : Type*} [UniformSpace β] {f : β → Matrix m n 𝕜} :
UniformContinuous f ↔ ∀ i j, UniformContinuous fun x => f x i j | by
simp only [UniformContinuous, Matrix.uniformity, Filter.tendsto_iInf, Filter.tendsto_comap_iff]
apply Iff.intro <;> intro a <;> apply a | theorem | Matrix.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Matrix.lean | [] | [
"Filter.tendsto_comap_iff",
"Filter.tendsto_iInf",
"Matrix",
"Matrix.uniformity",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpaceOfCompactR1 [TopologicalSpace γ] [CompactSpace γ] [R1Space γ] : UniformSpace γ | where
uniformity := 𝓝ˢ (diagonal γ)
symm := continuous_swap.tendsto_nhdsSet fun _ => Eq.symm
comp := by
/-
This is the difficult part of the proof. We need to prove that, for each neighborhood `W`
of the diagonal `Δ`, there exists a smaller neighborhood `V` such that `V ○ V ⊆ W`.
-/
... | def | uniformSpaceOfCompactR1 | Topology.UniformSpace | Mathlib/Topology/UniformSpace/OfCompactT2.lean | [] | [
"ClusterPt",
"CompactSpace",
"Inseparable",
"IsOpen",
"IsOpen.mem_nhds",
"IsOpen.prod",
"IsOpen.union",
"R1Space",
"Set",
"TopologicalSpace",
"UniformSpace",
"closure_compl",
"comap_id'",
"disjoint_nested_nhds_of_not_inseparable",
"exists_clusterPt_of_compactSpace",
"iSup_const",
"iS... | The unique uniform structure inducing a given compact topological structure. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ofFun [AddCommMonoid M] [PartialOrder M]
(d : X → X → M) (refl : ∀ x, d x x = 0)
(symm : ∀ x y, d x y = d y x) (triangle : ∀ x y z, d x z ≤ d x y + d y z)
(half : ∀ ε > (0 : M), ∃ δ > (0 : M), ∀ x < δ, ∀ y < δ, x + y < ε) :
UniformSpace X | .ofCore
{ uniformity := ⨅ r > 0, 𝓟 { x | d x.1 x.2 < r }
refl := le_iInf₂ fun r hr => principal_mono.2 <| by simp [Set.subset_def, *]
symm := tendsto_iInf_iInf fun r => tendsto_iInf_iInf fun _ => tendsto_principal_principal.2
fun x hx => by rwa [mem_setOf, symm]
comp := le_iInf₂ fun r hr ... | def | UniformSpace.ofFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/OfFun.lean | [] | [
"AddCommMonoid",
"PartialOrder",
"Set.subset_def",
"UniformSpace",
"refl",
"symm",
"uniformity"
] | Define a `UniformSpace` using a "distance" function. The function can be, e.g., the
distance in a (usual or extended) metric space or an absolute value on a ring. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hasBasis_ofFun [AddCommMonoid M] [LinearOrder M]
(h₀ : ∃ x : M, 0 < x) (d : X → X → M) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
(triangle : ∀ x y z, d x z ≤ d x y + d y z)
(half : ∀ ε > (0 : M), ∃ δ > (0 : M), ∀ x < δ, ∀ y < δ, x + y < ε) :
𝓤[.ofFun d refl symm triangle half].HasBasis ((0 ... | hasBasis_biInf_principal'
(fun ε₁ h₁ ε₂ h₂ => ⟨min ε₁ ε₂, lt_min h₁ h₂, fun _x hx => lt_of_lt_of_le hx (min_le_left _ _),
fun _x hx => lt_of_lt_of_le hx (min_le_right _ _)⟩) h₀ | theorem | UniformSpace.hasBasis_ofFun | Topology.UniformSpace | Mathlib/Topology/UniformSpace/OfFun.lean | [] | [
"AddCommMonoid",
"LinearOrder",
"lt_min",
"lt_of_lt_of_le",
"min_le_left",
"min_le_right",
"refl",
"symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ofFunOfHasBasis [t : TopologicalSpace X] [AddCommMonoid M] [LinearOrder M]
(d : X → X → M) (refl : ∀ x, d x x = 0)
(symm : ∀ x y, d x y = d y x) (triangle : ∀ x y z, d x z ≤ d x y + d y z)
(half : ∀ ε > (0 : M), ∃ δ > (0 : M), ∀ x < δ, ∀ y < δ, x + y < ε)
(basis : ∀ x, (𝓝 x).HasBasis (fun ε ↦ 0 < ε) (f... | where
toTopologicalSpace := t
nhds_eq_comap_uniformity x :=
(basis x).eq_of_same_basis <|
(hasBasis_ofFun (basis x).ex_mem d refl symm triangle half).comap (Prod.mk x)
__ := ofFun d refl symm triangle half | def | UniformSpace.ofFunOfHasBasis | Topology.UniformSpace | Mathlib/Topology/UniformSpace/OfFun.lean | [] | [
"AddCommMonoid",
