UniformConvergenceTopology.lean

/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv

#align_import topology.uniform_space.uniform_convergence_topology from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"

/-!
# Topology and uniform structure of uniform convergence

This files endows `α → β` with the topologies / uniform structures of
- uniform convergence on `α`
- uniform convergence on a specified family `𝔖` of sets of `α`, also called `𝔖`-convergence

Since `α → β` is already endowed with the topologies and uniform structures of pointwise
convergence, we introduce type aliases `UniformFun α β` (denoted `α →ᵤ β`) and
`UniformOnFun α β 𝔖` (denoted `α →ᵤ[𝔖] β`) and we actually endow *these* with the structures
of uniform and `𝔖`-convergence respectively.

Usual examples of the second construction include :
- the topology of compact convergence, when `𝔖` is the set of compacts of `α`
- the strong topology on the dual of a topological vector space (TVS) `E`, when `𝔖` is the set of
  Von Neumann bounded subsets of `E`
- the weak-* topology on the dual of a TVS `E`, when `𝔖` is the set of singletons of `E`.

This file contains a lot of technical facts, so it is heavily commented, proofs included!

## Main definitions

* `UniformFun.gen`: basis sets for the uniformity of uniform convergence. These are sets
  of the form `S(V) := {(f, g) | ∀ x : α, (f x, g x) ∈ V}` for some `V : Set (β × β)`
* `UniformFun.uniformSpace`: uniform structure of uniform convergence. This is the
  `UniformSpace` on `α →ᵤ β` whose uniformity is generated by the sets `S(V)` for `V ∈ 𝓤 β`.
  We will denote this uniform space as `𝒰(α, β, uβ)`, both in the comments and as a local notation
  in the Lean code, where `uβ` is the uniform space structure on `β`.
  This is declared as an instance on `α →ᵤ β`.
* `UniformOnFun.uniformSpace`: uniform structure of `𝔖`-convergence, where
  `𝔖 : Set (Set α)`. This is the infimum, for `S ∈ 𝔖`, of the pullback of `𝒰 S β` by the map of
  restriction to `S`. We will denote it `𝒱(α, β, 𝔖, uβ)`, where `uβ` is the uniform space structure
  on `β`.
  This is declared as an instance on `α →ᵤ[𝔖] β`.

## Main statements

### Basic properties

* `UniformFun.uniformContinuous_eval`: evaluation is uniformly continuous on `α →ᵤ β`.
* `UniformFun.t2Space`: the topology of uniform convergence on `α →ᵤ β` is T₂ if
  `β` is T₂.
* `UniformFun.tendsto_iff_tendstoUniformly`: `𝒰(α, β, uβ)` is
  indeed the uniform structure of uniform convergence
* `UniformOnFun.uniformContinuous_eval_of_mem`: evaluation at a point contained in a
  set of `𝔖` is uniformly continuous on `α →ᵤ[𝔖] β`
* `UniformOnFun.t2Space_of_covering`: the topology of `𝔖`-convergence on `α →ᵤ[𝔖] β` is T₂ if
  `β` is T₂ and `𝔖` covers `α`
* `UniformOnFun.tendsto_iff_tendstoUniformlyOn`:
  `𝒱(α, β, 𝔖 uβ)` is indeed the uniform structure of `𝔖`-convergence

### Functoriality and compatibility with product of uniform spaces

In order to avoid the need for filter bases as much as possible when using these definitions,
we develop an extensive API for manipulating these structures abstractly. As usual in the topology
section of mathlib, we first state results about the complete lattices of `UniformSpace`s on
fixed types, and then we use these to deduce categorical-like results about maps between two
uniform spaces.

We only describe these in the harder case of `𝔖`-convergence, as the names of the corresponding
results for uniform convergence can easily be guessed.

#### Order statements

* `UniformOnFun.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)`.
* `UniformOnFun.iInf_eq`: 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)`.
* `UniformOnFun.comap_eq`: if `u` is a uniform structures on `β` and `f : γ → β`, then
  `𝒱(α, γ, 𝔖, comap f u) = comap (fun g ↦ f ∘ g) 𝒱(α, γ, 𝔖, u₁)`.

An interesting note about these statements is that they are proved without ever unfolding the basis
definition of the uniform structure of uniform convergence! Instead, we build a
(not very interesting) Galois connection `UniformFun.gc` and then rely on the Galois
connection API to do most of the work.

#### Morphism statements (unbundled)

* `UniformOnFun.postcomp_uniformContinuous`: if `f : γ → β` is uniformly
  continuous, then `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous.
* `UniformOnFun.postcomp_uniformInducing`: if `f : γ → β` is a uniform
  inducing, then `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing.
* `UniformOnFun.precomp_uniformContinuous`: let `f : γ → α`, `𝔖 : Set (Set α)`,
  `𝔗 : Set (Set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`. Then, the function
  `(fun g ↦ g ∘ f) : (α →ᵤ[𝔖] β) → (γ →ᵤ[𝔗] β)` is uniformly continuous.

#### Isomorphism statements (bundled)

* `UniformOnFun.congrRight`: turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism
  `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)` by post-composing.
* `UniformOnFun.congrLeft`: 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.
* `UniformOnFun.uniformEquivPiComm`: 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`.

#### Important use cases

* If `G` is a uniform group, then `α →ᵤ[𝔖] G` is a uniform group: since `(/) : G × G → G` is
  uniformly continuous, `UniformOnFun.postcomp_uniformContinuous` tells us that
  `((/) ∘ —) : (α →ᵤ[𝔖] G × G) → (α →ᵤ[𝔖] G)` is uniformly continuous. By precomposing with
  `UniformOnFun.uniformEquivProdArrow`, this gives that
  `(/) : (α →ᵤ[𝔖] G) × (α →ᵤ[𝔖] G) → (α →ᵤ[𝔖] G)` is also uniformly continuous
* The transpose of a continuous linear map is continuous for the strong topologies: since
  continuous linear maps are uniformly continuous and map bounded sets to bounded sets,
  this is just a special case of `UniformOnFun.precomp_uniformContinuous`.

