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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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filter (𝓕 : filter $ β × β) : filter ((α →ᵤ β) × (α →ᵤ β)) | (uniform_fun.basis α β 𝓕).filter | def | uniform_fun.filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter",
"uniform_fun.basis"
] | For `𝓕 : filter (β × β)`, this is the filter generated by the filter basis
`uniform_convergence.basis α β 𝓕`. For `𝓕 = 𝓤 β`, this will be the uniformity of uniform
convergence on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc : galois_connection lower_adjoint
(λ 𝓕, uniform_fun.filter α β 𝓕) | begin
intros 𝓐 𝓕,
symmetry,
calc 𝓐 ≤ uniform_fun.filter α β 𝓕
↔ (uniform_fun.basis α β 𝓕).sets ⊆ 𝓐.sets :
by rw [uniform_fun.filter, ← filter_basis.generate, sets_iff_generate]
... ↔ ∀ U ∈ 𝓕, uniform_fun.gen α β U ∈ 𝓐 : image_subset_iff
... ↔ ∀ U ∈ 𝓕, {uv | ∀ x, (uv, x) ∈
{t :... | lemma | uniform_fun.gc | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter",
"filter_basis.generate",
"forall₂_congr",
"galois_connection",
"lower_adjoint",
"uniform_fun.basis",
"uniform_fun.filter",
"uniform_fun.gen"
] | The function `uniform_convergence.filter α β : filter (β × β) → filter ((α →ᵤ β) × (α →ᵤ β))`
has a lower adjoint `l` (in the sense of `galois_connection`). The exact definition of `l` is not
interesting; we will only use that it exists (in `uniform_convergence.mono` and
`uniform_convergence.infi_eq`) and that
`l (filt... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_core : uniform_space.core (α →ᵤ β) | uniform_space.core.mk_of_basis (uniform_fun.basis α β (𝓤 β))
(λ U ⟨V, hV, hVU⟩ f, hVU ▸ λ x, refl_mem_uniformity hV)
(λ U ⟨V, hV, hVU⟩, hVU ▸ ⟨uniform_fun.gen α β (prod.swap ⁻¹' V),
⟨prod.swap ⁻¹' V, tendsto_swap_uniformity hV, rfl⟩, λ uv huv x, huv x⟩)
(λ U ⟨V, hV, hVU⟩, hVU ▸ let ⟨W, hW, hWV⟩ := comp_mem_u... | def | uniform_fun.uniform_core | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"comp_mem_uniformity_sets",
"prod.swap",
"refl_mem_uniformity",
"tendsto_swap_uniformity",
"uniform_fun.basis",
"uniform_space.core",
"uniform_space.core.mk_of_basis"
] | Core of the uniform structure of uniform convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity :
(𝓤 (α →ᵤ β)).has_basis (λ V, V ∈ 𝓤 β)
(uniform_fun.gen α β) | (uniform_fun.is_basis_gen α β (𝓤 β)).has_basis | lemma | uniform_fun.has_basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.gen",
"uniform_fun.is_basis_gen"
] | By definition, the uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}`
for `V ∈ 𝓤 β` as a filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis {ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)}
(h : (𝓤 β).has_basis p s) :
(𝓤 (α →ᵤ β)).has_basis p (uniform_fun.gen α β ∘ s) | (uniform_fun.has_basis_uniformity α β).to_has_basis
(λ U hU, let ⟨i, hi, hiU⟩ := h.mem_iff.mp hU in ⟨i, hi, λ uv huv x, hiU (huv x)⟩)
(λ i hi, ⟨s i, h.mem_of_mem hi, subset_refl _⟩) | lemma | uniform_fun.has_basis_uniformity_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"subset_refl",
"uniform_fun.gen",
"uniform_fun.has_basis_uniformity"
] | The uniformity of `α →ᵤ β` admits the family `{(f, g) | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as
a filter basis, for any basis `𝓑` of `𝓤 β` (in the case `𝓑 = (𝓤 β).as_basis` this is true by
definition). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_of_basis (f) {p : ι → Prop} {s : ι → set (β × β)}
(h : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis p (λ i, {g | (f, g) ∈ uniform_fun.gen α β (s i)}) | nhds_basis_uniformity' (uniform_fun.has_basis_uniformity_of_basis α β h) | lemma | uniform_fun.has_basis_nhds_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"nhds_basis_uniformity'",
"uniform_fun.gen",
"uniform_fun.has_basis_uniformity_of_basis"
] | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓑` as a filter
basis, for any basis `𝓑` of `𝓤 β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds (f) :
(𝓝 f).has_basis (λ V, V ∈ 𝓤 β) (λ V, {g | (f, g) ∈ uniform_fun.gen α β V}) | uniform_fun.has_basis_nhds_of_basis α β f (filter.basis_sets _) | lemma | uniform_fun.has_basis_nhds | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter.basis_sets",
"uniform_fun.gen",
"uniform_fun.has_basis_nhds_of_basis"
] | For `f : α →ᵤ β`, `𝓝 f` admits the family `{g | ∀ x, (f x, g x) ∈ V}` for `V ∈ 𝓤 β` as a
filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_eval (x : α) :
uniform_continuous (function.eval x ∘ to_fun : (α →ᵤ β) → β) | begin
change _ ≤ _,
rw [map_le_iff_le_comap,
(uniform_fun.has_basis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)],
exact λ U hU, ⟨U, hU, λ uv huv, huv x⟩
end | lemma | uniform_fun.uniform_continuous_eval | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"function.eval",
"uniform_continuous",
"uniform_fun.has_basis_uniformity"
] | Evaluation at a fixed point is uniformly continuous on `α →ᵤ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono : monotone (@uniform_fun.uniform_space α γ) | λ u₁ u₂ hu, (uniform_fun.gc α γ).monotone_u hu | lemma | uniform_fun.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"monotone",
"uniform_fun.gc"
] | If `u₁` and `u₂` are two uniform structures on `γ` and `u₁ ≤ u₂`, then
`𝒰(α, γ, u₁) ≤ 𝒰(α, γ, u₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_eq {u : ι → uniform_space γ} :
(𝒰(α, γ, ⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i) | begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
ext : 1,
change uniform_fun.filter α γ (𝓤[⨅ i, u i]) = 𝓤[⨅ i, 𝒰(α, γ, u i)],
rw [infi_uniformity, infi_uniformity],
exact (uniform_fun.gc α γ).u_infi
end | lemma | uniform_fun.infi_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_uniformity",
"uniform_fun.filter",
"uniform_fun.gc",
"uniform_space"
] | If `u` is a family of uniform structures on `γ`, then
`𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_eq {u₁ u₂ : uniform_space γ} :
(𝒰(α, γ, u₁ ⊓ u₂)) = (𝒰(α, γ, u₁)) ⊓ (𝒰(α, γ, u₂)) | begin
-- This follows directly from the fact that the upper adjoint in a Galois connection maps
-- infimas to infimas.