"LinearOrder",
"TopologicalSpace",
"UniformSpace",
"comap",
"nhds_eq_comap_uniformity",
"refl",
"symm"
] | Define a `UniformSpace` using a "distance" function. The function can be, e.g., the
distance in a (usual or extended) metric space or an absolute value on a ring. We assume that
there is a preexisting topology, for which the neighborhoods can be expressed using the "distance",
and we make sure that the uniform space st... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instUniformSpace : UniformSpace (Path x y) | .comap ((↑) : _ → C(I, X)) ContinuousMap.compactConvergenceUniformSpace | instance | Path.instUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"ContinuousMap.compactConvergenceUniformSpace",
"Path",
"UniformSpace",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isUniformEmbedding_coe : IsUniformEmbedding ((↑) : Path x y → C(I, X)) | where
comap_uniformity := rfl
injective := ContinuousMap.coe_injective' | theorem | Path.isUniformEmbedding_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"ContinuousMap.coe_injective'",
"IsUniformEmbedding",
"Path"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous (γ : Path x y) : UniformContinuous γ | CompactSpace.uniformContinuous_of_continuous <| map_continuous _ | theorem | Path.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"CompactSpace.uniformContinuous_of_continuous",
"Path",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_extend (γ : Path x y) : UniformContinuous γ.extend | γ.uniformContinuous.comp <| LipschitzWith.projIcc _ |>.uniformContinuous | theorem | Path.uniformContinuous_extend | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"LipschitzWith.projIcc",
"Path",
"UniformContinuous"
] | Given a path `γ`, it extension to the real line `γ.extend : C(ℝ, X)`
is a uniformly continuous function. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_extend_left : UniformContinuous (Path.extend : Path x y → C(ℝ, X)) | ContinuousMap.projIccCM.uniformContinuous_comp_left.comp isUniformEmbedding_coe.uniformContinuous | theorem | Path.uniformContinuous_extend_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"Path",
"Path.extend",
"UniformContinuous"
] | The function sending a path `γ` to its extension `γ.extend : ℝ → X`
is uniformly continuous in `γ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
_root_.Filter.HasBasis.uniformityPath {ι : Sort*} {p : ι → Prop} {U : ι → Set (X × X)}
(hU : (𝓤 X).HasBasis p U) :
(𝓤 (Path x y)).HasBasis p fun i ↦ {γ | ∀ t, (γ.1 t, γ.2 t) ∈ U i} | hU.compactConvergenceUniformity_of_compact.comap _ | theorem | Filter.HasBasis.uniformityPath | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"Path",
"Set"
] | If `{U i | p i}` form a basis of entourages of `X`,
then the entourages `{V i | p i}`, `V i = {(γ₁, γ₂) | ∀ t, (γ₁ t, γ₂ t) ∈ U i}`,
form a basis of entourages of paths between `x` and `y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
hasBasis_uniformity :
(𝓤 (Path x y)).HasBasis (· ∈ 𝓤 X) ({γ | ∀ t, (γ.1 t, γ.2 t) ∈ ·}) | (𝓤 X).basis_sets.uniformityPath | theorem | Path.hasBasis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"Path"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_symm : UniformContinuous (Path.symm : Path x y → Path y x) | hasBasis_uniformity.uniformContinuous_iff hasBasis_uniformity |>.mpr fun U hU ↦
⟨U, hU, fun _ _ h x ↦ h (σ x)⟩ | theorem | Path.uniformContinuous_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"Path",
"Path.symm",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_trans :
UniformContinuous (Path.trans : Path x y → Path y z → Path x z).uncurry | hasBasis_uniformity.uniformity_prod hasBasis_uniformity
|>.uniformContinuous_iff hasBasis_uniformity |>.mpr fun U hU ↦
⟨(U, U), ⟨hU, hU⟩, fun ⟨_, _⟩ ⟨_, _⟩ ⟨h₁, h₂⟩ t ↦ by
