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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Equicontinuous.tendsto_uniformFun_iff_pi [CompactSpace X]
(F_eqcont : Equicontinuous F) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformFun.ofFun ∘ F) ℱ (𝓝 <| UniformFun.ofFun f) ↔
Tendsto F ℱ (𝓝 f) | by
-- Assume `ℱ` is non-trivial.
rcases ℱ.eq_or_neBot with rfl | ℱ_ne
· simp
constructor <;> intro H
-- The forward direction is always true, the interesting part is the converse.
· exact UniformFun.uniformContinuous_toFun.continuous.tendsto _ |>.comp H
-- To prove it, assume that `F` tends to `f` *pointw... | theorem | Equicontinuous.tendsto_uniformFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"Equicontinuous",
"Filter",
"Filter.map_comap_of_mem",
"Set",
"Subtype.range_coe",
"UniformFun.ofFun",
"closure",
"comap",
"continuous_id",
"equicontinuous_iff_range",
"mem_closure_of_tendsto",
"nhds_induced",
"subset_closure"
] | Let `X` be a compact topological space, `α` a uniform space, `F : ι → (X → α)` an
equicontinuous family, and `ℱ` a filter on `ι`. Then, `F` tends *uniformly* to `f : X → α` along
`ℱ` iff it tends to `f` *pointwise* along `ℱ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.comap_uniformOnFun_eq {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
(UniformOnFun.uniformSpace X α 𝔖).comap F =
(Pi.uniformSpace _).comap ((⋃₀ 𝔖).restrict ∘ F) | by
-- Recall that the uniform structure on `X →ᵤ[𝔖] α` is the one induced by all the maps
-- `K.restrict : (X →ᵤ[𝔖] α) → (K →ᵤ α)` for `K ∈ 𝔖`. Its pullback along `F`, which is
-- the LHS of our goal, is thus the uniform structure induced by the maps
-- `K.restrict ∘ F : ι → (K →ᵤ α)` for `K ∈ 𝔖`.
have H1... | theorem | EquicontinuousOn.comap_uniformOnFun_eq | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"EquicontinuousOn",
"Equiv.coe_fn_mk",
"Equiv.coe_fn_symm_mk",
"IsCompact",
"Pi.uniformSpace",
"Pi.uniformSpace_comap_restrict_sUnion",
"Set",
"UniformFun",
"UniformFun.ofFun",
"UniformFun.uniformSpace",
"UniformOnFun.ofFun",
"UniformOnFun.toFun",
"UniformOnFun.uniformSpace... | Let `X` be a topological space, `𝔖` a family of compact subsets of `X`, `α` a uniform space,
and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the uniform
structures of uniform convergence on `𝔖` and pointwise convergence on `⋃₀ 𝔖` induce the same
uniform structure on `ι`.
In particular... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi' [UniformSpace ι]
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
IsUniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
IsUniformInducing ((⋃₀ 𝔖).restrict ∘ F) | by
rw [isUniformInducing_iff_uniformSpace, isUniformInducing_iff_uniformSpace,
← EquicontinuousOn.comap_uniformOnFun_eq 𝔖_compact F_eqcont]
rfl | lemma | EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.comap_uniformOnFun_eq",
"IsCompact",
"IsUniformInducing",
"Set",
"UniformOnFun.ofFun",
"UniformSpace",
"isUniformInducing_iff_uniformSpace"
] | Let `X` be a topological space, `𝔖` a family of compact subsets of `X`, `α` a uniform space,
and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the uniform
structures of uniform convergence on `𝔖` and pointwise convergence on `⋃₀ 𝔖` induce the same
uniform structure on `ι`.
In particular... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi [UniformSpace ι]
{𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
IsUniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
IsUniformInducing F | by
rw [eq_univ_iff_forall] at 𝔖_covers
-- This obviously follows from the previous lemma, we formalize it by going through the
-- isomorphism of uniform spaces between `(⋃₀ 𝔖) → α` and `X → α`.
let φ : ((⋃₀ 𝔖) → α) ≃ᵤ (X → α) := UniformEquiv.piCongrLeft (β := fun _ ↦ α)
(Equiv.subtypeUnivEquiv 𝔖_covers)... | lemma | EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'",
"Equiv.subtypeUnivEquiv",
"IsCompact",
"IsUniformInducing",
"Set",
"UniformEquiv.piCongrLeft",
"UniformOnFun.ofFun",
"UniformSpace"
] | Let `X` be a topological space, `𝔖` a covering of `X` by compact subsets, `α` a uniform space,
and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the uniform
structures of uniform convergence on `𝔖` and pointwise convergence induce the same
uniform structure on `ι`.
This is a specializati... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.inducing_uniformOnFun_iff_pi' [TopologicalSpace ι]
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
IsInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
IsInducing ((⋃₀ 𝔖).restrict ∘ F) | by
rw [isInducing_iff, isInducing_iff]
change (_ = ((UniformOnFun.uniformSpace X α 𝔖).comap F).toTopologicalSpace) ↔
(_ = ((Pi.uniformSpace _).comap ((⋃₀ 𝔖).restrict ∘ F)).toTopologicalSpace)
rw [← EquicontinuousOn.comap_uniformOnFun_eq 𝔖_compact F_eqcont] | lemma | EquicontinuousOn.inducing_uniformOnFun_iff_pi' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.comap_uniformOnFun_eq",
"IsCompact",
"Pi.uniformSpace",
"Set",
"TopologicalSpace",
"UniformOnFun.ofFun",
"UniformOnFun.uniformSpace",
"comap"
] | Let `X` be a topological space, `𝔖` a family of compact subsets of `X`, `α` a uniform space,
and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the topologies
of uniform convergence on `𝔖` and pointwise convergence on `⋃₀ 𝔖` induce the same topology on `ι`.
In particular, pointwise conv... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.isInducing_uniformOnFun_iff_pi [TopologicalSpace ι]
{𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
IsInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
IsInducing F | by
rw [eq_univ_iff_forall] at 𝔖_covers
-- This obviously follows from the previous lemma, we formalize it by going through the
-- homeomorphism between `(⋃₀ 𝔖) → α` and `X → α`.
let φ : ((⋃₀ 𝔖) → α) ≃ₜ (X → α) := Homeomorph.piCongrLeft (Y := fun _ ↦ α)
(Equiv.subtypeUnivEquiv 𝔖_covers)
rw [Equicontinu... | lemma | EquicontinuousOn.isInducing_uniformOnFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.inducing_uniformOnFun_iff_pi'",
"Equiv.subtypeUnivEquiv",
"Homeomorph.piCongrLeft",
"IsCompact",
"Set",
"TopologicalSpace",
"UniformOnFun.ofFun"
] | Let `X` be a topological space, `𝔖` a covering of `X` by compact subsets, `α` a uniform space,
and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the topologies
of uniform convergence on `𝔖` and pointwise convergence induce the same topology on `ι`.