## TODO

* Show that the uniform structure of `𝔖`-convergence is exactly the structure of `𝔖'`-convergence,
  where `𝔖'` is the ***noncovering*** bornology (i.e ***not*** what `Bornology` currently refers
  to in mathlib) generated by `𝔖`.

## References

* [N. Bourbaki, *General Topology, Chapter X*][bourbaki1966]

## Tags

uniform convergence
-/


noncomputable section

open scoped Classical Topology Uniformity
open Set Filter

section TypeAlias

/-- The type of functions from `α` to `β` equipped with the uniform structure and topology of
uniform convergence. We denote it `α →ᵤ β`. -/
def UniformFun (α β : Type*) :=
  α → β
#align uniform_fun UniformFun

/-- The type of functions from `α` to `β` equipped with the uniform structure and topology of
uniform convergence on some family `𝔖` of subsets of `α`. We denote it `α →ᵤ[𝔖] β`. -/
@[nolint unusedArguments]
def UniformOnFun (α β : Type*) (_ : Set (Set α)) :=
  α → β
#align uniform_on_fun UniformOnFun

@[inherit_doc] scoped[UniformConvergence] notation:25 α " →ᵤ " β:0 => UniformFun α β

@[inherit_doc] scoped[UniformConvergence] notation:25 α " →ᵤ[" 𝔖 "] " β:0 => UniformOnFun α β 𝔖

open UniformConvergence

variable {α β : Type*} {𝔖 : Set (Set α)}

instance [Nonempty β] : Nonempty (α →ᵤ β) := Pi.instNonempty

instance [Nonempty β] : Nonempty (α →ᵤ[𝔖] β) := Pi.instNonempty

instance [Subsingleton β] : Subsingleton (α →ᵤ β) :=
  inferInstanceAs <| Subsingleton <| α → β

instance [Subsingleton β] : Subsingleton (α →ᵤ[𝔖] β) :=
  inferInstanceAs <| Subsingleton <| α → β

/-- Reinterpret `f : α → β` as an element of `α →ᵤ β`. -/
def UniformFun.ofFun : (α → β) ≃ (α →ᵤ β) :=
  ⟨fun x => x, fun x => x, fun _ => rfl, fun _ => rfl⟩
#align uniform_fun.of_fun UniformFun.ofFun

/-- Reinterpret `f : α → β` as an element of `α →ᵤ[𝔖] β`. -/
def UniformOnFun.ofFun (𝔖) : (α → β) ≃ (α →ᵤ[𝔖] β) :=
  ⟨fun x => x, fun x => x, fun _ => rfl, fun _ => rfl⟩
#align uniform_on_fun.of_fun UniformOnFun.ofFun

/-- Reinterpret `f : α →ᵤ β` as an element of `α → β`. -/
def UniformFun.toFun : (α →ᵤ β) ≃ (α → β) :=
  UniformFun.ofFun.symm
#align uniform_fun.to_fun UniformFun.toFun

/-- Reinterpret `f : α →ᵤ[𝔖] β` as an element of `α → β`. -/
def UniformOnFun.toFun (𝔖) : (α →ᵤ[𝔖] β) ≃ (α → β) :=
  (UniformOnFun.ofFun 𝔖).symm
#align uniform_on_fun.to_fun UniformOnFun.toFun

@[simp] lemma UniformFun.toFun_ofFun (f : α → β) : toFun (ofFun f) = f := rfl
@[simp] lemma UniformFun.ofFun_toFun (f : α →ᵤ β) : ofFun (toFun f) = f := rfl
@[simp] lemma UniformOnFun.toFun_ofFun (f : α → β) : toFun 𝔖 (ofFun 𝔖 f) = f := rfl
@[simp] lemma UniformOnFun.ofFun_toFun (f : α →ᵤ[𝔖] β) : ofFun 𝔖 (toFun 𝔖 f) = f := rfl

-- Note: we don't declare a `CoeFun` instance because Lean wouldn't insert it when writing
-- `f x` (because of definitional equality with `α → β`).
end TypeAlias

open UniformConvergence

namespace UniformFun

variable (α β : Type*) {γ ι : Type*}
variable {s s' : Set α} {x : α} {p : Filter ι} {g : ι → α}

/-- Basis sets for the uniformity of uniform convergence: `gen α β V` is the set of pairs `(f, g)`
of functions `α →ᵤ β` such that `∀ x, (f x, g x) ∈ V`. -/
protected def gen (V : Set (β × β)) : Set ((α →ᵤ β) × (α →ᵤ β)) :=
  { uv : (α →ᵤ β) × (α →ᵤ β) | ∀ x, (toFun uv.1 x, toFun uv.2 x) ∈ V }
#align uniform_fun.gen UniformFun.gen

/-- If `𝓕` is a filter on `β × β`, then the set of all `UniformFun.gen α β V` for
`V ∈ 𝓕` is a filter basis on `(α →ᵤ β) × (α →ᵤ β)`. This will only be applied to `𝓕 = 𝓤 β` when
`β` is equipped with a `UniformSpace` structure, but it is useful to define it for any filter in
order to be able to state that it has a lower adjoint (see `UniformFun.gc`). -/
protected theorem isBasis_gen (𝓑 : Filter <| β × β) :
    IsBasis (fun V : Set (β × β) => V ∈ 𝓑) (UniformFun.gen α β) :=
  ⟨⟨univ, univ_mem⟩, @fun U V hU hV =>
    ⟨U ∩ V, inter_mem hU hV, fun _ huv => ⟨fun x => (huv x).left, fun x => (huv x).right⟩⟩⟩
#align uniform_fun.is_basis_gen UniformFun.isBasis_gen