rw [inf_eq_infi, inf_eq_infi, uniform_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | lemma | uniform_fun.inf_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"inf_eq_infi",
"infi_congr",
"uniform_fun.infi_eq",
"uniform_space"
] | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq {f : γ → β} :
(𝒰(α, γ, ‹uniform_space β›.comap f)) = (𝒰(α, β, _)).comap ((∘) f) | begin
letI : uniform_space γ := ‹uniform_space β›.comap f,
ext : 1,
change (uniform_fun.filter α γ ((𝓤 β).comap _)) =
(uniform_fun.filter α β ((𝓤 β))).comap _,
-- We have the following four Galois connection which form a square diagram, and we want
-- to show that the square of upper adjoints is commuta... | lemma | uniform_fun.comap_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"filter.gc_map_comap",
"galois_connection.u_comm_of_l_comm",
"uniform_fun.filter",
"uniform_fun.gc",
"uniform_space"
] | If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒰(α, γ, comap f u) = comap (λ g, f ∘ g) 𝒰(α, γ, u₁)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) | -- This is a direct consequence of `uniform_convergence.comap_eq`
uniform_continuous_iff.mpr $
calc 𝒰(α, γ, _)
≤ 𝒰(α, γ, ‹uniform_space β›.comap f) :
uniform_fun.mono (uniform_continuous_iff.mp hf)
... = (𝒰(α, β, _)).comap ((∘) f) :
uniform_fun.comap_eq | lemma | uniform_fun.postcomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_fun.comap_eq",
"uniform_fun.mono",
"uniform_space"
] | Post-composition by a uniformly continuous function is uniformly continuous on `α →ᵤ β`.
More precisely, if `f : γ → β` is uniformly continuous, then `(λ g, f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)`
is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun ∘ ((∘) f) ∘ to_fun : (α →ᵤ γ) → (α →ᵤ β)) | -- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change comap (prod.map (of_fun ∘ (∘) f ∘ to_fun) (of_fun ∘ (∘) f ∘ to_fun)) _ = _,
rw [← uniformity_comap] at ⊢ hf,
congr,
rw [← uniform_space_eq hf, uniform_fu... | lemma | uniform_fun.postcomp_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.comap_eq",
"uniform_inducing",
"uniform_space",
"uniform_space_eq",
"uniformity_comap"
] | Post-composition by a uniform inducing is a uniform inducing for the
uniform structures of uniform convergence.
More precisely, if `f : γ → β` is a uniform inducing, then `(λ g, f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)` is
a uniform inducing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ γ) ≃ᵤ (α →ᵤ β) | { uniform_continuous_to_fun :=
uniform_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.to_equiv) } | def | uniform_fun.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.Pi_congr_right",
"uniform_fun.postcomp_uniform_continuous",
"uniform_space"
] | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ γ) ≃ᵤ (α →ᵤ β)` by
post-composing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precomp_uniform_continuous {f : γ → α} :
uniform_continuous (λ g : α →ᵤ β, of_fun (g ∘ f)) | begin
-- Here we simply go back to filter bases.
rw uniform_continuous_iff,
change 𝓤 (α →ᵤ β) ≤ (𝓤 (γ →ᵤ β)).comap (prod.map (λ g : α →ᵤ β, g ∘ f) (λ g : α →ᵤ β, g ∘ f)),
rw (uniform_fun.has_basis_uniformity α β).le_basis_iff
((uniform_fun.has_basis_uniformity γ β).comap _),
exact λ U hU, ⟨U, hU, λ uv h... | lemma | uniform_fun.precomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_iff",
"uniform_fun.has_basis_uniformity"
] | Pre-composition by a any function is uniformly continuous for the uniform structures of
uniform convergence.
More precisely, for any `f : γ → α`, the function `(λ g, g ∘ f) : (α →ᵤ β) → (γ →ᵤ β)` is uniformly
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_left (e : γ ≃ α) :
(γ →ᵤ β) ≃ᵤ (α →ᵤ β) | { uniform_continuous_to_fun :=
uniform_fun.precomp_uniform_continuous,
uniform_continuous_inv_fun :=
uniform_fun.precomp_uniform_continuous,
.. equiv.arrow_congr e (equiv.refl _) } | def | uniform_fun.congr_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_congr",
"equiv.refl",
"uniform_fun.precomp_uniform_continuous"
] | Turn a bijection `γ ≃ α` into a uniform isomorphism
`(γ →ᵤ β) ≃ᵤ (α →ᵤ β)` by pre-composing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_to_fun : uniform_continuous (to_fun : (α →ᵤ β) → α → β) | begin
-- By definition of the product uniform structure, this is just `uniform_continuous_eval`.
rw uniform_continuous_pi,
intros x,
exact uniform_continuous_eval β x
end | lemma | uniform_fun.uniform_continuous_to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_pi"
] | The natural map `uniform_fun.to_fun` from `α →ᵤ β` to `α → β` is uniformly continuous.
In other words, the uniform structure of uniform convergence is finer than that of pointwise
convergence, aka the product uniform structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_uniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
tendsto F p (𝓝 f) ↔ tendsto_uniformly F f p | begin
rw [(uniform_fun.has_basis_nhds α β f).tendsto_right_iff, tendsto_uniformly],
exact iff.rfl,
end | lemma | uniform_fun.tendsto_iff_tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"tendsto_uniformly",
"uniform_fun.has_basis_nhds"
] | The topology of uniform convergence indeed gives the same notion of convergence as
`tendsto_uniformly`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ β × γ) ≃ᵤ ((α →ᵤ β) × (α →ᵤ γ)) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒰(α, β, uβ) × 𝒰(α, γ, uγ)) = 𝒰(α, β × γ, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `uniform_convergence.inf_eq` and `uniform_convergence.comap_eq`, which leaves us to check
-- that some square c... | def | uniform_fun.uniform_equiv_prod_arrow | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_prod_equiv_prod_arrow",
"uniform_fun.comap_eq",
"uniform_fun.inf_eq",
"uniform_space",
"uniform_space.comap_comap",
"uniform_space.comap_inf",
"uniformity_comap"
] | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ β × γ` and `(α →ᵤ β) × (α →ᵤ γ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_Pi_comm : uniform_equiv (α →ᵤ Π i, δ i) (Π i, α →ᵤ δ i) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒰(α, δ i, uδ i)) = 𝒰(α, (Π i, δ i), (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence.infi_eq` and `uniform_convergence.comap_eq`, which leaves us to check
-- that some squ... | def | uniform_fun.uniform_equiv_Pi_comm | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"Pi.uniform_space",
"equiv.Pi_comm",
"equiv.to_uniform_equiv_of_uniform_inducing",
"infi_congr",
"uniform_equiv",
"uniform_fun.comap_eq",
"uniform_fun.infi_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.of_core_eq_to_core",
"uniformity_comap"
] | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ (Π i, δ i)` and `Π i, α →ᵤ δ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen (𝔖) (S : set α) (V : set (β × β)) : set ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)) | {uv : (α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β) | ∀ x ∈ S, (uv.1 x, uv.2 x) ∈ V} | def | uniform_on_fun.gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [] | Basis sets for the uniformity of `𝔖`-convergence: for `S : set α` and `V : set (β × β)`,
`gen 𝔖 S V` is the set of pairs `(f, g)` of functions `α →ᵤ[𝔖] β` such that
`∀ x ∈ S, (f x, g x) ∈ V`. Note that the family `𝔖 : set (set α)` is only used to specify which
type alias of `α → β` to use here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_eq_preimage_restrict {𝔖} (S : set α) (V : set (β × β)) :
uniform_on_fun.gen 𝔖 S V =
(prod.map S.restrict S.restrict) ⁻¹' (uniform_fun.gen S β V) | begin
ext uv,
exact ⟨λ h ⟨x, hx⟩, h x hx, λ h x hx, h ⟨x, hx⟩⟩
end | lemma | uniform_on_fun.gen_eq_preimage_restrict | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.gen",
"uniform_on_fun.gen"
] | For `S : set α` and `V : set (β × β)`, we have
`uniform_on_fun.gen 𝔖 S V = (S.restrict × S.restrict) ⁻¹' (uniform_fun.gen S β V)`.