This is a specialization of `Equicontin... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 <| (⋃₀ 𝔖).restrict ... | by
-- Recall that the uniform structure on `X →ᵤ[𝔖] α` is the one induced by all the maps
-- `K.restrict : (X →ᵤ[𝔖] α) → (K →ᵤ α)` for `K ∈ 𝔖`.
-- Similarly, the uniform structure on `X → α` induced by the map
-- `(⋃₀ 𝔖).restrict : (X → α) → ((⋃₀ 𝔖) → α)` is equal to the one induced by
-- all maps `K.res... | theorem | EquicontinuousOn.tendsto_uniformOnFun_iff_pi' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"EquicontinuousOn",
"Filter",
"Filter.tendsto_comap_iff",
"IsCompact",
"Pi.induced_restrict_sUnion",
"Set",
"UniformOnFun.ofFun",
"UniformOnFun.topologicalSpace_eq",
"equicontinuous_restrict_iff",
"nhds_induced"
] | Let `X` be a topological space, `𝔖` a family of compact subsets of `X`,
`α` a uniform space, `F : ι → (X → α)` a family equicontinuous on each `K ∈ 𝔖`, and `ℱ` a filter
on `ι`. Then, `F` tends to `f : X → α` along `ℱ` *uniformly on each `K ∈ 𝔖`* iff it tends to `f`
*pointwise on `⋃₀ 𝔖`* along `ℱ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.tendsto_uniformOnFun_iff_pi
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
Tendsto F ℱ (𝓝 f) | by
rw [eq_univ_iff_forall] at 𝔖_covers
let φ : ((⋃₀ 𝔖) → α) ≃ₜ (X → α) := Homeomorph.piCongrLeft (Y := fun _ ↦ α)
(Equiv.subtypeUnivEquiv 𝔖_covers)
rw [EquicontinuousOn.tendsto_uniformOnFun_iff_pi' 𝔖_compact F_eqcont,
show restrict (⋃₀ 𝔖) ∘ F = φ.symm ∘ F by rfl, show restrict (⋃₀ 𝔖) f = φ.symm f ... | theorem | EquicontinuousOn.tendsto_uniformOnFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.tendsto_uniformOnFun_iff_pi'",
"Equiv.subtypeUnivEquiv",
"Filter",
"Homeomorph.piCongrLeft",
"IsCompact",
"Set",
"UniformOnFun.ofFun"
] | Let `X` be a topological space, `𝔖` a covering of `X` by compact subsets,
`α` a uniform space, `F : ι → (X → α)` a family equicontinuous on each `K ∈ 𝔖`, and `ℱ` a filter
on `ι`. Then, `F` tends to `f : X → α` along `ℱ` *uniformly on each `K ∈ 𝔖`* iff it tends to `f`
*pointwise* along `ℱ`.
This is a specialization ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.isClosed_range_pi_of_uniformOnFun'
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
(H : IsClosed (range <| UniformOnFun.ofFun 𝔖 ∘ F)) :
IsClosed (range <| (⋃₀ 𝔖).restrict ∘ F) | by
-- Do we have no equivalent of `nontriviality`?
rcases isEmpty_or_nonempty α with _ | _
· simp [isClosed_discrete]
-- This follows from the previous lemmas and the characterization of the closure using filters.
simp_rw [isClosed_iff_clusterPt, ← Filter.map_top, ← mapClusterPt_def,
mapClusterPt_iff_ultr... | theorem | EquicontinuousOn.isClosed_range_pi_of_uniformOnFun' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.tendsto_uniformOnFun_iff_pi'",
"Filter.map_top",
"IsClosed",
"IsCompact",
"Set",
"UniformOnFun.ofFun",
"isClosed_discrete",
"isClosed_iff_clusterPt",
"isEmpty_or_nonempty",
"mapClusterPt_def",
"mapClusterPt_iff_ultrafilter"
] | Let `X` be a topological space, `𝔖` a family of compact subsets of `X` and
`α` a uniform space. An equicontinuous subset of `X → α` is closed in the topology of uniform
convergence on all `K ∈ 𝔖` iff it is closed in the topology of pointwise convergence on `⋃₀ 𝔖`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (𝔖_covers : ⋃₀ 𝔖 = univ)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
IsClosed (range <| UniformOnFun.ofFun 𝔖 ∘ F) ↔
IsClosed (range F) | by
-- This follows from the previous lemmas and the characterization of the closure using filters.
simp_rw [isClosed_iff_clusterPt, ← Filter.map_top, ← mapClusterPt_def,
mapClusterPt_iff_ultrafilter, range_comp, (UniformOnFun.ofFun 𝔖).surjective.forall,
← EquicontinuousOn.tendsto_uniformOnFun_iff_pi 𝔖_com... | theorem | EquicontinuousOn.isClosed_range_uniformOnFun_iff_pi | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"EquicontinuousOn",
"EquicontinuousOn.tendsto_uniformOnFun_iff_pi",
"Filter.map_top",
"IsClosed",
"IsCompact",
"Set",
"UniformOnFun.ofFun",
"isClosed_iff_clusterPt",
"mapClusterPt_def",
"mapClusterPt_iff_ultrafilter"
] | Let `X` be a topological space, `𝔖` a covering of `X` by compact subsets, and
`α` a uniform space. An equicontinuous subset of `X → α` is closed in the topology of uniform
convergence on all `K ∈ 𝔖` iff it is closed in the topology of pointwise convergence.
This is a specialization of `EquicontinuousOn.isClosed_rang... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ArzelaAscoli.compactSpace_of_closed_inducing' [TopologicalSpace ι] {𝔖 : Set (Set X)}
(𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_ind : IsInducing (UniformOnFun.ofFun 𝔖 ∘ F))
(F_cl : IsClosed <| range <| UniformOnFun.ofFun 𝔖 ∘ F)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
(F_pointwiseCompact : ∀ K ∈ 𝔖, ... | by
-- By equicontinuity, we know that the topology on `ι` is also the one induced by
-- `restrict (⋃₀ 𝔖) ∘ F`.
have : IsInducing (restrict (⋃₀ 𝔖) ∘ F) := by
rwa [EquicontinuousOn.inducing_uniformOnFun_iff_pi' 𝔖_compact F_eqcont] at F_ind
-- Thus, we just have to check that the range of this map is compac... | theorem | ArzelaAscoli.compactSpace_of_closed_inducing' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"EquicontinuousOn",
"EquicontinuousOn.inducing_uniformOnFun_iff_pi'",
"EquicontinuousOn.isClosed_range_pi_of_uniformOnFun'",
"IsClosed",
"IsCompact",
"IsCompact.of_isClosed_subset",
"Set",
"TopologicalSpace",
"UniformOnFun.ofFun",
"forall_sUnion",
"isCompact_univ_iff",
"isCom... | A version of the **Arzela-Ascoli theorem**.