by_cases ht : (t : ℝ) ≤ 2⁻¹ <;> simp [Path.trans_apply, ht, h₁ _, h₂ _]⟩ | theorem | Path.uniformContinuous_trans | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"Path",
"Path.trans",
"Path.trans_apply",
"UniformContinuous",
"by_cases"
] | The function `Path.trans` that concatenates two paths `γ₁ : Path x y` and `γ₂ : Path y z`
is uniformly continuous in `(γ₁, γ₂)`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instCompleteSpace [CompleteSpace X] : CompleteSpace (Path x y) | isUniformEmbedding_coe.completeSpace <| by simpa [Set.EqOn, range_coe]
using ContinuousMap.isComplete_setOf_eqOn (Function.update (fun _ : I ↦ y) 0 x) {0, 1} | instance | Path.instCompleteSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Path.lean | [] | [
"CompleteSpace",
"ContinuousMap.isComplete_setOf_eqOn",
"Function.update",
"Path",
"Set.EqOn"
] | The space of paths between two points in a complete uniform space
is a complete uniform space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Pi.uniformSpace : UniformSpace (∀ i, α i) | UniformSpace.ofCoreEq (⨅ i, UniformSpace.comap (eval i) (U i)).toCore
Pi.topologicalSpace <|
Eq.symm toTopologicalSpace_iInf | instance | Pi.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.topologicalSpace",
"UniformSpace",
"UniformSpace.comap",
"UniformSpace.ofCoreEq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformSpace_eq :
Pi.uniformSpace α = ⨅ i, UniformSpace.comap (eval i) (U i) | by
ext : 1; rfl | lemma | Pi.uniformSpace_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformity :
𝓤 (∀ i, α i) = ⨅ i : ι, (Filter.comap fun a => (a.1 i, a.2 i)) (𝓤 (α i)) | iInf_uniformity | theorem | Pi.uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Filter.comap",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_pi {β : Type*} [UniformSpace β] {f : β → ∀ i, α i} :
UniformContinuous f ↔ ∀ i, UniformContinuous fun x => f x i | by
simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp_def] | theorem | uniformContinuous_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformity",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_proj (i : ι) : UniformContinuous fun a : ∀ i : ι, α i => a i | uniformContinuous_pi.1 uniformContinuous_id i | theorem | Pi.uniformContinuous_proj | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"UniformContinuous",
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_precomp' (φ : ι' → ι) :
UniformContinuous (fun (f : (∀ i, α i)) (j : ι') ↦ f (φ j)) | uniformContinuous_pi.mpr fun j ↦ uniformContinuous_proj α (φ j) | theorem | Pi.uniformContinuous_precomp' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_precomp (φ : ι' → ι) :
UniformContinuous (· ∘ φ : (ι → β) → (ι' → β)) | Pi.uniformContinuous_precomp' _ φ | theorem | Pi.uniformContinuous_precomp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformContinuous_precomp'",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_postcomp' {β : ι → Type*} [∀ i, UniformSpace (β i)]
{g : ∀ i, α i → β i} (hg : ∀ i, UniformContinuous (g i)) :
UniformContinuous (fun (f : (∀ i, α i)) (i : ι) ↦ g i (f i)) | uniformContinuous_pi.mpr fun i ↦ (hg i).comp <| uniformContinuous_proj α i | theorem | Pi.uniformContinuous_postcomp' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_postcomp {α : Type*} [UniformSpace α] {g : α → β}
(hg : UniformContinuous g) : UniformContinuous (g ∘ · : (ι → α) → (ι → β)) | Pi.uniformContinuous_postcomp' _ fun _ ↦ hg | theorem | Pi.uniformContinuous_postcomp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformContinuous_postcomp'",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformSpace_comap_precomp' (φ : ι' → ι) :
UniformSpace.comap (fun g i' ↦ g (φ i')) (Pi.uniformSpace (fun i' ↦ α (φ i'))) =