/-- For `𝓕 : Filter (β × β)`, this is the set of all `UniformFun.gen α β V` for
`V ∈ 𝓕` as a bundled `FilterBasis` over `(α →ᵤ β) × (α →ᵤ β)`. This will only be applied to
`𝓕 = 𝓤 β` when `β` is equipped with a `UniformSpace` structure, but it is useful to define it for
any filter in order to be able to state that it has a lower adjoint
(see `UniformFun.gc`). -/
protected def basis (𝓕 : Filter <| β × β) : FilterBasis ((α →ᵤ β) × (α →ᵤ β)) :=
  (UniformFun.isBasis_gen α β 𝓕).filterBasis
#align uniform_fun.basis UniformFun.basis

/-- For `𝓕 : Filter (β × β)`, this is the filter generated by the filter basis
`UniformFun.basis α β 𝓕`. For `𝓕 = 𝓤 β`, this will be the uniformity of uniform
convergence on `α`. -/
protected def filter (𝓕 : Filter <| β × β) : Filter ((α →ᵤ β) × (α →ᵤ β)) :=
  (UniformFun.basis α β 𝓕).filter
#align uniform_fun.filter UniformFun.filter

--local notation "Φ" => fun (α β : Type*) (uvx : ((α →ᵤ β) × (α →ᵤ β)) × α) =>
  --(uvx.fst.fst uvx.2, uvx.1.2 uvx.2)

protected def phi (α β : Type*) (uvx : ((α →ᵤ β) × (α →ᵤ β)) × α) : β × β :=
  (uvx.fst.fst uvx.2, uvx.1.2 uvx.2)

set_option quotPrecheck false -- Porting note: error message suggested to do this
/- This is a lower adjoint to `UniformFun.filter` (see `UniformFun.gc`).
The exact definition of the lower adjoint `l` is not interesting; we will only use that it exists
(in `UniformFun.mono` and `UniformFun.iInf_eq`) and that
`l (Filter.map (Prod.map f f) 𝓕) = Filter.map (Prod.map ((∘) f) ((∘) f)) (l 𝓕)` for each
`𝓕 : Filter (γ × γ)` and `f : γ → α` (in `UniformFun.comap_eq`). -/
local notation "lowerAdjoint" => fun 𝓐 => map (UniformFun.phi α β) (𝓐 ×ˢ ⊤)

/-- The function `UniformFun.filter α β : Filter (β × β) → Filter ((α →ᵤ β) × (α →ᵤ β))`
has a lower adjoint `l` (in the sense of `GaloisConnection`). The exact definition of `l` is not
interesting; we will only use that it exists (in `UniformFun.mono` and
`UniformFun.iInf_eq`) and that
`l (Filter.map (Prod.map f f) 𝓕) = Filter.map (Prod.map ((∘) f) ((∘) f)) (l 𝓕)` for each
`𝓕 : Filter (γ × γ)` and `f : γ → α` (in `UniformFun.comap_eq`). -/
protected theorem gc : GaloisConnection lowerAdjoint fun 𝓕 => UniformFun.filter α β 𝓕 := by
  intro 𝓐 𝓕
  symm
  calc
    𝓐 ≤ UniformFun.filter α β 𝓕 ↔ (UniformFun.basis α β 𝓕).sets ⊆ 𝓐.sets :=
      by rw [UniformFun.filter, ← FilterBasis.generate, le_generate_iff]
    _ ↔ ∀ U ∈ 𝓕, UniformFun.gen α β U ∈ 𝓐 := image_subset_iff
    _ ↔ ∀ U ∈ 𝓕,
          { uv | ∀ x, (uv, x) ∈ { t : ((α →ᵤ β) × (α →ᵤ β)) × α | (t.1.1 t.2, t.1.2 t.2) ∈ U } } ∈
            𝓐 :=
      Iff.rfl
    _ ↔ ∀ U ∈ 𝓕,
          { uvx : ((α →ᵤ β) × (α →ᵤ β)) × α | (uvx.1.1 uvx.2, uvx.1.2 uvx.2) ∈ U } ∈
            𝓐 ×ˢ (⊤ : Filter α) :=
      forall₂_congr fun U _hU => mem_prod_top.symm
    _ ↔ lowerAdjoint 𝓐 ≤ 𝓕 := Iff.rfl
#align uniform_fun.gc UniformFun.gc

variable [UniformSpace β]

/-- Core of the uniform structure of uniform convergence. -/
protected def uniformCore : UniformSpace.Core (α →ᵤ β) :=
  UniformSpace.Core.mkOfBasis (UniformFun.basis α β (𝓤 β))
    (fun _ ⟨_, hV, hVU⟩ _ => hVU ▸ fun _ => refl_mem_uniformity hV)
    (fun _ ⟨V, hV, hVU⟩ =>
      hVU ▸
        ⟨UniformFun.gen α β (Prod.swap ⁻¹' V), ⟨Prod.swap ⁻¹' V, tendsto_swap_uniformity hV, rfl⟩,
          fun _ huv x => huv x⟩)
    fun _ ⟨_, hV, hVU⟩ =>
    hVU ▸
      let ⟨W, hW, hWV⟩ := comp_mem_uniformity_sets hV
      ⟨UniformFun.gen α β W, ⟨W, hW, rfl⟩, fun _ ⟨w, huw, hwv⟩ x => hWV ⟨w x, ⟨huw x, hwv x⟩⟩⟩
#align uniform_fun.uniform_core UniformFun.uniformCore

/-- Uniform structure of uniform convergence, declared as an instance on `α →ᵤ β`.
We will denote it `𝒰(α, β, uβ)` in the rest of this file. -/
instance uniformSpace : UniformSpace (α →ᵤ β) :=
  UniformSpace.ofCore (UniformFun.uniformCore α β)