Let `X` be a topological space, `𝔖` a family of compact subsets of `X`, `α` a uniform space,
and `F : ι → (X → α)`. Assume that:
* `F`, viewed as a function `ι → (X →ᵤ[𝔖] α)`, is closed and inducing
* `F` is equicontinuous on each `K ∈ 𝔖`
* For all `x ∈ ⋃₀ 𝔖`, the range ... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ArzelaAscoli.compactSpace_of_isClosedEmbedding [TopologicalSpace ι] {𝔖 : Set (Set X)}
(𝔖_compact : ∀ K ∈ 𝔖, IsCompact K) (F_clemb : IsClosedEmbedding (UniformOnFun.ofFun 𝔖 ∘ F))
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K)
(F_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i, F i x ∈ Q) :
... | compactSpace_of_closed_inducing' 𝔖_compact F_clemb.isInducing F_clemb.isClosed_range
F_eqcont F_pointwiseCompact | theorem | ArzelaAscoli.compactSpace_of_isClosedEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"CompactSpace",
"EquicontinuousOn",
"IsCompact",
"Set",
"TopologicalSpace",
"UniformOnFun.ofFun"
] | A version of the **Arzela-Ascoli theorem**.
Let `X, ι` be topological spaces, `𝔖` a covering of `X` by compact subsets, `α` a uniform space,
and `F : ι → (X → α)`. Assume that:
* `F`, viewed as a function `ι → (X →ᵤ[𝔖] α)`, is a closed embedding (in other words, `ι`
identifies to a closed subset of `X →ᵤ[𝔖] α` th... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ArzelaAscoli.isCompact_closure_of_isClosedEmbedding [TopologicalSpace ι] [T2Space α]
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_clemb : IsClosedEmbedding (UniformOnFun.ofFun 𝔖 ∘ F))
{s : Set ι} (s_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn (F ∘ ((↑) : s → ι)) K)
(s_pointwiseCompact : ∀ K ∈ 𝔖,... | by
-- We apply `ArzelaAscoli.compactSpace_of_isClosedEmbedding` to the map
-- `F ∘ (↑) : closure s → (X → α)`, for which all the hypotheses are easily verified.
rw [isCompact_iff_compactSpace]
have : ∀ K ∈ 𝔖, ∀ x ∈ K, Continuous (eval x ∘ F) := fun K hK x hx ↦
UniformOnFun.uniformContinuous_eval_of_mem _ _... | theorem | ArzelaAscoli.isCompact_closure_of_isClosedEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"ArzelaAscoli.compactSpace_of_isClosedEmbedding",
"Continuous",
"EquicontinuousOn",
"IsCompact",
"Set",
"T2Space",
"TopologicalSpace",
"UniformOnFun.ofFun",
"UniformOnFun.uniformContinuous_eval_of_mem",
"closure",
"closure_minimal",
"continuous_pi",
"isCompact_iff_compactSpace"
] | A version of the **Arzela-Ascoli theorem**.
Let `X, ι` be topological spaces, `𝔖` a covering of `X` by compact subsets, `α` a T2 uniform space,
`F : ι → (X → α)`, and `s` a subset of `ι`. Assume that:
* `F`, viewed as a function `ι → (X →ᵤ[𝔖] α)`, is a closed embedding (in other words, `ι`
identifies to a closed s... | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ArzelaAscoli.isCompact_of_equicontinuous
(S : Set C(X, α)) (hS1 : IsCompact (ContinuousMap.toFun '' S))
(hS2 : Equicontinuous ((↑) : S → X → α)) : IsCompact S | by
suffices h : IsInducing (Equiv.Set.image _ S DFunLike.coe_injective) by
rw [isCompact_iff_compactSpace] at hS1 ⊢
exact (Equiv.toHomeomorphOfIsInducing _ h).symm.compactSpace
rw [← IsInducing.subtypeVal.of_comp_iff, ← EquicontinuousOn.isInducing_uniformOnFun_iff_pi _ _ _]
· exact ContinuousMap.isUniform... | theorem | ArzelaAscoli.isCompact_of_equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Ascoli.lean | [] | [
"DFunLike.coe_injective",
"Equicontinuous",
"EquicontinuousOn.isInducing_uniformOnFun_iff_pi",
"Equiv.Set.image",
"Equiv.toHomeomorphOfIsInducing",
"IsCompact",
"Set",
"isCompact_iff_compactSpace",
"isCompact_singleton"
] | A version of the **Arzela-Ascoli theorem**.