This is the crucial fact for proving that the family `uniform_on_fun.gen S V` for `S ∈ 𝔖` and
`V ∈ 𝓤 β` is indeed a basis for the uniformity `α →ᵤ[𝔖] β` endowed with `𝒱(α, β, 𝔖, uβ)`
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_mono {𝔖} {S S' : set α} {V V' : set (β × β)} (hS : S' ⊆ S) (hV : V ⊆ V') :
uniform_on_fun.gen 𝔖 S V ⊆ uniform_on_fun.gen 𝔖 S' V' | λ uv h x hx, hV (h x $ hS hx) | lemma | uniform_on_fun.gen_mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_on_fun.gen"
] | `uniform_on_fun.gen` is antitone in the first argument and monotone in the second. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_basis_gen (𝔖 : set (set α)) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
(𝓑 : filter_basis $ β × β) :
is_basis (λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓑)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) | ⟨h.prod 𝓑.nonempty, λ U₁V₁ U₂V₂ h₁ h₂,
let ⟨U₃, hU₃, hU₁₃, hU₂₃⟩ := h' U₁V₁.1 h₁.1 U₂V₂.1 h₂.1 in
let ⟨V₃, hV₃, hV₁₂₃⟩ := 𝓑.inter_sets h₁.2 h₂.2 in ⟨⟨U₃, V₃⟩, ⟨⟨hU₃, hV₃⟩, λ uv huv,
⟨(λ x hx, (hV₁₂₃ $ huv x $ hU₁₃ hx).1), (λ x hx, (hV₁₂₃ $ huv x $ hU₂₃ hx).2)⟩⟩⟩⟩ | lemma | uniform_on_fun.is_basis_gen | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"filter_basis",
"uniform_on_fun.gen"
] | If `𝔖 : set (set α)` is nonempty and directed and `𝓑` is a filter basis on `β × β`, then the
family `uniform_on_fun.gen 𝔖 S V` for `S ∈ 𝔖` and `V ∈ 𝓑` is a filter basis on
`(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] β)`.
We will show in `has_basis_uniformity_of_basis` that, if `𝓑` is a basis for `𝓤 β`, then the
corresponding filt... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space_eq :
uniform_on_fun.topological_space α β 𝔖 = ⨅ (s : set α) (hs : s ∈ 𝔖),
topological_space.induced s.restrict (uniform_fun.topological_space s β) | begin
simp only [uniform_on_fun.topological_space, to_topological_space_infi,
to_topological_space_infi, to_topological_space_comap],
refl
end | lemma | uniform_on_fun.topological_space_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"to_topological_space_comap",
"to_topological_space_infi",
"topological_space.induced",
"topological_space_eq"
] | The topology of `𝔖`-convergence is the infimum, for `S ∈ 𝔖`, of topology induced by the map
of `S.restrict : (α →ᵤ[𝔖] β) → (↥S →ᵤ β)` of restriction to `S`, where `↥S →ᵤ β` is endowed with
the topology of uniform convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity_of_basis_aux₁ {p : ι → Prop} {s : ι → set (β × β)}
(hb : has_basis (𝓤 β) p s) (S : set α) :
(@uniformity (α →ᵤ[𝔖] β) ((uniform_fun.uniform_space S β).comap S.restrict)).has_basis
p (λ i, uniform_on_fun.gen 𝔖 S (s i)) | begin
simp_rw [uniform_on_fun.gen_eq_preimage_restrict, uniformity_comap],
exact (uniform_fun.has_basis_uniformity_of_basis S β hb).comap _
end | lemma | uniform_on_fun.has_basis_uniformity_of_basis_aux₁ | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.has_basis_uniformity_of_basis",
"uniform_on_fun.gen",
"uniform_on_fun.gen_eq_preimage_restrict",
"uniformity",
"uniformity_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_uniformity_of_basis_aux₂ (h : directed_on (⊆) 𝔖) {p : ι → Prop}
{s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
directed_on ((λ s : set α, (uniform_fun.uniform_space s β).comap
(s.restrict : (α →ᵤ β) → s →ᵤ β)) ⁻¹'o ge) 𝔖 | h.mono $ λ s t hst,
((uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb _).le_basis_iff
(uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb _)).mpr
(λ V hV, ⟨V, hV, uniform_on_fun.gen_mono hst subset_rfl⟩) | lemma | uniform_on_fun.has_basis_uniformity_of_basis_aux₂ | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"uniform_on_fun.gen_mono",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₁"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_uniformity_of_basis (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, uniform_on_fun.gen 𝔖 Si.1 (s Si.2)) | begin
simp only [infi_uniformity],
exact has_basis_binfi_of_directed h (λ S, (uniform_on_fun.gen 𝔖 S) ∘ s) _
(λ S hS, uniform_on_fun.has_basis_uniformity_of_basis_aux₁ α β 𝔖 hb S)
(uniform_on_fun.has_basis_uniformity_of_basis_aux₂ α β 𝔖 h' hb)
end | lemma | uniform_on_fun.has_basis_uniformity_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"infi_uniformity",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₁",
"uniform_on_fun.has_basis_uniformity_of_basis_aux₂"
] | If `𝔖 : set (set α)` is nonempty and directed and `𝓑` is a filter basis of `𝓤 β`, then the
uniformity of `α →ᵤ[𝔖] β` admits the family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and
`V ∈ 𝓑` as a filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_uniformity (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓤 (α →ᵤ[𝔖] β)).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, uniform_on_fun.gen 𝔖 SV.1 SV.2) | uniform_on_fun.has_basis_uniformity_of_basis α β 𝔖 h h' (𝓤 β).basis_sets | lemma | uniform_on_fun.has_basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis"
] | If `𝔖 : set (set α)` is nonempty and directed, then the uniformity of `α →ᵤ[𝔖] β` admits the
family `{(f, g) | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds_of_basis (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖)
{p : ι → Prop} {s : ι → set (β × β)} (hb : has_basis (𝓤 β) p s) :
(𝓝 f).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 Si.1 (s Si.2)}) | begin
letI : uniform_space (α → β) := uniform_on_fun.uniform_space α β 𝔖,
exact nhds_basis_uniformity (uniform_on_fun.has_basis_uniformity_of_basis α β 𝔖 h h' hb)
end | lemma | uniform_on_fun.has_basis_nhds_of_basis | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"nhds_basis_uniformity",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_uniformity_of_basis",
"uniform_space"
] | For `f : α →ᵤ[𝔖] β`, where `𝔖 : set (set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓑` as a filter basis, for any basis
`𝓑` of `𝓤 β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_nhds (f : α →ᵤ[𝔖] β) (h : 𝔖.nonempty) (h' : directed_on (⊆) 𝔖) :