/-- Topology of uniform convergence, declared as an instance on `α →ᵤ β`. -/
instance topologicalSpace : TopologicalSpace (α →ᵤ β) :=
  inferInstance

local notation "𝒰(" α ", " β ", " u ")" => @UniformFun.uniformSpace α β u

/-- 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. -/
protected theorem hasBasis_uniformity :
    (𝓤 (α →ᵤ β)).HasBasis (· ∈ 𝓤 β) (UniformFun.gen α β) :=
  (UniformFun.isBasis_gen α β (𝓤 β)).hasBasis
#align uniform_fun.has_basis_uniformity UniformFun.hasBasis_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). -/
protected theorem hasBasis_uniformity_of_basis {ι : Sort*} {p : ι → Prop} {s : ι → Set (β × β)}
    (h : (𝓤 β).HasBasis p s) : (𝓤 (α →ᵤ β)).HasBasis p (UniformFun.gen α β ∘ s) :=
  (UniformFun.hasBasis_uniformity α β).to_hasBasis
    (fun _ hU =>
      let ⟨i, hi, hiU⟩ := h.mem_iff.mp hU
      ⟨i, hi, fun _ huv x => hiU (huv x)⟩)
    fun i hi => ⟨s i, h.mem_of_mem hi, subset_refl _⟩
#align uniform_fun.has_basis_uniformity_of_basis UniformFun.hasBasis_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`. -/
protected theorem hasBasis_nhds_of_basis (f) {p : ι → Prop} {s : ι → Set (β × β)}
    (h : HasBasis (𝓤 β) p s) :
    (𝓝 f).HasBasis p fun i => { g | (f, g) ∈ UniformFun.gen α β (s i) } :=
  nhds_basis_uniformity' (UniformFun.hasBasis_uniformity_of_basis α β h)
#align uniform_fun.has_basis_nhds_of_basis UniformFun.hasBasis_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. -/
protected theorem hasBasis_nhds (f) :
    (𝓝 f).HasBasis (fun V => V ∈ 𝓤 β) fun V => { g | (f, g) ∈ UniformFun.gen α β V } :=
  UniformFun.hasBasis_nhds_of_basis α β f (Filter.basis_sets _)
#align uniform_fun.has_basis_nhds UniformFun.hasBasis_nhds

variable {α}

/-- Evaluation at a fixed point is uniformly continuous on `α →ᵤ β`. -/
theorem uniformContinuous_eval (x : α) :
    UniformContinuous (Function.eval x ∘ toFun : (α →ᵤ β) → β) := by
  change _ ≤ _
  rw [map_le_iff_le_comap,
    (UniformFun.hasBasis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)]
  exact fun U hU => ⟨U, hU, fun uv huv => huv x⟩
#align uniform_fun.uniform_continuous_eval UniformFun.uniformContinuous_eval

variable {β}

@[simp]
protected lemma mem_gen {f g : α →ᵤ β} {V : Set (β × β)} :
    (f, g) ∈ UniformFun.gen α β V ↔ ∀ x, (toFun f x, toFun g x) ∈ V :=
  .rfl

/-- If `u₁` and `u₂` are two uniform structures on `γ` and `u₁ ≤ u₂`, then
`𝒰(α, γ, u₁) ≤ 𝒰(α, γ, u₂)`. -/
protected theorem mono : Monotone (@UniformFun.uniformSpace α γ) := fun _ _ hu =>
  (UniformFun.gc α γ).monotone_u hu
#align uniform_fun.mono UniformFun.mono

/-- If `u` is a family of uniform structures on `γ`, then
`𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i)`. -/
protected theorem iInf_eq {u : ι → UniformSpace γ} : 𝒰(α, γ, (⨅ i, u i)) = ⨅ i, 𝒰(α, γ, u i) := by
  -- This follows directly from the fact that the upper adjoint in a Galois connection maps
  -- infimas to infimas.
  ext : 1
  change UniformFun.filter α γ 𝓤[⨅ i, u i] = 𝓤[⨅ i, 𝒰(α, γ, u i)]
  rw [iInf_uniformity, iInf_uniformity]
  exact (UniformFun.gc α γ).u_iInf
#align uniform_fun.infi_eq UniformFun.iInf_eq

/-- If `u₁` and `u₂` are two uniform structures on `γ`, then
`𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂)`. -/
protected theorem inf_eq {u₁ u₂ : UniformSpace γ} :
    𝒰(α, γ, u₁ ⊓ u₂) = 𝒰(α, γ, u₁) ⊓ 𝒰(α, γ, u₂) := by
  -- This follows directly from the fact that the upper adjoint in a Galois connection maps
  -- infimas to infimas.
  rw [inf_eq_iInf, inf_eq_iInf, UniformFun.iInf_eq]
  refine' iInf_congr fun i => _
  cases i <;> rfl
#align uniform_fun.inf_eq UniformFun.inf_eq

/-- Post-composition by a uniform inducing function is
a uniform inducing function for the uniform structures of uniform convergence.

More precisely, if `f : γ → β` is uniform inducing,
then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is uniform inducing. -/
protected theorem postcomp_uniformInducing [UniformSpace γ] {f : γ → β} (hf : UniformInducing f) :
    UniformInducing (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) :=
  ⟨((UniformFun.hasBasis_uniformity _ _).comap _).eq_of_same_basis <|
    UniformFun.hasBasis_uniformity_of_basis _ _ (hf.basis_uniformity (𝓤 β).basis_sets)⟩
#align uniform_fun.postcomp_uniform_inducing UniformFun.postcomp_uniformInducing

/-- Post-composition by a uniform embedding is
a uniform embedding for the uniform structures of uniform convergence.