If an equicontinuous family of continuous functions is compact in the pointwise topology, then it
is compact in the compact open topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsOpen.relComp [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
{s : SetRel α β} {t : SetRel β γ} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ○ t) | by
conv =>
arg 1; equals ⋃ b, (fun p => (p.1, b)) ⁻¹' s ∩ (fun p => (b, p.2)) ⁻¹' t => ext ⟨_, _⟩; simp
exact isOpen_iUnion fun a ↦ hs.preimage (by fun_prop) |>.inter <| ht.preimage (by fun_prop) | lemma | IsOpen.relComp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel",
"TopologicalSpace",
"isOpen_iUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.relInv [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsOpen s) : IsOpen s.inv | hs.preimage continuous_swap | lemma | IsOpen.relInv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel",
"TopologicalSpace",
"continuous_swap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.relImage [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsOpen s) {t : Set α} : IsOpen (s.image t) | by
simp_rw [SetRel.image, ← exists_prop, Set.setOf_exists]
exact isOpen_biUnion fun _ _ => hs.preimage <| .prodMk_right _ | lemma | IsOpen.relImage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"Set",
"Set.setOf_exists",
"SetRel",
"SetRel.image",
"TopologicalSpace",
"isOpen_biUnion"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsOpen.relPreimage [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsOpen s) {t : Set β} : IsOpen (s.preimage t) | hs.relInv.relImage | lemma | IsOpen.relPreimage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"Set",
"SetRel",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.relInv [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsClosed s) : IsClosed s.inv | hs.preimage continuous_swap | lemma | IsClosed.relInv | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"SetRel",
"TopologicalSpace",
"continuous_swap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.relImage_of_finite [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsClosed s) {t : Set α} (ht : t.Finite) : IsClosed (s.image t) | by
simp_rw [SetRel.image, ← exists_prop, Set.setOf_exists]
exact ht.isClosed_biUnion fun _ _ => hs.preimage <| .prodMk_right _ | lemma | IsClosed.relImage_of_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"Set",
"Set.setOf_exists",
"SetRel",
"SetRel.image",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsClosed.relPreimage_of_finite [TopologicalSpace α] [TopologicalSpace β]
{s : SetRel α β} (hs : IsClosed s) {t : Set β} (ht : t.Finite) : IsClosed (s.preimage t) | hs.relInv.relImage_of_finite ht | lemma | IsClosed.relPreimage_of_finite | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"Set",
"SetRel",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eventually_uniformity_iterate_comp_subset {s : SetRel α α} (hs : s ∈ 𝓤 α) (n : ℕ) :
∀ᶠ t in (𝓤 α).smallSets, (t ○ ·)^[n] t ⊆ s | by
suffices ∀ᶠ t in (𝓤 α).smallSets, t ⊆ s ∧ (t ○ ·)^[n] t ⊆ s from (eventually_and.1 this).2
induction n generalizing s with
| zero => simpa
| succ _ ihn =>
rcases comp_mem_uniformity_sets hs with ⟨t, htU, hts⟩
refine (ihn htU).mono fun U hU => ?_
rw [Function.iterate_succ_apply']
have := isRe... | theorem | eventually_uniformity_iterate_comp_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Function.iterate_succ_apply'",
"SetRel",
"SetRel.comp_subset_comp",
"comp_mem_uniformity_sets",
"isRefl_of_mem_uniformity",
"trans"
] | If `s ∈ 𝓤 α`, then for any natural `n`, for a subset `t` of a sufficiently small set in `𝓤 α`,
we have `t ○ t ○ ... ○ t ⊆ s` (`n` compositions). | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
eventually_uniformity_comp_subset {s : SetRel α α} (hs : s ∈ 𝓤 α) :
∀ᶠ t in (𝓤 α).smallSets, t ○ t ⊆ s | eventually_uniformity_iterate_comp_subset hs 1 | theorem | eventually_uniformity_comp_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"eventually_uniformity_iterate_comp_subset"
] | If `s ∈ 𝓤 α`, then for a subset `t` of a sufficiently small set in `𝓤 α`,
we have `t ○ t ⊆ s`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_ball (x : α) {V : SetRel α α} (hV : IsOpen V) : IsOpen (ball x V) | hV.preimage <| .prodMk_right _ | lemma | UniformSpace.isOpen_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isClosed_ball (x : α) {V : SetRel α α} (hV : IsClosed V) : IsClosed (ball x V) | hV.preimage <| .prodMk_right _ | lemma | UniformSpace.isClosed_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
hasBasis_nhds_prod (x y : α) :
HasBasis (𝓝 (x, y)) (fun s => s ∈ 𝓤 α ∧ SetRel.IsSymm s) fun s => ball x s ×ˢ ball y s | by
rw [nhds_prod_eq]
apply (hasBasis_nhds x).prod_same_index (hasBasis_nhds y)
rintro U V ⟨U_in, U_symm⟩ ⟨V_in, V_symm⟩
exact ⟨U ∩ V, ⟨(𝓤 α).inter_sets U_in V_in, inferInstance⟩, ball_inter_left x U V,
ball_inter_right y U V⟩ | theorem | UniformSpace.hasBasis_nhds_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel.IsSymm",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_uniformity_prod {a b : α} :
𝓝 (a, b) =
(𝓤 α).lift' fun s : SetRel α α => { y : α | (y, a) ∈ s } ×ˢ { y : α | (b, y) ∈ s } | by
rw [nhds_prod_eq, nhds_nhds_eq_uniformity_uniformity_prod, lift_lift'_same_eq_lift']
· exact fun s => monotone_const.set_prod monotone_preimage
· refine fun t => Monotone.set_prod ?_ monotone_const
exact monotone_preimage (f := fun y => (y, a)) | theorem | nhds_eq_uniformity_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Monotone.set_prod",
"SetRel",
"monotone_const",
"nhds_nhds_eq_uniformity_uniformity_prod",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhdset_of_mem_uniformity {d : SetRel α α} (s : SetRel α α) (hd : d ∈ 𝓤 α) :
∃ t : SetRel α α, IsOpen t ∧ s ⊆ t ∧
t ⊆ { p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d } | by
let cl_d := { p : α × α | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d }
have : ∀ p ∈ s, ∃ t, t ⊆ cl_d ∧ IsOpen t ∧ p ∈ t := fun ⟨x, y⟩ hp =>
mem_nhds_iff.mp <|
show cl_d ∈ 𝓝 (x, y) by
rw [nhds_eq_uniformity_prod, mem_lift'_sets]
· exact ⟨d, hd, fun ⟨a, b⟩ ⟨ha, hb⟩ => ⟨x, y, ha, hp,... | theorem | nhdset_of_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel",
"isOpen_iUnion",
"nhds_eq_uniformity_prod"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_le_uniformity (x : α) : 𝓝 (x, x) ≤ 𝓤 α | by