(𝓝 f).has_basis
(λ SV : set α × set (β × β), SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β)
(λ SV, {g | (g, f) ∈ uniform_on_fun.gen 𝔖 SV.1 SV.2}) | uniform_on_fun.has_basis_nhds_of_basis α β 𝔖 f h h' (filter.basis_sets _) | lemma | uniform_on_fun.has_basis_nhds | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"directed_on",
"filter.basis_sets",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_nhds_of_basis"
] | For `f : α →ᵤ[𝔖] β`, where `𝔖 : set (set α)` is nonempty and directed, `𝓝 f` admits the
family `{g | ∀ x ∈ S, (f x, g x) ∈ V}` for `S ∈ 𝔖` and `V ∈ 𝓤 β` as a filter basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_restrict (h : s ∈ 𝔖) :
uniform_continuous (uniform_fun.of_fun ∘ (s.restrict : (α → β) → (s → β)) ∘ (to_fun 𝔖)) | begin
change _ ≤ _,
simp only [uniform_on_fun.uniform_space, map_le_iff_le_comap, infi_uniformity],
exact infi₂_le s h
end | lemma | uniform_on_fun.uniform_continuous_restrict | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_uniformity",
"infi₂_le",
"uniform_continuous",
"uniform_fun.of_fun"
] | If `S ∈ 𝔖`, then the restriction to `S` is a uniformly continuous map from `α →ᵤ[𝔖] β` to
`↥S →ᵤ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mono ⦃u₁ u₂ : uniform_space γ⦄ (hu : u₁ ≤ u₂) ⦃𝔖₁ 𝔖₂ : set (set α)⦄
(h𝔖 : 𝔖₂ ⊆ 𝔖₁) :
𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂) | calc 𝒱(α, γ, 𝔖₁, u₁)
≤ 𝒱(α, γ, 𝔖₂, u₁) : infi_le_infi_of_subset h𝔖
... ≤ 𝒱(α, γ, 𝔖₂, u₂) : infi₂_mono
(λ i hi, uniform_space.comap_mono $ uniform_fun.mono hu) | lemma | uniform_on_fun.mono | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_le_infi_of_subset",
"infi₂_mono",
"uniform_fun.mono",
"uniform_space",
"uniform_space.comap_mono"
] | Let `u₁`, `u₂` be two uniform structures on `γ` and `𝔖₁ 𝔖₂ : set (set α)`. If `u₁ ≤ u₂` and
`𝔖₂ ⊆ 𝔖₁` then `𝒱(α, γ, 𝔖₁, u₁) ≤ 𝒱(α, γ, 𝔖₂, u₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
uniform_continuous ((function.eval x : (α → β) → β) ∘ to_fun 𝔖) | (uniform_fun.uniform_continuous_eval β (⟨x, hxs⟩ : s)).comp
(uniform_on_fun.uniform_continuous_restrict α β 𝔖 hs) | lemma | uniform_on_fun.uniform_continuous_eval_of_mem | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"function.eval",
"uniform_continuous",
"uniform_fun.uniform_continuous_eval",
"uniform_on_fun.uniform_continuous_restrict"
] | If `x : α` is in some `S ∈ 𝔖`, then evaluation at `x` is uniformly continuous on
`α →ᵤ[𝔖] β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infi_eq {u : ι → uniform_space γ} :
𝒱(α, γ, 𝔖, ⨅ i, u i) =
⨅ i, 𝒱(α, γ, 𝔖, u i) | begin
simp_rw [uniform_on_fun.uniform_space, uniform_fun.infi_eq, uniform_space.comap_infi],
rw infi_comm,
exact infi_congr (λ s, infi_comm)
end | lemma | uniform_on_fun.infi_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_comm",
"infi_congr",
"uniform_fun.infi_eq",
"uniform_space",
"uniform_space.comap_infi"
] | If `u` is a family of uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, (⨅ i, u i)) = ⨅ i, 𝒱(α, γ, 𝔖, u i)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_eq {u₁ u₂ : uniform_space γ} :
𝒱(α, γ, 𝔖, u₁ ⊓ u₂) =
𝒱(α, γ, 𝔖, u₁) ⊓
𝒱(α, γ, 𝔖, u₂) | begin
rw [inf_eq_infi, inf_eq_infi, uniform_on_fun.infi_eq],
refine infi_congr (λ i, _),
cases i; refl
end | lemma | uniform_on_fun.inf_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"inf_eq_infi",
"infi_congr",
"uniform_on_fun.infi_eq",
"uniform_space"
] | If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒱(α, γ, 𝔖, u₁ ⊓ u₂) = 𝒱(α, γ, 𝔖, u₁) ⊓ 𝒱(α, γ, 𝔖, u₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq {f : γ → β} :
𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) =
𝒱(α, β, 𝔖, _).comap ((∘) f) | begin
-- We reduce this to `uniform_convergence.comap_eq` using the fact that `comap` distributes
-- on `infi`.
simp_rw [uniform_on_fun.uniform_space, uniform_space.comap_infi,
uniform_fun.comap_eq, ← uniform_space.comap_comap],
refl -- by definition, `∀ S ∈ 𝔖, (f ∘ —) ∘ S.restrict = S.restrict ∘ (... | lemma | uniform_on_fun.comap_eq | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_fun.comap_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi"
] | If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒱(α, γ, 𝔖, comap f u) = comap (λ g, f ∘ g) 𝒱(α, γ, 𝔖, u₁)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_continuous [uniform_space γ] {f : γ → β}
(hf : uniform_continuous f):
uniform_continuous (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) | begin
-- This is a direct consequence of `uniform_convergence.comap_eq`
rw uniform_continuous_iff,
calc 𝒱(α, γ, 𝔖, _)
≤ 𝒱(α, γ, 𝔖, ‹uniform_space β›.comap f) :
uniform_on_fun.mono (uniform_continuous_iff.mp hf) (subset_rfl)
... = 𝒱(α, β, 𝔖, _).comap ((∘) f) :
uniform_on_fun.comap_eq
... | lemma | uniform_on_fun.postcomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"subset_rfl",
"uniform_continuous",
"uniform_continuous_iff",
"uniform_on_fun.comap_eq",
"uniform_on_fun.mono",
"uniform_space"
] | Post-composition by a uniformly continuous function is uniformly continuous for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is uniformly continuous, then
`(λ g, f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
postcomp_uniform_inducing [uniform_space γ] {f : γ → β}
(hf : uniform_inducing f):
uniform_inducing (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) | -- This is a direct consequence of `uniform_convergence.comap_eq`
begin
split,
replace hf : (𝓤 β).comap (prod.map f f) = _ := hf.comap_uniformity,
change comap (prod.map (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖) (of_fun 𝔖 ∘ (∘) f ∘ to_fun 𝔖)) _ = _,
rw [← uniformity_comap] at ⊢ hf,
congr,
rw [← uniform_space_eq hf... | lemma | uniform_on_fun.postcomp_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_inducing",
"uniform_on_fun.comap_eq",
"uniform_space",
"uniform_space_eq",
"uniformity_comap"
] | Post-composition by a uniform inducing is a uniform inducing for the
uniform structures of `𝔖`-convergence.