More precisely, if `f : γ → β` is a uniform embedding,
then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is a uniform embedding. -/
protected theorem postcomp_uniformEmbedding [UniformSpace γ] {f : γ → β} (hf : UniformEmbedding f) :
    UniformEmbedding (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) where
  toUniformInducing := UniformFun.postcomp_uniformInducing hf.toUniformInducing
  inj _ _ H := funext fun _ ↦ hf.inj (congrFun H _)

-- Porting note: had to add a type annotation at `((f ∘ ·) : ((α → γ) → (α → β)))`
/-- If `u` is a uniform structures on `β` and `f : γ → β`, then
`𝒰(α, γ, comap f u) = comap (fun g ↦ f ∘ g) 𝒰(α, γ, u₁)`. -/
protected theorem comap_eq {f : γ → β} :
    𝒰(α, γ, ‹UniformSpace β›.comap f) = 𝒰(α, β, _).comap (f ∘ ·) := by
  letI : UniformSpace γ := .comap f ‹_›
  exact (UniformFun.postcomp_uniformInducing (f := f) ⟨rfl⟩).comap_uniformSpace.symm
#align uniform_fun.comap_eq UniformFun.comap_eq

/-- Post-composition by a uniformly continuous function is uniformly continuous on `α →ᵤ β`.

More precisely, if `f : γ → β` is uniformly continuous, then `(fun g ↦ f ∘ g) : (α →ᵤ γ) → (α →ᵤ β)`
is uniformly continuous. -/
protected theorem postcomp_uniformContinuous [UniformSpace γ] {f : γ → β}
    (hf : UniformContinuous f) :
    UniformContinuous (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) := by
  -- This is a direct consequence of `UniformFun.comap_eq`
    refine uniformContinuous_iff.mpr ?_
    exact (UniformFun.mono (uniformContinuous_iff.mp hf)).trans_eq UniformFun.comap_eq
    -- Porting note: the original calc proof below gives a deterministic timeout
    --calc
    --  𝒰(α, γ, _) ≤ 𝒰(α, γ, ‹UniformSpace β›.comap f) :=
    --    UniformFun.mono (uniformContinuous_iff.mp hf)
    --  _ = 𝒰(α, β, _).comap (f ∘ ·) := @UniformFun.comap_eq α β γ _ f
#align uniform_fun.postcomp_uniform_continuous UniformFun.postcomp_uniformContinuous

/-- Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ γ) ≃ᵤ (α →ᵤ β)` by
post-composing. -/
protected def congrRight [UniformSpace γ] (e : γ ≃ᵤ β) : (α →ᵤ γ) ≃ᵤ (α →ᵤ β) :=
  { Equiv.piCongrRight fun _ => e.toEquiv with
    uniformContinuous_toFun := UniformFun.postcomp_uniformContinuous e.uniformContinuous
    uniformContinuous_invFun := UniformFun.postcomp_uniformContinuous e.symm.uniformContinuous }
#align uniform_fun.congr_right UniformFun.congrRight

/-- Pre-composition by any function is uniformly continuous for the uniform structures of
uniform convergence.

More precisely, for any `f : γ → α`, the function `(· ∘ f) : (α →ᵤ β) → (γ →ᵤ β)` is uniformly
continuous. -/
protected theorem precomp_uniformContinuous {f : γ → α} :
    UniformContinuous fun g : α →ᵤ β => ofFun (toFun g ∘ f) := by
  -- Here we simply go back to filter bases.
  rw [uniformContinuous_iff]
  change
    𝓤 (α →ᵤ β) ≤ (𝓤 (γ →ᵤ β)).comap (Prod.map (fun g : α →ᵤ β => g ∘ f) fun g : α →ᵤ β => g ∘ f)
  rw [(UniformFun.hasBasis_uniformity α β).le_basis_iff
      ((UniformFun.hasBasis_uniformity γ β).comap _)]
  exact fun U hU => ⟨U, hU, fun uv huv x => huv (f x)⟩
#align uniform_fun.precomp_uniform_continuous UniformFun.precomp_uniformContinuous

/-- Turn a bijection `γ ≃ α` into a uniform isomorphism
`(γ →ᵤ β) ≃ᵤ (α →ᵤ β)` by pre-composing. -/
protected def congrLeft (e : γ ≃ α) : (γ →ᵤ β) ≃ᵤ (α →ᵤ β) where
  toEquiv := e.arrowCongr (.refl _)
  uniformContinuous_toFun := UniformFun.precomp_uniformContinuous
  uniformContinuous_invFun := UniformFun.precomp_uniformContinuous
#align uniform_fun.congr_left UniformFun.congrLeft

/-- The natural map `UniformFun.toFun` 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. -/
protected theorem uniformContinuous_toFun : UniformContinuous (toFun : (α →ᵤ β) → α → β) := by
  -- By definition of the product uniform structure, this is just `uniform_continuous_eval`.
  rw [uniformContinuous_pi]
  intro x
  exact uniformContinuous_eval β x
#align uniform_fun.uniform_continuous_to_fun UniformFun.uniformContinuous_toFun

/-- The topology of uniform convergence is T₂. -/
instance [T2Space β] : T2Space (α →ᵤ β) :=
  .of_injective_continuous toFun.injective UniformFun.uniformContinuous_toFun.continuous

/-- The topology of uniform convergence indeed gives the same notion of convergence as
`TendstoUniformly`. -/
protected theorem tendsto_iff_tendstoUniformly {F : ι → α →ᵤ β} {f : α →ᵤ β} :
    Tendsto F p (𝓝 f) ↔ TendstoUniformly (toFun ∘ F) (toFun f) p := by
  rw [(UniformFun.hasBasis_nhds α β f).tendsto_right_iff, TendstoUniformly]
  simp only [mem_setOf, UniformFun.gen, Function.comp_def]
#align uniform_fun.tendsto_iff_tendsto_uniformly UniformFun.tendsto_iff_tendstoUniformly