intro V V_in
rcases comp_symm_mem_uniformity_sets V_in with ⟨w, w_in, w_symm, w_sub⟩
have : ball x w ×ˢ ball x w ∈ 𝓝 (x, x) := by
rw [nhds_prod_eq]
exact prod_mem_prod (ball_mem_nhds x w_in) (ball_mem_nhds x w_in)
apply mem_of_superset this
rintro ⟨u, v⟩ ⟨u_in, v_in⟩
exact w_sub (mem_comp_of_mem... | theorem | nhds_le_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"comp_symm_mem_uniformity_sets",
"nhds_prod_eq"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
iSup_nhds_le_uniformity : ⨆ x : α, 𝓝 (x, x) ≤ 𝓤 α | iSup_le nhds_le_uniformity | theorem | iSup_nhds_le_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"iSup_le",
"nhds_le_uniformity"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhdsSet_diagonal_le_uniformity : 𝓝ˢ (diagonal α) ≤ 𝓤 α | (nhdsSet_diagonal α).trans_le iSup_nhds_le_uniformity | theorem | nhdsSet_diagonal_le_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"iSup_nhds_le_uniformity",
"nhdsSet_diagonal"
] | Entourages are neighborhoods of the diagonal. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformSpace.has_seq_basis [IsCountablyGenerated <| 𝓤 α] :
∃ V : ℕ → SetRel α α, HasAntitoneBasis (𝓤 α) V ∧ ∀ n, SetRel.IsSymm (V n) | let ⟨U, hsym, hbasis⟩ := (@UniformSpace.hasBasis_symmetric α _).exists_antitone_subbasis
⟨U, hbasis, fun n => (hsym n).2⟩ | theorem | UniformSpace.has_seq_basis | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"UniformSpace.hasBasis_symmetric"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_eq_uniformity (s : Set <| α × α) :
closure s = ⋂ V ∈ {V | V ∈ 𝓤 α ∧ SetRel.IsSymm V}, V ○ s ○ V | by
ext ⟨x, y⟩
simp +contextual only
[mem_closure_iff_nhds_basis (UniformSpace.hasBasis_nhds_prod x y), mem_iInter, mem_setOf_eq,
and_imp, mem_comp_comp, ← mem_inter_iff, inter_comm, Set.Nonempty] | theorem | closure_eq_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"Set.Nonempty",
"SetRel.IsSymm",
"UniformSpace.hasBasis_nhds_prod",
"closure",
"mem_closure_iff_nhds_basis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_hasBasis_closed :
HasBasis (𝓤 α) (fun V : SetRel α α => V ∈ 𝓤 α ∧ IsClosed V) id | by
refine Filter.hasBasis_self.2 fun t h => ?_
rcases comp_comp_symm_mem_uniformity_sets h with ⟨w, w_in, w_symm, r⟩
refine ⟨closure w, mem_of_superset w_in subset_closure, isClosed_closure, ?_⟩
refine Subset.trans ?_ r
rw [closure_eq_uniformity]
apply iInter_subset_of_subset
apply iInter_subset
exact ⟨... | theorem | uniformity_hasBasis_closed | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"SetRel",
"closure_eq_uniformity",
"comp_comp_symm_mem_uniformity_sets",
"isClosed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_eq_uniformity_closure : 𝓤 α = (𝓤 α).lift' closure | Eq.symm <| uniformity_hasBasis_closed.lift'_closure_eq_self fun _ => And.right | theorem | uniformity_eq_uniformity_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformity_closure {p : ι → Prop} {U : ι → SetRel α α}
(h : (𝓤 α).HasBasis p U) : (𝓤 α).HasBasis p fun i => closure (U i) | (@uniformity_eq_uniformity_closure α _).symm ▸ h.lift'_closure | theorem | Filter.HasBasis.uniformity_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"closure",
"symm",
"uniformity_eq_uniformity_closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_hasBasis_closure : HasBasis (𝓤 α) (fun V : SetRel α α => V ∈ 𝓤 α) closure | (𝓤 α).basis_sets.uniformity_closure | theorem | uniformity_hasBasis_closure | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"closure"
] | Closed entourages form a basis of the uniformity filter. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
closure_eq_inter_uniformity {t : SetRel α α} : closure t = ⋂ d ∈ 𝓤 α, d ○ (t ○ d) | calc
closure t = ⋂ (V) (_ : V ∈ 𝓤 α ∧ SetRel.IsSymm V), V ○ t ○ V := closure_eq_uniformity t
_ = ⋂ V ∈ 𝓤 α, V ○ t ○ V :=
Eq.symm <|
UniformSpace.hasBasis_symmetric.biInter_mem fun _ _ hV => by dsimp at *; gcongr
_ = ⋂ V ∈ 𝓤 α, V ○ (t ○ V) := by simp [SetRel.comp_assoc] | theorem | closure_eq_inter_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"SetRel.comp_assoc",
"closure",
"closure_eq_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_eq_uniformity_interior : 𝓤 α = (𝓤 α).lift' interior | le_antisymm
(le_iInf₂ fun d hd => by
let ⟨s, hs, hs_comp⟩ := comp3_mem_uniformity hd
let ⟨t, ht, hst, ht_comp⟩ := nhdset_of_mem_uniformity s hs
have : s ⊆ interior d :=
calc
s ⊆ t := hst
_ ⊆ interior d :=
ht.subset_interior_iff.mpr fun x (hx : x ∈ t) =>
... | theorem | uniformity_eq_uniformity_interior | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"comp3_mem_uniformity",
"interior",
"interior_subset",
"le_antisymm",
"nhdset_of_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
interior_mem_uniformity {s : SetRel α α} (hs : s ∈ 𝓤 α) : interior s ∈ 𝓤 α | by
rw [uniformity_eq_uniformity_interior]; exact mem_lift' hs | theorem | interior_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"interior",
"uniformity_eq_uniformity_interior"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_uniformity_isClosed {s : SetRel α α} (h : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, IsClosed t ∧ t ⊆ s | let ⟨t, ⟨ht_mem, htc⟩, hts⟩ := uniformity_hasBasis_closed.mem_iff.1 h
⟨t, ht_mem, htc, hts⟩ | theorem | mem_uniformity_isClosed | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsClosed",
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isOpen_iff_isOpen_ball_subset {s : Set α} :
IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, IsOpen V ∧ ball x V ⊆ s | by
rw [isOpen_iff_ball_subset]
constructor <;> intro h x hx
· obtain ⟨V, hV, hV'⟩ := h x hx
exact
⟨interior V, interior_mem_uniformity hV, isOpen_interior,
(ball_mono interior_subset x).trans hV'⟩
· obtain ⟨V, hV, -, hV'⟩ := h x hx
exact ⟨V, hV, hV'⟩ | theorem | isOpen_iff_isOpen_ball_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"Set",
"interior_mem_uniformity",
"interior_subset",
"isOpen_iff_ball_subset",
"isOpen_interior",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
closure_ball_subset {x : α} {V : SetRel α α} : closure (ball x V) ⊆ ball x (closure V) | (Continuous.prodMk_right x).closure_preimage_subset V | theorem | closure_ball_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Continuous.prodMk_right",