⨅ i', UniformSpace.comap (eval (φ i')) (U (φ i')) | by
simp [Pi.uniformSpace_eq, UniformSpace.comap_iInf, ← UniformSpace.comap_comap, comp_def] | lemma | Pi.uniformSpace_comap_precomp' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformSpace",
"Pi.uniformSpace_eq",
"UniformSpace.comap",
"UniformSpace.comap_comap",
"UniformSpace.comap_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformSpace_comap_precomp (φ : ι' → ι) :
UniformSpace.comap (· ∘ φ) (Pi.uniformSpace (fun _ ↦ β)) =
⨅ i', UniformSpace.comap (eval (φ i')) ‹UniformSpace β› | uniformSpace_comap_precomp' (fun _ ↦ β) φ | lemma | Pi.uniformSpace_comap_precomp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformContinuous_restrict (S : Set ι) :
UniformContinuous (S.restrict : (∀ i : ι, α i) → (∀ i : S, α i)) | Pi.uniformContinuous_precomp' _ ((↑) : S → ι) | lemma | Pi.uniformContinuous_restrict | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformContinuous_precomp'",
"Set",
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformSpace_comap_restrict (S : Set ι) :
UniformSpace.comap (S.restrict) (Pi.uniformSpace (fun i : S ↦ α i)) =
⨅ i ∈ S, UniformSpace.comap (eval i) (U i) | by
simp +unfoldPartialApp
[← iInf_subtype'', ← uniformSpace_comap_precomp' _ ((↑) : S → ι), Set.restrict] | lemma | Pi.uniformSpace_comap_restrict | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformSpace",
"Set",
"Set.restrict",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchy_pi_iff [Nonempty ι] {l : Filter (∀ i, α i)} :
Cauchy l ↔ ∀ i, Cauchy (map (eval i) l) | by
simp_rw +instances [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace, cauchy_comap_uniformSpace] | lemma | cauchy_pi_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Cauchy",
"Filter",
"Pi.uniformSpace_eq",
"cauchy_comap_uniformSpace",
"cauchy_iInf_uniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cauchy_pi_iff' {l : Filter (∀ i, α i)} [l.NeBot] :
Cauchy l ↔ ∀ i, Cauchy (map (eval i) l) | by
simp_rw +instances [Pi.uniformSpace_eq, cauchy_iInf_uniformSpace', cauchy_comap_uniformSpace] | lemma | cauchy_pi_iff' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Cauchy",
"Filter",
"Pi.uniformSpace_eq",
"cauchy_comap_uniformSpace",
"cauchy_iInf_uniformSpace'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Cauchy.pi [Nonempty ι] {l : ∀ i, Filter (α i)} (hl : ∀ i, Cauchy (l i)) :
Cauchy (Filter.pi l) | by
have := fun i ↦ (hl i).1
simpa [cauchy_pi_iff] | lemma | Cauchy.pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Cauchy",
"Filter",
"Filter.pi",
"cauchy_pi_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.complete [∀ i, CompleteSpace (α i)] : CompleteSpace (∀ i, α i) | where
complete {f} hf := by
have := hf.1
simp_rw [cauchy_pi_iff', cauchy_iff_exists_le_nhds] at hf
choose x hx using hf
use x
rwa [nhds_pi, le_pi] | instance | Pi.complete | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"CompleteSpace",
"cauchy_iff_exists_le_nhds",
"cauchy_pi_iff'",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Pi.uniformSpace_comap_restrict_sUnion (𝔖 : Set (Set ι)) :
UniformSpace.comap (⋃₀ 𝔖).restrict (Pi.uniformSpace (fun i : (⋃₀ 𝔖) ↦ α i)) =
⨅ S ∈ 𝔖, UniformSpace.comap S.restrict (Pi.uniformSpace (fun i : S ↦ α i)) | by
simp_rw [Pi.uniformSpace_comap_restrict α, iInf_sUnion] | lemma | Pi.uniformSpace_comap_restrict_sUnion | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"Pi.uniformSpace",
"Pi.uniformSpace_comap_restrict",
"Set",
"UniformSpace.comap",
"iInf_sUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
CompleteSpace.iInf {ι X : Type*} {u : ι → UniformSpace X}
(hu : ∀ i, @CompleteSpace X (u i))
(ht : ∃ t, @T2Space X t ∧ ∀ i, (u i).toTopologicalSpace ≤ t) :
@CompleteSpace X (⨅ i, u i) | by
-- We can assume `X` is nonempty.