/-- 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)`. -/
protected def uniformEquivProdArrow [UniformSpace 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
  -- `UniformFun.inf_eq` and `UniformFun.comap_eq`, which leaves us to check
  -- that some square commutes.
  Equiv.toUniformEquivOfUniformInducing (Equiv.arrowProdEquivProdArrow _ _ _) <| by
    constructor
    change
      comap (Prod.map (Equiv.arrowProdEquivProdArrow _ _ _) (Equiv.arrowProdEquivProdArrow _ _ _))
          _ = _
    simp_rw [UniformFun]
    rw [← uniformity_comap]
    congr
    unfold instUniformSpaceProd
    rw [UniformSpace.comap_inf, ← UniformSpace.comap_comap, ← UniformSpace.comap_comap]
    have := (@UniformFun.inf_eq α (β × γ)
      (UniformSpace.comap Prod.fst ‹_›) (UniformSpace.comap Prod.snd ‹_›)).symm
    rwa [UniformFun.comap_eq, UniformFun.comap_eq] at this
#align uniform_fun.uniform_equiv_prod_arrow UniformFun.uniformEquivProdArrow

-- the relevant diagram commutes by definition
variable (α) (δ : ι → Type*) [∀ i, UniformSpace (δ i)]

/-- 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`. -/
protected def uniformEquivPiComm : UniformEquiv (α →ᵤ ∀ i, δ i) (∀ 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
    -- `UniformFun.iInf_eq` and `UniformFun.comap_eq`, which leaves us to check
    -- that some square commutes.
    @Equiv.toUniformEquivOfUniformInducing
    _ _ 𝒰(α, ∀ i, δ i, Pi.uniformSpace δ)
    (@Pi.uniformSpace ι (fun i => α → δ i) fun i => 𝒰(α, δ i, _)) (Equiv.piComm _) <| by
      refine @UniformInducing.mk ?_ ?_ ?_ ?_ ?_ ?_
      change comap (Prod.map Function.swap Function.swap) _ = _
      rw [← uniformity_comap]
      congr
      unfold Pi.uniformSpace
      rw [UniformSpace.ofCoreEq_toCore, UniformSpace.ofCoreEq_toCore,
        UniformSpace.comap_iInf, UniformFun.iInf_eq]
      refine' iInf_congr fun i => _
      rw [← UniformSpace.comap_comap, UniformFun.comap_eq]
      rfl
#align uniform_fun.uniform_equiv_Pi_comm UniformFun.uniformEquivPiComm

-- Like in the previous lemma, the diagram actually commutes by definition

/-- The set of continuous functions is closed in the uniform convergence topology.
This is a simple wrapper over `TendstoUniformly.continuous`. -/
theorem isClosed_setOf_continuous [TopologicalSpace α] :
    IsClosed {f : α →ᵤ β | Continuous (toFun f)} := by
  refine isClosed_iff_forall_filter.2 fun f u _ hu huf ↦ ?_
  rw [← tendsto_id', UniformFun.tendsto_iff_tendstoUniformly] at huf
  exact huf.continuous (le_principal_iff.mp hu)

end UniformFun

namespace UniformOnFun

variable {α β : Type*} {γ ι : Type*}
variable {s s' : Set α} {x : α} {p : Filter ι} {g : ι → α}

local notation "𝒰(" α ", " β ", " u ")" => @UniformFun.uniformSpace α β u

/-- 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. -/
protected def 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, (toFun 𝔖 uv.1 x, toFun 𝔖 uv.2 x) ∈ V }
#align uniform_on_fun.gen UniformOnFun.gen

/-- For `S : Set α` and `V : Set (β × β)`, we have
`UniformOnFun.gen 𝔖 S V = (S.restrict × S.restrict) ⁻¹' (UniformFun.gen S β V)`.
This is the crucial fact for proving that the family `UniformOnFun.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)`
the uniform structure of `𝔖`-convergence, as defined in `UniformOnFun.uniformSpace`. -/
protected theorem gen_eq_preimage_restrict {𝔖} (S : Set α) (V : Set (β × β)) :
    UniformOnFun.gen 𝔖 S V =
      Prod.map (S.restrict ∘ UniformFun.toFun) (S.restrict ∘ UniformFun.toFun) ⁻¹'
        UniformFun.gen S β V := by
  ext uv
  exact ⟨fun h ⟨x, hx⟩ => h x hx, fun h x hx => h ⟨x, hx⟩⟩
#align uniform_on_fun.gen_eq_preimage_restrict UniformOnFun.gen_eq_preimage_restrict

/-- `UniformOnFun.gen` is antitone in the first argument and monotone in the second. -/
protected theorem gen_mono {𝔖} {S S' : Set α} {V V' : Set (β × β)} (hS : S' ⊆ S) (hV : V ⊆ V') :
    UniformOnFun.gen 𝔖 S V ⊆ UniformOnFun.gen 𝔖 S' V' := fun _uv h x hx => hV (h x <| hS hx)
#align uniform_on_fun.gen_mono UniformOnFun.gen_mono

/-- If `𝔖 : Set (Set α)` is nonempty and directed and `𝓑` is a filter basis on `β × β`, then the
family `UniformOnFun.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 filter is the uniformity of `α →ᵤ[𝔖] β`. -/
protected theorem isBasis_gen (𝔖 : Set (Set α)) (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖)
    (𝓑 : FilterBasis <| β × β) :
    IsBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓑) fun SV =>
      UniformOnFun.gen 𝔖 SV.1 SV.2 :=
  ⟨h.prod 𝓑.nonempty, fun {U₁V₁ U₂V₂} h₁ h₂ =>
    let ⟨U₃, hU₃, hU₁₃, hU₂₃⟩ := h' U₁V₁.1 h₁.1 U₂V₂.1 h₂.1
    let ⟨V₃, hV₃, hV₁₂₃⟩ := 𝓑.inter_sets h₁.2 h₂.2
    ⟨⟨U₃, V₃⟩,
      ⟨⟨hU₃, hV₃⟩, fun _ H =>
        ⟨fun x hx => (hV₁₂₃ <| H x <| hU₁₃ hx).1, fun x hx => (hV₁₂₃ <| H x <| hU₂₃ hx).2⟩⟩⟩⟩
#align uniform_on_fun.is_basis_gen UniformOnFun.isBasis_gen

variable (α β) [UniformSpace β] (𝔖 : Set (Set α))