"SetRel",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Dense.biUnion_uniformity_ball {s : Set α} {U : SetRel α α} (hs : Dense s) (hU : U ∈ 𝓤 α) :
⋃ x ∈ s, ball x U = univ | by
refine iUnion₂_eq_univ_iff.2 fun y => ?_
rcases hs.inter_nhds_nonempty (mem_nhds_right y hU) with ⟨x, hxs, hxy : (x, y) ∈ U⟩
exact ⟨x, hxs, hxy⟩ | theorem | Dense.biUnion_uniformity_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Dense",
"Set",
"SetRel",
"mem_nhds_right"
] | The uniform neighborhoods of all points of a dense set cover the whole space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
DenseRange.iUnion_uniformity_ball {ι : Type*} {xs : ι → α}
(xs_dense : DenseRange xs) {U : SetRel α α} (hU : U ∈ uniformity α) :
⋃ i, UniformSpace.ball (xs i) U = univ | by
rw [← biUnion_range (f := xs) (g := fun x ↦ UniformSpace.ball x U)]
exact Dense.biUnion_uniformity_ball xs_dense hU | lemma | DenseRange.iUnion_uniformity_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Dense.biUnion_uniformity_ball",
"DenseRange",
"SetRel",
"UniformSpace.ball",
"uniformity"
] | The uniform neighborhoods of all points of a dense indexed collection cover the whole space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_hasBasis_open : HasBasis (𝓤 α) (fun V : SetRel α α => V ∈ 𝓤 α ∧ IsOpen V) id | hasBasis_self.2 fun s hs =>
⟨interior s, interior_mem_uniformity hs, isOpen_interior, interior_subset⟩ | theorem | uniformity_hasBasis_open | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel",
"interior_mem_uniformity",
"isOpen_interior"
] | Open elements of `𝓤 α` form a basis of `𝓤 α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.mem_uniformity_iff {p : β → Prop} {s : β → SetRel α α}
(h : (𝓤 α).HasBasis p s) {t : SetRel α α} :
t ∈ 𝓤 α ↔ ∃ i, p i ∧ ∀ a b, (a, b) ∈ s i → (a, b) ∈ t | h.mem_iff.trans <| by simp only [Prod.forall, subset_def] | theorem | Filter.HasBasis.mem_uniformity_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_hasBasis_open_symmetric :
HasBasis (𝓤 α) (fun V : SetRel α α => V ∈ 𝓤 α ∧ IsOpen V ∧ SetRel.IsSymm V) id | by
simp only [← and_assoc]
refine uniformity_hasBasis_open.restrict fun s hs => ⟨SetRel.symmetrize s, ?_⟩
exact
⟨⟨symmetrize_mem_uniformity hs.1, IsOpen.inter hs.2 (hs.2.preimage continuous_swap)⟩,
inferInstance, SetRel.symmetrize_subset_self⟩ | theorem | uniformity_hasBasis_open_symmetric | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"IsOpen.inter",
"SetRel",
"SetRel.IsSymm",
"continuous_swap"
] | Open elements `s : SetRel α α` of `𝓤 α` such that `(x, y) ∈ s ↔ (y, x) ∈ s` form a basis
of `𝓤 α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comp_open_symm_mem_uniformity_sets {s : SetRel α α} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, IsOpen t ∧ SetRel.IsSymm t ∧ t ○ t ⊆ s | by
obtain ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs
obtain ⟨u, ⟨hu₁, hu₂, hu₃⟩, hu₄ : u ⊆ t⟩ := uniformity_hasBasis_open_symmetric.mem_iff.mp ht₁
exact ⟨u, hu₁, hu₂, hu₃, (SetRel.comp_subset_comp hu₄ hu₄).trans ht₂⟩ | theorem | comp_open_symm_mem_uniformity_sets | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"IsOpen",
"SetRel",
"SetRel.IsSymm",
"SetRel.comp_subset_comp",
"comp_mem_uniformity_sets",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.le_def {u₁ u₂ : UniformSpace α} : u₁ ≤ u₂ ↔ 𝓤[u₁] ≤ 𝓤[u₂] | Iff.rfl | theorem | UniformSpace.le_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.sInf_le {tt : Set (UniformSpace α)} {t : UniformSpace α}
(h : t ∈ tt) : sInf tt ≤ t | show ⨅ u ∈ tt, 𝓤[u] ≤ 𝓤[t] from iInf₂_le t h | theorem | UniformSpace.sInf_le | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.le_sInf {tt : Set (UniformSpace α)} {t : UniformSpace α}
(h : ∀ t' ∈ tt, t ≤ t') : t ≤ sInf tt | show 𝓤[t] ≤ ⨅ u ∈ tt, 𝓤[u] from le_iInf₂ h | theorem | UniformSpace.le_sInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.isGLB_sInf {tt : Set (UniformSpace α)} : IsGLB tt (sInf tt) | ⟨fun _ ↦ UniformSpace.sInf_le, fun _ ↦ UniformSpace.le_sInf⟩ | theorem | UniformSpace.isGLB_sInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformSpace",
"UniformSpace.sInf_le"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
iInf_uniformity {ι : Sort*} {u : ι → UniformSpace α} : 𝓤[iInf u] = ⨅ i, 𝓤[u i] | iInf_range | theorem | iInf_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inf_uniformity {u v : UniformSpace α} : 𝓤[u ⊓ v] = 𝓤[u] ⊓ 𝓤[v] | rfl | theorem | inf_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
bot_uniformity : 𝓤[(⊥ : UniformSpace α)] = 𝓟 SetRel.id | rfl | lemma | bot_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel.id",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
top_uniformity : 𝓤[(⊤ : UniformSpace α)] = ⊤ | rfl | lemma | top_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inhabitedUniformSpace : Inhabited (UniformSpace α) | ⟨⊥⟩ | instance | inhabitedUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
inhabitedUniformSpaceCore : Inhabited (UniformSpace.Core α) | ⟨@UniformSpace.toCore _ default⟩ | instance | inhabitedUniformSpaceCore | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace.Core",
"UniformSpace.toCore"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.comap (f : α → β) (u : UniformSpace β) : UniformSpace α | where
uniformity := 𝓤[u].comap fun p : α × α => (f p.1, f p.2)
symm := by
simp only [tendsto_comap_iff]
exact tendsto_swap_uniformity.comp tendsto_comap
comp := le_trans
(by
rw [comap_lift'_eq, comap_lift'_eq2]
· exact lift'_mono' fun s _ ⟨a₁, a₂⟩ ⟨x, h₁, h₂⟩ => ⟨f x, h₁, h₂⟩
· exac... | abbrev | UniformSpace.comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"comap",
"le_trans",
"monotone_id",
"nhds_eq_comap_uniformity",
"nhds_induced",
"symm",
"uniformity"
] | Given `f : α → β` and a uniformity `u` on `β`, the inverse image of `u` under `f`
is the inverse image in the filter sense of the induced function `α × α → β × β`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_comap {_ : UniformSpace β} (f : α → β) :
𝓤[UniformSpace.comap f ‹_›] = comap (Prod.map f f) (𝓤 β) | rfl | theorem | uniformity_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"UniformSpace.comap",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_preimage {f : α → β} {U : SetRel β β} {x : α} :
UniformSpace.ball x (Prod.map f f ⁻¹' U) = f ⁻¹' UniformSpace.ball (f x) U | by
ext : 1