More precisely, if `f : γ → β` is a uniform inducing, then
`(λ g, f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_right [uniform_space γ] (e : γ ≃ᵤ β) :
(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) | { uniform_continuous_to_fun :=
uniform_on_fun.postcomp_uniform_continuous e.uniform_continuous,
uniform_continuous_inv_fun :=
uniform_on_fun.postcomp_uniform_continuous e.symm.uniform_continuous,
.. equiv.Pi_congr_right (λ a, e.to_equiv) } | def | uniform_on_fun.congr_right | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.Pi_congr_right",
"uniform_on_fun.postcomp_uniform_continuous",
"uniform_space"
] | Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
by post-composing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
precomp_uniform_continuous {𝔗 : set (set γ)} {f : γ → α}
(hf : 𝔗 ⊆ (image f) ⁻¹' 𝔖) :
uniform_continuous (λ g : α →ᵤ[𝔖] β, of_fun 𝔗 (g ∘ f)) | begin
-- Since `comap` distributes on `infi`, it suffices to prove that
-- `⨅ s ∈ 𝔖, comap s.restrict 𝒰(↥s, β, uβ) ≤ ⨅ t ∈ 𝔗, comap (t.restrict ∘ (— ∘ f)) 𝒰(↥t, β, uβ)`.
simp_rw [uniform_continuous_iff, uniform_on_fun.uniform_space, uniform_space.comap_infi,
← uniform_space.comap_comap],
-- For ... | lemma | uniform_on_fun.precomp_uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"infi_le_of_le",
"le_infi₂",
"uniform_continuous",
"uniform_continuous_iff",
"uniform_fun.precomp_uniform_continuous",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.comap_mono"
] | Let `f : γ → α`, `𝔖 : set (set α)`, `𝔗 : set (set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`.
Then, the function `(λ g, g ∘ f) : (α →ᵤ[𝔖] β) → (γ →ᵤ[𝔗] β)` is uniformly continuous.
Note that one can easily see that assuming `∀ T ∈ 𝔗, ∃ S ∈ 𝔖, f '' T ⊆ S` would work too, but
we will get this for free when we pr... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_left {𝔗 : set (set γ)} (e : γ ≃ α)
(he : 𝔗 ⊆ (image e) ⁻¹' 𝔖) (he' : 𝔖 ⊆ (preimage e) ⁻¹' 𝔗) :
(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β) | { uniform_continuous_to_fun :=
uniform_on_fun.precomp_uniform_continuous
begin
intros s hs,
change e.symm '' s ∈ 𝔗,
rw ← preimage_equiv_eq_image_symm,
exact he' hs
end,
uniform_continuous_inv_fun :=
uniform_on_fun.precomp_uniform_continuous he,
.. equiv.arrow_congr e (equiv.... | def | uniform_on_fun.congr_left | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_congr",
"equiv.refl",
"uniform_on_fun.precomp_uniform_continuous"
] | Turn a bijection `e : γ ≃ α` such that we have both `∀ T ∈ 𝔗, e '' T ∈ 𝔖` and
`∀ S ∈ 𝔖, e ⁻¹' S ∈ 𝔗` into a uniform isomorphism `(γ →ᵤ[𝔗] β) ≃ᵤ (α →ᵤ[𝔖] β)` by pre-composing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space_of_covering [t2_space β] (h : ⋃₀ 𝔖 = univ) :
t2_space (α →ᵤ[𝔖] β) | { t2 :=
begin
intros f g hfg,
obtain ⟨x, hx⟩ := not_forall.mp (mt funext hfg),
obtain ⟨s, hs, hxs⟩ : ∃ s ∈ 𝔖, x ∈ s := mem_sUnion.mp (h.symm ▸ true.intro),
exact separated_by_continuous (uniform_continuous_eval_of_mem β 𝔖 hxs hs).continuous hx
end } | lemma | uniform_on_fun.t2_space_of_covering | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"continuous",
"separated_by_continuous",
"t2_space"
] | If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_to_fun (h : ⋃₀ 𝔖 = univ) :
uniform_continuous (to_fun 𝔖 : (α →ᵤ[𝔖] β) → α → β) | begin
rw uniform_continuous_pi,
intros x,
obtain ⟨s : set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := sUnion_eq_univ_iff.mp h x,
exact uniform_continuous_eval_of_mem β 𝔖 hxs hs
end | lemma | uniform_on_fun.uniform_continuous_to_fun | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"uniform_continuous",
"uniform_continuous_pi"
] | If `𝔖` covers `α`, the natural map `uniform_on_fun.to_fun` from `α →ᵤ[𝔖] β` to `α → β` is
uniformly continuous.
In other words, if `𝔖` covers `α`, then the uniform structure of `𝔖`-convergence is finer than
that of pointwise convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_iff_tendsto_uniformly_on {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
tendsto F p (𝓝 f) ↔
∀ s ∈ 𝔖, tendsto_uniformly_on F f p s | begin
rw [uniform_on_fun.topological_space_eq, nhds_infi, tendsto_infi],
refine forall_congr (λ s, _),
rw [nhds_infi, tendsto_infi],
refine forall_congr (λ hs, _),
rw [nhds_induced, tendsto_comap_iff, tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe,
uniform_fun.tendsto_iff_tendsto_uniformly],
refl... | lemma | uniform_on_fun.tendsto_iff_tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"nhds_induced",
"nhds_infi",
"tendsto_uniformly_on",
"tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe",
"uniform_fun.tendsto_iff_tendsto_uniformly",
"uniform_on_fun.topological_space_eq"
] | Convergence in the topology of `𝔖`-convergence means uniform convergence on `S` (in the sense
of `tendsto_uniformly_on`) for all `S ∈ 𝔖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_prod_arrow [uniform_space γ] :
(α →ᵤ[𝔖] β × γ) ≃ᵤ ((α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (𝒱(α, β, 𝔖, uβ) × 𝒱(α, γ, 𝔖, uγ)) = 𝒱(α, β × γ, 𝔖, uβ × uγ)`.
-- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
-- `uniform_convergence_on.inf_eq` and `uniform_convergence_on.comap_eq`, which leaves us to check
-- ... | def | uniform_on_fun.uniform_equiv_prod_arrow | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"equiv.arrow_prod_equiv_prod_arrow",
"inf_uniformity",
"uniform_on_fun.comap_eq",
"uniform_on_fun.inf_eq",
"uniform_on_fun.of_fun",
"uniform_space",
"uniformity_comap",
"uniformity_prod"
] | The natural bijection between `α → β × γ` and `(α → β) × (α → γ)`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] β × γ` and `(α →ᵤ[𝔖] β) × (α →ᵤ[𝔖] γ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_Pi_comm :
(α →ᵤ[𝔖] Π i, δ i) ≃ᵤ (Π i, α →ᵤ[𝔖] δ i) | -- Denote `φ` this bijection. We want to show that
-- `comap φ (Π i, 𝒱(α, δ i, 𝔖, uδ i)) = 𝒱(α, (Π i, δ i), 𝔖, (Π i, uδ i))`.
-- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
-- `uniform_convergence_on.infi_eq` and `uniform_convergence_on.comap_eq`, which leaves us to check
--... | def | uniform_on_fun.uniform_equiv_Pi_comm | topology.uniform_space | src/topology/uniform_space/uniform_convergence_topology.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.pi",
"topology.uniform_space.equiv"
] | [
"Pi.uniform_space",
"equiv.Pi_comm",
"infi_congr",
"uniform_on_fun.comap_eq",
"uniform_on_fun.infi_eq",
"uniform_space.comap_comap",
"uniform_space.comap_infi",
"uniform_space.of_core_eq_to_core",
"uniformity_comap"
] | The natural bijection between `α → Π i, δ i` and `Π i, α → δ i`, upgraded to a uniform
isomorphism between `α →ᵤ[𝔖] (Π i, δ i)` and `Π i, α →ᵤ[𝔖] δ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing (f : α → β) : Prop | (comap_uniformity : comap (λx:α×α, (f x.1, f x.2)) (𝓤 β) = 𝓤 α) | structure | uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [] | A map `f : α → β` between uniform spaces is called *uniform inducing* if the uniformity filter
on `α` is the pullback of the uniformity filter on `β` under `prod.map f f`. If `α` is a separated
space, then this implies that `f` is injective, hence it is a `uniform_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.comap_uniform_space {f : α → β} (hf : uniform_inducing f) :
‹uniform_space β›.comap f = ‹uniform_space α› | uniform_space_eq hf.1 | lemma | uniform_inducing.comap_uniform_space | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing",
"uniform_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing_iff' {f : α → β} :
uniform_inducing f ↔ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α | by rw [uniform_inducing_iff, uniform_continuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; refl | lemma | uniform_inducing_iff' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_inducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_inducing f ↔
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j... | by simp [uniform_inducing_iff', h.uniform_continuous_iff h', (h'.comap _).le_basis_iff h,
subset_def] | lemma | filter.has_basis.uniform_inducing_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing",
"uniform_inducing_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.mk' {f : α → β} (h : ∀ s, s ∈ 𝓤 α ↔
∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : uniform_inducing f | ⟨by simp [eq_comm, filter.ext_iff, subset_def, h]⟩ | lemma | uniform_inducing.mk' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"filter.ext_iff",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing_id : uniform_inducing (@id α) | ⟨by rw [← prod.map_def, prod.map_id, comap_id]⟩ | lemma | uniform_inducing_id | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"prod.map_def",
"prod.map_id",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.comp {g : β → γ} (hg : uniform_inducing g)
{f : α → β} (hf : uniform_inducing f) : uniform_inducing (g ∘ f) | ⟨by rw [← hf.1, ← hg.1, comap_comap]⟩ | lemma | uniform_inducing.comp | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.basis_uniformity {f : α → β} (hf : uniform_inducing f)
{ι : Sort*} {p : ι → Prop} {s : ι → set (β × β)} (H : (𝓤 β).has_basis p s) :
(𝓤 α).has_basis p (λ i, prod.map f f ⁻¹' s i) | hf.1 ▸ H.comap _ | lemma | uniform_inducing.basis_uniformity | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.cauchy_map_iff {f : α → β} (hf : uniform_inducing f) {F : filter α} :
cauchy (map f F) ↔ cauchy F | by simp only [cauchy, map_ne_bot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | lemma | uniform_inducing.cauchy_map_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"filter",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing_of_compose {f : α → β} {g : β → γ} (hf : uniform_continuous f)
(hg : uniform_continuous g) (hgf : uniform_inducing (g ∘ f)) : uniform_inducing f | begin
refine ⟨le_antisymm _ hf.le_comap⟩,
rw [← hgf.1, ← prod.map_def, ← prod.map_def, ← prod.map_comp_map f f g g,
← @comap_comap _ _ _ _ (prod.map f f)],
exact comap_mono hg.le_comap
end | lemma | uniform_inducing_of_compose | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"prod.map_comp_map",
"prod.map_def",
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.uniform_continuous {f : α → β}
(hf : uniform_inducing f) : uniform_continuous f | (uniform_inducing_iff'.1 hf).1 | lemma | uniform_inducing.uniform_continuous | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.uniform_continuous_iff {f : α → β} {g : β → γ} (hg : uniform_inducing g) :
uniform_continuous f ↔ uniform_continuous (g ∘ f) | by { dsimp only [uniform_continuous, tendsto],
rw [← hg.comap_uniformity, ← map_le_iff_le_comap, filter.map_map] } | lemma | uniform_inducing.uniform_continuous_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"filter.map_map",
"uniform_continuous",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.inducing {f : α → β} (h : uniform_inducing f) : inducing f | begin
unfreezingI { obtain rfl := h.comap_uniform_space },
letI := uniform_space.comap f _,
exact ⟨rfl⟩
end | lemma | uniform_inducing.inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"inducing",
"uniform_inducing",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_inducing e₁) (h₂ : uniform_inducing e₂) :
uniform_inducing (λp:α×β, (e₁ p.1, e₂ p.2)) | ⟨by simp [(∘), uniformity_prod, h₁.comap_uniformity.symm, h₂.comap_uniformity.symm,
comap_inf, comap_comap]⟩ | lemma | uniform_inducing.prod | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing",
"uniform_space",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.dense_inducing {f : α → β} (h : uniform_inducing f) (hd : dense_range f) :
dense_inducing f | { dense := hd,
induced := h.inducing.induced } | lemma | uniform_inducing.dense_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense",
"dense_inducing",
"dense_range",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.injective [t0_space α] {f : α → β} (h : uniform_inducing f) :
injective f | h.inducing.injective | lemma | uniform_inducing.injective | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding (f : α → β) extends uniform_inducing f : Prop | (inj : function.injective f) | structure | uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_inducing"
] | A map `f : α → β` between uniform spaces is a *uniform embedding* if it is uniform inducing and
injective. If `α` is a separated space, then the latter assumption follows from the former. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_iff' {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧ comap (prod.map f f) (𝓤 β) ≤ 𝓤 α | by rw [uniform_embedding_iff, and_comm, uniform_inducing_iff'] | theorem | uniform_embedding_iff' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_embedding",
"uniform_inducing_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_embedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s'... | by rw [uniform_embedding_iff, and_comm, h.uniform_inducing_iff h'] | theorem | filter.has_basis.uniform_embedding_iff' | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_embedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).has_basis p s) (h' : (𝓤 β).has_basis p' s') {f : α → β} :
uniform_embedding f ↔ injective f ∧ uniform_continuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) | by simp only [h.uniform_embedding_iff' h', h.uniform_continuous_iff h', exists_prop] | theorem | filter.has_basis.uniform_embedding_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"exists_prop",
"uniform_continuous",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_subtype_val {p : α → Prop} :