nontriviality X
rcases ht with ⟨t, ht, hut⟩
-- The diagonal map `(X, ⨅ i, u i) → ∀ i, (X, u i)` is a uniform embedding.
have : @IsUniformInducing X (ι → X) (⨅ i, u i) (Pi.uniformSpace (U := u)) (const ι) := by
simp_rw [isUniformInducing_iff, iInf_uniformity, Pi.uniform... | theorem | CompleteSpace.iInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Pi.lean | [] | [
"CompleteSpace",
"Filter.comap_comap",
"Filter.comap_iInf",
"IsClosed.isComplete",
"IsUniformInducing",
"Pi.topologicalSpace",
"Pi.uniformSpace",
"Pi.uniformity",
"T2Space",
"UniformSpace",
"comap_id'",
"completeSpace_iff_isComplete_range",
"continuous_apply",
"iInf_uniformity",
"induced... | An infimum of complete uniformities is complete,
as long as the whole family is bounded by some common T2 topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_sum_smul_approximation_of_mem_uniformity [TopologicalSpace R]
[MonoidWithZero R] [MulActionWithZero R V] (f : C(X × Y, V)) (hS : S ∈ 𝓤 V) :
∃ (n : ℕ) (g : Fin n → C(X, R)) (h : Fin n → C(Y, V)),
∀ x y, (f (x, y), ∑ i, g i x • h i y) ∈ S | by
have hS' : {(f, g) | ∀ y, (f y, g y) ∈ S} ∈ 𝓤 C(Y, V) :=
(mem_compactConvergence_entourage_iff _).mpr
⟨_, _, isCompact_univ, hS, by simp only [Set.mem_univ, true_implies, subset_refl]⟩
obtain ⟨n, U, v, hv⟩ := exists_finite_sum_const_indicator_approximation_of_mem_nhds_diagonal
f.curry (nhdsSet_dia... | lemma | ContinuousMap.exists_finite_sum_smul_approximation_of_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"MonoidWithZero",
"MulActionWithZero",
"Set.mem_univ",
"TopologicalSpace",
"by_cases",
"continuous_const",
"isCompact_univ",
"nhdsSet_diagonal_le_uniformity",
"subset_refl"
] | A continuous function on `X × Y`, taking values in an `R`-module with a uniform structure,
can be uniformly approximated by sums of functions of the form `(x, y) ↦ f x • g y`.
Note that no continuity properties are assumed either for multiplication on `R`, or for the scalar
multiplication of `R` on `V`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
exists_finite_sum_mul_approximation_of_mem_uniformity [Ring R] [UniformSpace R]
[IsUniformAddGroup R] (f : C(X × Y, R)) {S : Set (R × R)} (hS : S ∈ 𝓤 R) :
∃ (n : ℕ) (g : Fin n → C(X, R)) (h : Fin n → C(Y, R)),
∀ x y, (f (x, y), ∑ i, g i x * h i y) ∈ S | exists_finite_sum_smul_approximation_of_mem_uniformity f hS | lemma | ContinuousMap.exists_finite_sum_mul_approximation_of_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"IsUniformAddGroup",
"Ring",
"Set",
"UniformSpace"
] | A continuous function on `X × Y`, taking values in a ring `R` equipped with a uniformity
compatible with addition, can be uniformly approximated by sums of functions of the form
`(x, y) ↦ f x * g y`.