simp only [UniformSpace.ball, mem_preimage, Prod.map_apply] | lemma | ball_preimage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel",
"UniformSpace.ball"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace_comap_id {α : Type*} : UniformSpace.comap (id : α → α) = id | by
ext : 2
rw [uniformity_comap, Prod.map_id, comap_id] | theorem | uniformSpace_comap_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace.comap",
"comap_id",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.comap_comap {α β γ} {uγ : UniformSpace γ} {f : α → β} {g : β → γ} :
UniformSpace.comap (g ∘ f) uγ = UniformSpace.comap f (UniformSpace.comap g uγ) | by
ext1
simp only [uniformity_comap, Filter.comap_comap, Prod.map_comp_map] | theorem | UniformSpace.comap_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Filter.comap_comap",
"UniformSpace",
"UniformSpace.comap",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.comap_inf {α γ} {u₁ u₂ : UniformSpace γ} {f : α → γ} :
(u₁ ⊓ u₂).comap f = u₁.comap f ⊓ u₂.comap f | UniformSpace.ext Filter.comap_inf | theorem | UniformSpace.comap_inf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Filter.comap_inf",
"UniformSpace",
"UniformSpace.ext",
"comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.comap_iInf {ι α γ} {u : ι → UniformSpace γ} {f : α → γ} :
(⨅ i, u i).comap f = ⨅ i, (u i).comap f | by
ext : 1
simp [uniformity_comap, iInf_uniformity] | theorem | UniformSpace.comap_iInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"comap",
"iInf_uniformity",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.comap_mono {α γ} {f : α → γ} :
Monotone fun u : UniformSpace γ => u.comap f | fun _ _ hu =>
Filter.comap_mono hu | theorem | UniformSpace.comap_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Filter.comap_mono",
"Monotone",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_iff_le_comap {α β} {uα : UniformSpace α} {uβ : UniformSpace β}
{f : α → β} : UniformContinuous f ↔ uα ≤ uβ.comap f | Filter.map_le_iff_le_comap
@[deprecated (since := "2026-05-23")]
alias uniformContinuous_iff := uniformContinuous_iff_le_comap | theorem | uniformContinuous_iff_le_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Filter.map_le_iff_le_comap",
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
le_iff_uniformContinuous_id {u v : UniformSpace α} :
u ≤ v ↔ @UniformContinuous _ _ u v id | by
rw [uniformContinuous_iff_le_comap, uniformSpace_comap_id, id] | theorem | le_iff_uniformContinuous_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace",
"uniformContinuous_iff_le_comap",
"uniformSpace_comap_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_comap {f : α → β} [u : UniformSpace β] :
@UniformContinuous α β (UniformSpace.comap f u) u f | tendsto_comap | theorem | uniformContinuous_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_comap' {f : γ → β} {g : α → γ} [v : UniformSpace β] [u : UniformSpace α]
(h : UniformContinuous (f ∘ g)) : @UniformContinuous α γ u (UniformSpace.comap f v) g | tendsto_comap_iff.2 h | theorem | uniformContinuous_comap' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
to_nhds_mono {u₁ u₂ : UniformSpace α} (h : u₁ ≤ u₂) (a : α) :
@nhds _ (@UniformSpace.toTopologicalSpace _ u₁) a ≤
@nhds _ (@UniformSpace.toTopologicalSpace _ u₂) a | by
rw [@nhds_eq_uniformity α u₁ a, @nhds_eq_uniformity α u₂ a]; exact lift'_mono h le_rfl | theorem | UniformSpace.to_nhds_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"le_rfl",
"nhds_eq_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_mono {u₁ u₂ : UniformSpace α} (h : u₁ ≤ u₂) :
@UniformSpace.toTopologicalSpace _ u₁ ≤ @UniformSpace.toTopologicalSpace _ u₂ | le_of_nhds_le_nhds <| to_nhds_mono h | theorem | UniformSpace.toTopologicalSpace_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"le_of_nhds_le_nhds"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_comap {f : α → β} {u : UniformSpace β} :
@UniformSpace.toTopologicalSpace _ (UniformSpace.comap f u) =
TopologicalSpace.induced f (@UniformSpace.toTopologicalSpace β u) | rfl | theorem | UniformSpace.toTopologicalSpace_comap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"TopologicalSpace.induced",
"UniformSpace",
"UniformSpace.comap"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformSpace_eq_bot {u : UniformSpace α} : u = ⊥ ↔ SetRel.id ∈ 𝓤[u] | le_bot_iff.symm.trans le_principal_iff | lemma | UniformSpace.uniformSpace_eq_bot | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"SetRel.id",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.Filter.HasBasis.uniformSpace_eq_bot {ι p} {s : ι → SetRel α α}
{u : UniformSpace α} (h : 𝓤[u].HasBasis p s) :
u = ⊥ ↔ ∃ i, p i ∧ Pairwise fun x y : α ↦ (x, y) ∉ s i | by
simp [uniformSpace_eq_bot, h.mem_iff, subset_def, Pairwise, not_imp_not] | lemma | Filter.HasBasis.uniformSpace_eq_bot | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Pairwise",
"SetRel",
"UniformSpace",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_bot : @UniformSpace.toTopologicalSpace α ⊥ = ⊥ | rfl | theorem | UniformSpace.toTopologicalSpace_bot | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_top : @UniformSpace.toTopologicalSpace α ⊤ = ⊤ | rfl | theorem | UniformSpace.toTopologicalSpace_top | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_iInf {ι : Sort*} {u : ι → UniformSpace α} :
(iInf u).toTopologicalSpace = ⨅ i, (u i).toTopologicalSpace | TopologicalSpace.ext_nhds fun a ↦ by simp only [@nhds_eq_comap_uniformity _ (iInf u), nhds_iInf,
iInf_uniformity, @nhds_eq_comap_uniformity _ (u _), Filter.comap_iInf] | theorem | UniformSpace.toTopologicalSpace_iInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Filter.comap_iInf",
"UniformSpace",
"iInf_uniformity",
"nhds_eq_comap_uniformity",
"nhds_iInf"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_sInf {s : Set (UniformSpace α)} :
(sInf s).toTopologicalSpace = ⨅ i ∈ s, @UniformSpace.toTopologicalSpace α i | by
rw [sInf_eq_iInf]
simp only [← toTopologicalSpace_iInf] | theorem | UniformSpace.toTopologicalSpace_sInf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toTopologicalSpace_inf {u v : UniformSpace α} :
(u ⊓ v).toTopologicalSpace = u.toTopologicalSpace ⊓ v.toTopologicalSpace | rfl | theorem | UniformSpace.toTopologicalSpace_inf | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.continuous (hf : UniformContinuous f) : Continuous f | continuous_iff_le_induced.mpr <| UniformSpace.toTopologicalSpace_mono <|