uniform_embedding (subtype.val : subtype p → α) | { comap_uniformity := rfl,
inj := subtype.val_injective } | lemma | uniform_embedding_subtype_val | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"subtype.val_injective",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_subtype_coe {p : α → Prop} :
uniform_embedding (coe : subtype p → α) | uniform_embedding_subtype_val | lemma | uniform_embedding_subtype_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_embedding_subtype_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_set_inclusion {s t : set α} (hst : s ⊆ t) :
uniform_embedding (inclusion hst) | { comap_uniformity :=
by { erw [uniformity_subtype, uniformity_subtype, comap_comap], congr },
inj := inclusion_injective hst } | lemma | uniform_embedding_set_inclusion | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniformity_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding.comp {g : β → γ} (hg : uniform_embedding g)
{f : α → β} (hf : uniform_embedding f) : uniform_embedding (g ∘ f) | { inj := hg.inj.comp hf.inj,
..hg.to_uniform_inducing.comp hf.to_uniform_inducing } | lemma | uniform_embedding.comp | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.uniform_embedding {α β : Type*} [uniform_space α] [uniform_space β] (f : α ≃ β)
(h₁ : uniform_continuous f) (h₂ : uniform_continuous f.symm) : uniform_embedding f | uniform_embedding_iff'.2 ⟨f.injective, h₁, by rwa [← equiv.prod_congr_apply, ← map_equiv_symm]⟩ | lemma | equiv.uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_continuous",
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_inl : uniform_embedding (sum.inl : α → α ⊕ β) | begin
refine ⟨⟨_⟩, sum.inl_injective⟩,
rw [sum.uniformity, comap_sup, comap_map, comap_eq_bot_iff_compl_range.2 _, sup_bot_eq],
{ refine mem_map.2 (univ_mem' _),
simp },
{ exact sum.inl_injective.prod_map sum.inl_injective }
end | theorem | uniform_embedding_inl | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"sum.inl_injective",
"sum.uniformity",
"sup_bot_eq",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_inr : uniform_embedding (sum.inr : β → α ⊕ β) | begin
refine ⟨⟨_⟩, sum.inr_injective⟩,
rw [sum.uniformity, comap_sup, comap_eq_bot_iff_compl_range.2 _, comap_map, bot_sup_eq],
{ exact sum.inr_injective.prod_map sum.inr_injective },
{ refine mem_map.2 (univ_mem' _),
simp },
end | theorem | uniform_embedding_inr | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"bot_sup_eq",
"sum.inr_injective",
"sum.uniformity",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.uniform_embedding [t0_space α] {f : α → β}
(hf : uniform_inducing f) :
uniform_embedding f | ⟨hf, hf.injective⟩ | theorem | uniform_inducing.uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_embedding",
"uniform_inducing"
] | If the domain of a `uniform_inducing` map `f` is a `separated_space`, then `f` is injective,
hence it is a `uniform_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_iff_uniform_inducing [t0_space α] {f : α → β} :
uniform_embedding f ↔ uniform_inducing f | ⟨uniform_embedding.to_uniform_inducing, uniform_inducing.uniform_embedding⟩ | theorem | uniform_embedding_iff_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"t0_space",
"uniform_embedding",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_uniformity_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
comap (prod.map f f) (𝓤 β) = 𝓟 id_rel | begin
refine le_antisymm _ (@refl_le_uniformity α (uniform_space.comap f ‹_›)),
calc comap (prod.map f f) (𝓤 β) ≤ comap (prod.map f f) (𝓟 s) : comap_mono (le_principal_iff.2 hs)
... = 𝓟 (prod.map f f ⁻¹' s) : comap_principal
... ≤ 𝓟 id_rel : principal_mono.2 _,
rintro ⟨x, y⟩, simpa [not_imp_not] using @hf... | lemma | comap_uniformity_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"id_rel",
"not_imp_not",
"pairwise",
"refl_le_uniformity",
"uniform_space.comap"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is uniform inducing with respect to the discrete uniformity on `α`:
the preimage of `𝓤 β` under `prod.map f f` is the principal filter generated by the diagonal in
`α × α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding_of_spaced_out {α} {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
@uniform_embedding α β ⊥ ‹_› f | begin
letI : uniform_space α := ⊥, haveI := discrete_topology_bot α,
haveI : separated_space α := separated_iff_t2.2 infer_instance,
exact uniform_inducing.uniform_embedding ⟨comap_uniformity_of_spaced_out hs hf⟩
end | lemma | uniform_embedding_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"discrete_topology_bot",
"pairwise",
"separated_space",
"uniform_embedding",
"uniform_inducing.uniform_embedding",
"uniform_space"
] | If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
`s ∈ 𝓤 β`, then `f` is a uniform embedding with respect to the discrete uniformity on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding.embedding {f : α → β} (h : uniform_embedding f) : embedding f | { induced := h.to_uniform_inducing.inducing.induced,
inj := h.inj } | lemma | uniform_embedding.embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding.dense_embedding {f : α → β} (h : uniform_embedding f) (hd : dense_range f) :
dense_embedding f | { dense := hd,
inj := h.inj,
induced := h.embedding.induced } | lemma | uniform_embedding.dense_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense",
"dense_embedding",
"dense_range",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_of_spaced_out {α} [topological_space α] [discrete_topology α]
[separated_space β] {f : α → β} {s : set (β × β)} (hs : s ∈ 𝓤 β)
(hf : pairwise (λ x y, (f x, f y) ∉ s)) :
closed_embedding f | begin
unfreezingI { rcases (discrete_topology.eq_bot α) with rfl }, letI : uniform_space α := ⊥,
exact { closed_range := is_closed_range_of_spaced_out hs hf,
.. (uniform_embedding_of_spaced_out hs hf).embedding }
end | lemma | closed_embedding_of_spaced_out | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"closed_embedding",
"discrete_topology",
"embedding",
"is_closed_range_of_spaced_out",
"pairwise",
"separated_space",
"topological_space",
"uniform_embedding_of_spaced_out",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_image_mem_nhds_of_uniform_inducing
{s : set (α×α)} {e : α → β} (b : β)
(he₁ : uniform_inducing e) (he₂ : dense_inducing e) (hs : s ∈ 𝓤 α) :
∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b | have s ∈ comap (λp:α×α, (e p.1, e p.2)) (𝓤 β),
from he₁.comap_uniformity.symm ▸ hs,
let ⟨t₁, ht₁u, ht₁⟩ := this in
have ht₁ : ∀p:α×α, (e p.1, e p.2) ∈ t₁ → p ∈ s, from ht₁,
let ⟨t₂, ht₂u, ht₂s, ht₂c⟩ := comp_symm_of_uniformity ht₁u in
let ⟨t, htu, hts, htc⟩ := comp_symm_of_uniformity ht₂u in
have preimage e {b' | (b... | lemma | closure_image_mem_nhds_of_uniform_inducing | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"closure",
"closure_eq_cluster_pts",
"comp_symm_of_uniformity",
"dense_inducing",
"mem_nhds_left",
"monotone_const",
"nhds_eq_uniformity",
"prod_mk_mem_comp_rel",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_subtype_emb (p : α → Prop) {e : α → β} (ue : uniform_embedding e)
(de : dense_embedding e) : uniform_embedding (dense_embedding.subtype_emb p e) | { comap_uniformity := by simp [comap_comap, (∘), dense_embedding.subtype_emb,
uniformity_subtype, ue.comap_uniformity.symm],
inj := (de.subtype p).inj } | lemma | uniform_embedding_subtype_emb | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"dense_embedding",
"dense_embedding.subtype_emb",
"uniform_embedding",
"uniformity_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding.prod {α' : Type*} {β' : Type*} [uniform_space α'] [uniform_space β']