/-- Uniform structure of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`,
declared as an instance on `α →ᵤ[𝔖] β`. It is defined as the infimum, for `S ∈ 𝔖`, of the pullback
by `S.restrict`, the map of restriction to `S`, of the uniform structure `𝒰(s, β, uβ)` on
`↥S →ᵤ β`. We will denote it `𝒱(α, β, 𝔖, uβ)`, where `uβ` is the uniform structure on `β`. -/
instance uniformSpace : UniformSpace (α →ᵤ[𝔖] β) :=
  ⨅ (s : Set α) (_ : s ∈ 𝔖),
    .comap (UniformFun.ofFun ∘ s.restrict ∘ UniformOnFun.toFun 𝔖) 𝒰(s, β, _)

local notation "𝒱(" α ", " β ", " 𝔖 ", " u ")" => @UniformOnFun.uniformSpace α β u 𝔖

/-- Topology of `𝔖`-convergence, i.e uniform convergence on the elements of `𝔖`, declared as an
instance on `α →ᵤ[𝔖] β`. -/
instance topologicalSpace : TopologicalSpace (α →ᵤ[𝔖] β) :=
  𝒱(α, β, 𝔖, _).toTopologicalSpace

/-- 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. -/
protected theorem topologicalSpace_eq :
    UniformOnFun.topologicalSpace α β 𝔖 =
      ⨅ (s : Set α) (_ : s ∈ 𝔖), TopologicalSpace.induced
        (UniformFun.ofFun ∘ s.restrict ∘ toFun 𝔖) (UniformFun.topologicalSpace s β) := by
  simp only [UniformOnFun.topologicalSpace, UniformSpace.toTopologicalSpace_iInf]
  rfl
#align uniform_on_fun.topological_space_eq UniformOnFun.topologicalSpace_eq

protected theorem hasBasis_uniformity_of_basis_aux₁ {p : ι → Prop} {s : ι → Set (β × β)}
    (hb : HasBasis (𝓤 β) p s) (S : Set α) :
    (@uniformity (α →ᵤ[𝔖] β) ((UniformFun.uniformSpace S β).comap S.restrict)).HasBasis p fun i =>
      UniformOnFun.gen 𝔖 S (s i) := by
  simp_rw [UniformOnFun.gen_eq_preimage_restrict, uniformity_comap]
  exact (UniformFun.hasBasis_uniformity_of_basis S β hb).comap _
#align uniform_on_fun.has_basis_uniformity_of_basis_aux₁ UniformOnFun.hasBasis_uniformity_of_basis_aux₁

protected theorem hasBasis_uniformity_of_basis_aux₂ (h : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop}
    {s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
    DirectedOn
      ((fun s : Set α => (UniformFun.uniformSpace s β).comap (s.restrict : (α →ᵤ β) → s →ᵤ β)) ⁻¹'o
        GE.ge)
      𝔖 :=
  h.mono fun _ _ hst =>
    ((UniformOnFun.hasBasis_uniformity_of_basis_aux₁ α β 𝔖 hb _).le_basis_iff
          (UniformOnFun.hasBasis_uniformity_of_basis_aux₁ α β 𝔖 hb _)).mpr
      fun V hV => ⟨V, hV, UniformOnFun.gen_mono hst subset_rfl⟩
#align uniform_on_fun.has_basis_uniformity_of_basis_aux₂ UniformOnFun.hasBasis_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. -/
protected theorem hasBasis_uniformity_of_basis (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖)
    {p : ι → Prop} {s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
    (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
      UniformOnFun.gen 𝔖 Si.1 (s Si.2) := by
  simp only [iInf_uniformity]
  exact
    hasBasis_biInf_of_directed h (fun S => UniformOnFun.gen 𝔖 S ∘ s) _
      (fun S _hS => UniformOnFun.hasBasis_uniformity_of_basis_aux₁ α β 𝔖 hb S)
      (UniformOnFun.hasBasis_uniformity_of_basis_aux₂ α β 𝔖 h' hb)
#align uniform_on_fun.has_basis_uniformity_of_basis UniformOnFun.hasBasis_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. -/
protected theorem hasBasis_uniformity (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖) :
    (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β) fun SV =>
      UniformOnFun.gen 𝔖 SV.1 SV.2 :=
  UniformOnFun.hasBasis_uniformity_of_basis α β 𝔖 h h' (𝓤 β).basis_sets
#align uniform_on_fun.has_basis_uniformity UniformOnFun.hasBasis_uniformity

variable {α β}

/-- Let `t i` be a nonempty directed subfamily of `𝔖`
such that every `s ∈ 𝔖` is included in some `t i`.
Let `V` bounded by `p` be a basis of entourages of `β`.