uniformContinuous_iff_le_comap.1 hf | theorem | UniformContinuous.continuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Continuous",
"UniformContinuous",
"UniformSpace.toTopologicalSpace_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.uniformContinuousOn (hf : UniformContinuous f) :
UniformContinuousOn f s | tendsto_inf_left hf | lemma | UniformContinuous.uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.mono (hf : UniformContinuousOn f s) (ht : t ⊆ s) :
UniformContinuousOn f t | Tendsto.mono_left hf (inf_le_inf le_rfl (by simp [ht])) | lemma | UniformContinuousOn.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuousOn",
"le_rfl"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.congr {f g : α → β} {s : Set α}
(hf : UniformContinuousOn f s) (h : EqOn f g s) :
UniformContinuousOn g s | by
apply hf.congr'
apply EventuallyEq.filter_mono _ inf_le_right
filter_upwards [mem_principal_self _] with ⟨a, b⟩ ⟨ha, hb⟩ using by simp [h ha, h hb] | lemma | UniformContinuousOn.congr | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuousOn.comp {g : β → γ} {t : Set β} (hg : UniformContinuousOn g t)
(hf : UniformContinuousOn f s) (hst : MapsTo f s t) : UniformContinuousOn (g ∘ f) s | by
change Tendsto ((fun x ↦ (g x.1, g x.2)) ∘ (fun x ↦ (f x.1, f x.2))) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 γ)
apply Tendsto.comp hg
refine tendsto_inf.2 ⟨hf, tendsto_inf_right ?_⟩
simp only [tendsto_principal, mem_prod, eventually_principal, and_imp, Prod.forall]
exact fun a b ha hb ↦ ⟨hst ha, hst hb⟩ | lemma | UniformContinuousOn.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.comp_uniformContinuousOn {g : β → γ}
(hg : UniformContinuous g) (hf : UniformContinuousOn f s) : UniformContinuousOn (g ∘ f) s | (hg.uniformContinuousOn (s := univ)).comp hf (mapsTo_univ _ _) | lemma | UniformContinuous.comp_uniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ULift.uniformSpace [UniformSpace α] : UniformSpace (ULift α) | UniformSpace.comap ULift.down ‹_› | instance | ULift.uniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace",
"UniformSpace.comap"
] | Uniform space structure on `ULift α`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
OrderDual.instUniformSpace [UniformSpace α] : UniformSpace (αᵒᵈ) | ‹UniformSpace α› | instance | OrderDual.instUniformSpace | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformSpace"
] | Uniform space structure on `αᵒᵈ`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuous.inf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ u₃ : UniformSpace β}
(h₁ : UniformContinuous[u₁, u₂] f) (h₂ : UniformContinuous[u₁, u₃] f) :
UniformContinuous[u₁, u₂ ⊓ u₃] f | tendsto_inf.mpr ⟨h₁, h₂⟩ | theorem | UniformContinuous.inf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.inf_dom_left {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β}
(hf : UniformContinuous[u₁, u₃] f) : UniformContinuous[u₁ ⊓ u₂, u₃] f | tendsto_inf_left hf | theorem | UniformContinuous.inf_dom_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.inf_dom_right {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β}
(hf : UniformContinuous[u₂, u₃] f) : UniformContinuous[u₁ ⊓ u₂, u₃] f | tendsto_inf_right hf | theorem | UniformContinuous.inf_dom_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_sInf_dom {f : α → β} {u₁ : Set (UniformSpace α)} {u₂ : UniformSpace β}
{u : UniformSpace α} (h₁ : u ∈ u₁) (hf : UniformContinuous[u, u₂] f) :
UniformContinuous[sInf u₁, u₂] f | by
delta UniformContinuous
rw [sInf_eq_iInf', iInf_uniformity]
exact tendsto_iInf' ⟨u, h₁⟩ hf | theorem | uniformContinuous_sInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"UniformContinuous",
"UniformSpace",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_sInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : Set (UniformSpace β)} :
UniformContinuous[u₁, sInf u₂] f ↔ ∀ u ∈ u₂, UniformContinuous[u₁, u] f | by
delta UniformContinuous
rw [sInf_eq_iInf', iInf_uniformity, tendsto_iInf, SetCoe.forall] | theorem | uniformContinuous_sInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Set",
"SetCoe.forall",
"UniformContinuous",
"UniformSpace",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_iInf_dom {f : α → β} {u₁ : ι → UniformSpace α} {u₂ : UniformSpace β}
{i : ι} (hf : UniformContinuous[u₁ i, u₂] f) : UniformContinuous[iInf u₁, u₂] f | by
delta UniformContinuous
rw [iInf_uniformity]
exact tendsto_iInf' i hf | theorem | uniformContinuous_iInf_dom | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_iInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : ι → UniformSpace β} :
UniformContinuous[u₁, iInf u₂] f ↔ ∀ i, UniformContinuous[u₁, u₂ i] f | by
delta UniformContinuous
rw [iInf_uniformity, tendsto_iInf] | theorem | uniformContinuous_iInf_rng | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"UniformContinuous",
"UniformSpace",
"iInf_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
discreteTopology_of_discrete_uniformity [hα : UniformSpace α]
(h : uniformity α = 𝓟 SetRel.id) : DiscreteTopology α | ⟨(UniformSpace.ext h.symm : ⊥ = hα) ▸ rfl⟩ | theorem | discreteTopology_of_discrete_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"DiscreteTopology",
"SetRel.id",
"UniformSpace",
"UniformSpace.ext",
"uniformity"
] | A uniform space with the discrete uniformity has the discrete topology. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_ofMul : UniformContinuous (ofMul : α → Additive α) | uniformContinuous_id | theorem | uniformContinuous_ofMul | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Additive",
"UniformContinuous",
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_toMul : UniformContinuous (toMul : Additive α → α) | uniformContinuous_id | theorem | uniformContinuous_toMul | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Additive",
"UniformContinuous",
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_ofAdd : UniformContinuous (ofAdd : α → Multiplicative α) | uniformContinuous_id | theorem | uniformContinuous_ofAdd | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Basic.lean | [] | [
"Multiplicative",
"UniformContinuous",
"uniformContinuous_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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