{e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_embedding e₁) (h₂ : uniform_embedding e₂) :
uniform_embedding (λp:α×β, (e₁ p.1, e₂ p.2)) | { inj := h₁.inj.prod_map h₂.inj,
..h₁.to_uniform_inducing.prod h₂.to_uniform_inducing } | lemma | uniform_embedding.prod | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_of_complete_image {m : α → β} {s : set α} (hm : uniform_inducing m)
(hs : is_complete (m '' s)) : is_complete s | begin
intros f hf hfs,
rw le_principal_iff at hfs,
obtain ⟨_, ⟨x, hx, rfl⟩, hyf⟩ : ∃ y ∈ m '' s, map m f ≤ 𝓝 y,
from hs (f.map m) (hf.map hm.uniform_continuous)
(le_principal_iff.2 (image_mem_map hfs)),
rw [map_le_iff_le_comap, ← nhds_induced, ← hm.inducing.induced] at hyf,
exact ⟨x, hx, hyf⟩
end | lemma | is_complete_of_complete_image | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"is_complete",
"nhds_induced",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete.complete_space_coe {s : set α} (hs : is_complete s) :
complete_space s | complete_space_iff_is_complete_univ.2 $
is_complete_of_complete_image uniform_embedding_subtype_coe.to_uniform_inducing $ by simp [hs] | lemma | is_complete.complete_space_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"is_complete",
"is_complete_of_complete_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_complete_image_iff {m : α → β} {s : set α} (hm : uniform_inducing m) :
is_complete (m '' s) ↔ is_complete s | begin
refine ⟨is_complete_of_complete_image hm, λ c, _⟩,
haveI : complete_space s := c.complete_space_coe,
set m' : s → β := m ∘ coe,
suffices : is_complete (range m'), by rwa [range_comp, subtype.range_coe] at this,
have hm' : uniform_inducing m' := hm.comp uniform_embedding_subtype_coe.to_uniform_inducing,
... | lemma | is_complete_image_iff | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"complete_space",
"filter.le_principal_iff",
"is_complete",
"subtype.range_coe",
"uniform_inducing"
] | A set is complete iff its image under a uniform inducing map is complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_iff_is_complete_range {f : α → β} (hf : uniform_inducing f) :
complete_space α ↔ is_complete (range f) | by rw [complete_space_iff_is_complete_univ, ← is_complete_image_iff hf, image_univ] | lemma | complete_space_iff_is_complete_range | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_univ",
"is_complete",
"is_complete_image_iff",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.is_complete_range [complete_space α] {f : α → β}
(hf : uniform_inducing f) :
is_complete (range f) | (complete_space_iff_is_complete_range hf).1 ‹_› | lemma | uniform_inducing.is_complete_range | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"is_complete",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_congr {e : α ≃ β} (he : uniform_embedding e) :
complete_space α ↔ complete_space β | by rw [complete_space_iff_is_complete_range he.to_uniform_inducing, e.range_eq_univ,
complete_space_iff_is_complete_univ] | lemma | complete_space_congr | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"complete_space_iff_is_complete_univ",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space_coe_iff_is_complete {s : set α} :
complete_space s ↔ is_complete s | (complete_space_iff_is_complete_range uniform_embedding_subtype_coe.to_uniform_inducing).trans $
by rw [subtype.range_coe] | lemma | complete_space_coe_iff_is_complete | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_range",
"is_complete",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.complete_space_coe [complete_space α] {s : set α} (hs : is_closed s) :
complete_space s | hs.is_complete.complete_space_coe | lemma | is_closed.complete_space_coe | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ulift.complete_space [h : complete_space α] : complete_space (ulift α) | begin
have : uniform_embedding (@equiv.ulift α), from ⟨⟨rfl⟩, ulift.down_injective⟩,
exact (complete_space_congr this).2 h,
end | instance | ulift.complete_space | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_congr",
"equiv.ulift",
"uniform_embedding"
] | The lift of a complete space to another universe is still complete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_space_extension {m : β → α} (hm : uniform_inducing m) (dense : dense_range m)
(h : ∀f:filter β, cauchy f → ∃x:α, map m f ≤ 𝓝 x) : complete_space α | ⟨assume (f : filter α), assume hf : cauchy f,
let
p : set (α × α) → set α → set α := λs t, {y : α| ∃x:α, x ∈ t ∧ (x, y) ∈ s},
g := (𝓤 α).lift (λs, f.lift' (p s))
in
have mp₀ : monotone p,
from assume a b h t s ⟨x, xs, xa⟩, ⟨x, xs, h xa⟩,
have mp₁ : ∀{s}, monotone (p s),
from assume s a b h x ⟨y, ya, yxs⟩, ⟨y, ... | lemma | complete_space_extension | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"cauchy",
"cluster_pt",
"comp_mem_uniformity_sets",
"comp_rel",
"complete_space",
"dense",
"dense_range",
"filter",
"filter.comap",
"le_infi",
"le_nhds_iff_adhp_of_cauchy",
"le_nhds_of_cauchy_adhp",
"lift",
"mem_nhds_left",
"monotone",
"monotone_const",
"prod.swap",
"prod_mk_mem_co... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_bounded_preimage {f : α → β} {s : set β} (hf : uniform_embedding f)
(hs : totally_bounded s) : totally_bounded (f ⁻¹' s) | λ t ht, begin
rw ← hf.comap_uniformity at ht,
rcases mem_comap.2 ht with ⟨t', ht', ts⟩,
rcases totally_bounded_iff_subset.1
(totally_bounded_subset (image_preimage_subset f s) hs) _ ht' with ⟨c, cs, hfc, hct⟩,
refine ⟨f ⁻¹' c, hfc.preimage (hf.inj.inj_on _), λ x h, _⟩,
have := hct (mem_image_of_mem f h), ... | lemma | totally_bounded_preimage | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"totally_bounded",
"totally_bounded_subset",
"uniform_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_space.sum [complete_space α] [complete_space β] :
complete_space (α ⊕ β) | begin
rw [complete_space_iff_is_complete_univ, ← range_inl_union_range_inr],
exact uniform_embedding_inl.to_uniform_inducing.is_complete_range.union
uniform_embedding_inr.to_uniform_inducing.is_complete_range
end | instance | complete_space.sum | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"complete_space",
"complete_space_iff_is_complete_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_comap {α : Type*} {β : Type*} {f : α → β} [u : uniform_space β]
(hf : function.injective f) : @uniform_embedding α β (uniform_space.comap f u) u f | @uniform_embedding.mk _ _ (uniform_space.comap f u) _ _
(@uniform_inducing.mk _ _ (uniform_space.comap f u) _ _ rfl) hf | lemma | uniform_embedding_comap | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"uniform_embedding",
"uniform_space",
"uniform_space.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.comap_uniform_space {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) : uniform_space α | (u.comap f).replace_topology h.induced | def | embedding.comap_uniform_space | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"topological_space",
"uniform_space"
] | Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
make sure that its topology is defeq to the original one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.to_uniform_embedding {α β} [topological_space α] [u : uniform_space β] (f : α → β)
(h : embedding f) :
@uniform_embedding α β (h.comap_uniform_space f) u f | { comap_uniformity := rfl,
inj := h.inj } | lemma | embedding.to_uniform_embedding | topology.uniform_space | src/topology/uniform_space/uniform_embedding.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation",
"topology.dense_embedding"
] | [
"embedding",
"topological_space",
"uniform_embedding",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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