Note that no assumption is needed relating the multiplication on `R` to the uniformity. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodMul : C(X, R) →ₗ[R] C(Y, R) →ₗ[R] C(X × Y, R) | LinearMap.mk₂ R (fun f g ↦ (f.comp .fst) * (g.comp .snd))
(fun f f' g ↦ by ext; simp [add_mul])
(fun r f g ↦ by ext; simp)
(fun f g g' ↦ by ext; simp [mul_add])
(fun r f g ↦ by ext; simp) | def | ContinuousMap.prodMul | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"LinearMap.mk₂"
] | The natural bilinear map sending `f, g` to the function `(x, y) ↦ f x * g y` on `X × Y`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
prodMul_apply (f : C(X, R)) (g : C(Y, R)) (p : X × Y) :
f.prodMul g p = f p.1 * g p.2 | (rfl) | lemma | ContinuousMap.prodMul_apply | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
prodMul_def (f : C(X, R)) (g : C(Y, R)) :
f.prodMul g = f.comp .fst * g.comp .snd | (rfl) | lemma | ContinuousMap.prodMul_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tensorHom : C(X, R) ⊗[R] C(Y, R) →ₗ[R] C(X × Y, R) | TensorProduct.lift prodMul | def | ContinuousMap.tensorHom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"TensorProduct.lift"
] | Tensor product version of `ContinuousMap.prodMul`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tensorHom_tmul (f : C(X, R)) (g : C(Y, R)) :
tensorHom (f ⊗ₜ g) = prodMul f g | by
rw [tensorHom, TensorProduct.lift.tmul] | lemma | ContinuousMap.tensorHom_tmul | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"TensorProduct.lift.tmul"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
denseRange_tensorHom [CompactSpace X] [T2Space X] [CompactSpace Y] [T2Space Y]
[TotallyDisconnectedSpace X] :
DenseRange (tensorHom : C(X, R) ⊗[R] C(Y, R) → C(X × Y, R)) | by
let : UniformSpace R := IsTopologicalAddGroup.rightUniformSpace R
let : IsUniformAddGroup R := isUniformAddGroup_of_addCommGroup
intro f
simp_rw [mem_closure_iff, Set.nonempty_def]
intro U hUo hUf
have := mem_nhds_uniformity_iff_right.mp (hUo.mem_nhds hUf)
obtain ⟨J, hJu, hJ'⟩ := (hasBasis_compactConve... | lemma | ContinuousMap.denseRange_tensorHom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/ProdApproximation.lean | [] | [
"CompactSpace",
"DenseRange",
"IsUniformAddGroup",
"Set.mem_of_subset_of_mem",
"Set.mem_setOf_eq",
"Set.nonempty_def",
"T2Space",
"TotallyDisconnectedSpace",
"UniformSpace",
"mem_closure_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Real.instCompleteSpace : CompleteSpace ℝ | by
apply complete_of_cauchySeq_tendsto
intro u hu
let c : CauSeq ℝ abs := ⟨u, Metric.cauchySeq_iff'.1 hu⟩
refine ⟨c.lim, fun s h => ?_⟩
rcases Metric.mem_nhds_iff.1 h with ⟨ε, ε0, hε⟩
have := c.equiv_lim ε ε0
simp only [mem_map, mem_atTop_sets]
exact this.imp fun N hN n hn => hε (hN n hn) | instance | Real.instCompleteSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Real.lean | [] | [
"CauSeq",
"CompleteSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_coe : Continuous ((↑) : ℝ≥0 → ℝ) | continuous_subtype_val | theorem | NNReal.continuous_coe | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Real.lean | [] | [
"Continuous",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.ContinuousMap.coeNNRealReal : C(ℝ≥0, ℝ) | ⟨(↑), continuous_coe⟩ | def | ContinuousMap.coeNNRealReal | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Real.lean | [] | [] | Embedding of `ℝ≥0` to `ℝ` as a bundled continuous map. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coeNNRealReal_zero : ContinuousMap.coeNNRealReal 0 = 0 | rfl | lemma | NNReal.coeNNRealReal_zero | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Real.lean | [] | [
"ContinuousMap.coeNNRealReal"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousMap.canLift {X : Type*} [TopologicalSpace X] :
CanLift C(X, ℝ) C(X, ℝ≥0) ContinuousMap.coeNNRealReal.comp fun f => ∀ x, 0 ≤ f x | where
prf f hf := ⟨⟨fun x => .mk (f x) (hf x), f.2.subtype_mk _⟩, DFunLike.ext' rfl⟩ | instance | NNReal.ContinuousMap.canLift | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Real.lean | [] | [
"CanLift",
"ContinuousMap.canLift",
"DFunLike.ext'",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.completelyNormalSpace_of_hasAntitoneBasis {ι : Type*} [LinearOrder ι]
{B : ι → SetRel α α} (hB : (uniformity α).HasAntitoneBasis B) : CompletelyNormalSpace α | where
completely_normal s t hSt hsT := by
let S (b : Bool) : Set α := b.casesOn (false := s) (true := t)
have hx (b : Bool) (x : S b) : ∃ i, Disjoint (ball x.1 ((B i).comp (B i).inv)) (S (!b)) := by
have hST : Disjoint (S b) (closure (S !b)) := b.casesOn (false := hsT) (true := hSt.symm)
rw [← dis... | theorem | UniformSpace.completelyNormalSpace_of_hasAntitoneBasis | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"CompletelyNormalSpace",
"Disjoint",
"Filter.disjoint_iff",
"LinearOrder",
"Set",
"Set.disjoint_iff",
"SetRel",
"SetRel.comp_subset_comp",
"SetRel.inv_mono",
"closure",
"comp_symm_mem_uniformity_sets",
"disjoint_nhdsSet_principal",
"le_total",
"mem_nhdsSet_iff_forall",
"nhdsSet",
"nhds... | If the uniformity has a linearly ordered basis, then the space is completely normal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.specializes_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
(h : (𝓤 α).HasBasis p s) {x y : α} : x ⤳ y ↔ ∀ i, p i → (x, y) ∈ s i | (nhds_basis_uniformity h).specializes_iff | theorem | Filter.HasBasis.specializes_iff_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Set",
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.inseparable_iff_uniformity {ι : Sort*} {p : ι → Prop} {s : ι → Set (α × α)}
(h : (𝓤 α).HasBasis p s) {x y : α} : Inseparable x y ↔ ∀ i, p i → (x, y) ∈ s i | specializes_iff_inseparable.symm.trans h.specializes_iff_uniformity | theorem | Filter.HasBasis.inseparable_iff_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Inseparable",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_iff_ker_uniformity {x y : α} : Inseparable x y ↔ (x, y) ∈ (𝓤 α).ker | (𝓤 α).basis_sets.inseparable_iff_uniformity | theorem | inseparable_iff_ker_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Inseparable.nhds_le_uniformity {x y : α} (h : Inseparable x y) :
𝓝 (x, y) ≤ 𝓤 α | by
rw [h.prod rfl]
apply nhds_le_uniformity | theorem | Inseparable.nhds_le_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Inseparable",
"nhds_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inseparable_iff_clusterPt_uniformity {x y : α} :
Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) | by
refine ⟨fun h ↦ .of_nhds_le h.nhds_le_uniformity, fun h ↦ ?_⟩
simp_rw [uniformity_hasBasis_closed.inseparable_iff_uniformity, isClosed_iff_clusterPt]
exact fun U ⟨hU, hUc⟩ ↦ hUc _ <| h.mono <| le_principal_iff.2 hU | theorem | inseparable_iff_clusterPt_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"ClusterPt",
"Inseparable",
"isClosed_iff_clusterPt"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_uniformity :
T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y | by
simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker] | theorem | t0Space_iff_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"T0Space",
"inseparable_iff_ker_uniformity",
"t0Space_iff_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_uniformity' :
T0Space α ↔ Pairwise fun x y ↦ ∃ r ∈ 𝓤 α, (x, y) ∉ r | by
simp [t0Space_iff_not_inseparable, inseparable_iff_ker_uniformity] | theorem | t0Space_iff_uniformity' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Pairwise",
"T0Space",
"inseparable_iff_ker_uniformity",
"t0Space_iff_not_inseparable"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
t0Space_iff_ker_uniformity : T0Space α ↔ (𝓤 α).ker = diagonal α | by
simp_rw [t0Space_iff_uniformity, subset_antisymm_iff, diagonal_subset_iff, subset_def,
Prod.forall, Filter.mem_ker, mem_diagonal_iff, iff_self_and]
exact fun _ x s hs ↦ refl_mem_uniformity hs | theorem | t0Space_iff_ker_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Filter.mem_ker",
"T0Space",
"refl_mem_uniformity",
"subset_antisymm_iff",
"t0Space_iff_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_of_uniformity {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y | t0Space_iff_uniformity.mp ‹T0Space α› x y @h | theorem | eq_of_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"T0Space",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_of_uniformity_basis {α : Type*} [UniformSpace α] [T0Space α] {ι : Sort*}
{p : ι → Prop} {s : ι → Set (α × α)} (hs : (𝓤 α).HasBasis p s) {x y : α}
(h : ∀ {i}, p i → (x, y) ∈ s i) : x = y | (hs.inseparable_iff_uniformity.2 @h).eq | theorem | eq_of_uniformity_basis | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"Set",
"T0Space",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_of_forall_symmetric {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → SetRel.IsSymm V → (x, y) ∈ V) : x = y | eq_of_uniformity_basis hasBasis_symmetric (by simpa) | theorem | eq_of_forall_symmetric | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"SetRel.IsSymm",
"T0Space",
"UniformSpace",
"eq_of_uniformity_basis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_of_clusterPt_uniformity [T0Space α] {x y : α} (h : ClusterPt (x, y) (𝓤 α)) : x = y | (inseparable_iff_clusterPt_uniformity.2 h).eq | theorem | eq_of_clusterPt_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"ClusterPt",
"T0Space"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.Tendsto.inseparable_iff_uniformity {β} {l : Filter β} [NeBot l] {f g : β → α}
{a b : α} (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) :
Inseparable a b ↔ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α) | by
refine ⟨fun h ↦ (ha.prodMk_nhds hb).mono_right h.nhds_le_uniformity, fun h ↦ ?_⟩
rw [inseparable_iff_clusterPt_uniformity]
exact (ClusterPt.of_le_nhds (ha.prodMk_nhds hb)).mono h | theorem | Filter.Tendsto.inseparable_iff_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"ClusterPt.of_le_nhds",
"Filter",
"Inseparable",
"inseparable_iff_clusterPt_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_of_spaced_out [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α) {s : Set α}
(hs : s.Pairwise fun x y => (x, y) ∉ V₀) : IsClosed s | by
rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩
apply isClosed_of_closure_subset
intro x hx
rw [mem_closure_iff_ball] at hx
rcases hx V₁_in with ⟨y, hy, hy'⟩
suffices x = y by rwa [this]
apply eq_of_forall_symmetric
intro V V_in _
rcases hx (inter_mem V₁_in V_in) with ⟨... | theorem | isClosed_of_spaced_out | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"IsClosed",
"Set",
"T0Space",
"comp_symm_mem_uniformity_sets",
"eq_of_forall_symmetric",
"isClosed_of_closure_subset"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_range_of_spaced_out {ι} [T0Space α] {V₀ : Set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{f : ι → α} (hf : Pairwise fun x y => (f x, f y) ∉ V₀) : IsClosed (range f) | isClosed_of_spaced_out V₀_in <| by
rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h
exact hf (ne_of_apply_ne f h) | theorem | isClosed_range_of_spaced_out | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Separation.lean | [] | [
"IsClosed",
"Pairwise",
"Set",
"T0Space",
"isClosed_of_spaced_out"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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