Then `UniformOnFun.gen 𝔖 (t i) (V j)` bounded by `p j` is a basis of entourages of `α →ᵤ[𝔖] β`. -/
protected theorem hasBasis_uniformity_of_covering_of_basis {ι ι' : Type*} [Nonempty ι]
    {t : ι → Set α} {p : ι' → Prop} {V : ι' → Set (β × β)} (ht : ∀ i, t i ∈ 𝔖)
    (hdir : Directed (· ⊆ ·) t) (hex : ∀ s ∈ 𝔖, ∃ i, s ⊆ t i) (hb : HasBasis (𝓤 β) p V) :
    (𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun i : ι × ι' ↦ p i.2) fun i ↦
      UniformOnFun.gen 𝔖 (t i.1) (V i.2) := by
  have hne : 𝔖.Nonempty := (range_nonempty t).mono (range_subset_iff.2 ht)
  have hd : DirectedOn (· ⊆ ·) 𝔖 := fun s₁ hs₁ s₂ hs₂ ↦ by
    rcases hex s₁ hs₁, hex s₂ hs₂ with ⟨⟨i₁, his₁⟩, i₂, his₂⟩
    rcases hdir i₁ i₂ with ⟨i, hi₁, hi₂⟩
    exact ⟨t i, ht _, his₁.trans hi₁, his₂.trans hi₂⟩
  refine (UniformOnFun.hasBasis_uniformity_of_basis α β 𝔖 hne hd hb).to_hasBasis
    (fun ⟨s, i'⟩ ⟨hs, hi'⟩ ↦ ?_) fun ⟨i, i'⟩ hi' ↦ ⟨(t i, i'), ⟨ht i, hi'⟩, Subset.rfl⟩
  rcases hex s hs with ⟨i, hi⟩
  exact ⟨(i, i'), hi', UniformOnFun.gen_mono hi Subset.rfl⟩

/-- If `t n` is a monotone sequence of sets in `𝔖`
such that each `s ∈ 𝔖` is included in some `t n`
and `V n` is an antitone basis of entourages of `β`,
then `UniformOnFun.gen 𝔖 (t n) (V n)` is an antitone basis of entourages of `α →ᵤ[𝔖] β`. -/
protected theorem hasAntitoneBasis_uniformity {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)]
    {t : ι → Set α} {V : ι → Set (β × β)}
    (ht : ∀ n, t n ∈ 𝔖) (hmono : Monotone t) (hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n)
    (hb : HasAntitoneBasis (𝓤 β) V) :
    (𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n ↦ UniformOnFun.gen 𝔖 (t n) (V n) := by
  have := hb.nonempty
  refine ⟨(UniformOnFun.hasBasis_uniformity_of_covering_of_basis 𝔖
    ht hmono.directed_le hex hb.1).to_hasBasis ?_ fun i _ ↦ ⟨(i, i), trivial, Subset.rfl⟩, ?_⟩
  · rintro ⟨k, l⟩ -
    rcases directed_of (· ≤ ·) k l with ⟨n, hkn, hln⟩
    exact ⟨n, trivial, UniformOnFun.gen_mono (hmono hkn) (hb.2 <| hln)⟩
  · exact fun k l h ↦ UniformOnFun.gen_mono (hmono h) (hb.2 h)

protected theorem isCountablyGenerated_uniformity [IsCountablyGenerated (𝓤 β)] {t : ℕ → Set α}
    (ht : ∀ n, t n ∈ 𝔖) (hmono : Monotone t) (hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n) :
    IsCountablyGenerated (𝓤 (α →ᵤ[𝔖] β)) :=
  let ⟨_V, hV⟩ := exists_antitone_basis (𝓤 β)
  (UniformOnFun.hasAntitoneBasis_uniformity 𝔖 ht hmono hex hV).isCountablyGenerated

variable (α β)

/-- 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`. -/
protected theorem hasBasis_nhds_of_basis (f : α →ᵤ[𝔖] β) (h : 𝔖.Nonempty)
    (h' : DirectedOn (· ⊆ ·) 𝔖) {p : ι → Prop} {s : ι → Set (β × β)} (hb : HasBasis (𝓤 β) p s) :
    (𝓝 f).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
      { g | (g, f) ∈ UniformOnFun.gen 𝔖 Si.1 (s Si.2) } :=
  letI : UniformSpace (α → β) := UniformOnFun.uniformSpace α β 𝔖
  nhds_basis_uniformity (UniformOnFun.hasBasis_uniformity_of_basis α β 𝔖 h h' hb)
#align uniform_on_fun.has_basis_nhds_of_basis UniformOnFun.hasBasis_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. -/
protected theorem hasBasis_nhds (f : α →ᵤ[𝔖] β) (h : 𝔖.Nonempty) (h' : DirectedOn (· ⊆ ·) 𝔖) :
    (𝓝 f).HasBasis (fun SV : Set α × Set (β × β) => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓤 β) fun SV =>
      { g | (g, f) ∈ UniformOnFun.gen 𝔖 SV.1 SV.2 } :=
  UniformOnFun.hasBasis_nhds_of_basis α β 𝔖 f h h' (Filter.basis_sets _)
#align uniform_on_fun.has_basis_nhds UniformOnFun.hasBasis_nhds

/-- If `S ∈ 𝔖`, then the restriction to `S` is a uniformly continuous map from `α →ᵤ[𝔖] β` to
`↥S →ᵤ β`. -/
protected theorem uniformContinuous_restrict (h : s ∈ 𝔖) :
    UniformContinuous (UniformFun.ofFun ∘ (s.restrict : (α → β) → s → β) ∘ toFun 𝔖) := by
  change _ ≤ _
  simp only [UniformOnFun.uniformSpace, map_le_iff_le_comap, iInf_uniformity]
  exact iInf₂_le s h
#align uniform_on_fun.uniform_continuous_restrict UniformOnFun.uniformContinuous_restrict

variable {α}

/-- A version of `UniformOnFun.hasBasis_uniformity_of_basis`
with weaker conclusion and weaker assumptions.

We make no assumptions about the set `𝔖`
but conclude only that the uniformity is equal to some indexed infimum. -/
protected theorem uniformity_eq_of_basis {ι : Sort*} {p : ι → Prop} {V : ι → Set (β × β)}
    (h : (𝓤 β).HasBasis p V) :
    𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ (i) (_ : p i), 𝓟 (UniformOnFun.gen 𝔖 s (V i)) := by
  simp_rw [iInf_uniformity, uniformity_comap,
    (UniformFun.hasBasis_uniformity_of_basis _ _ h).eq_biInf, comap_iInf, comap_principal,
    Function.comp_apply, UniformFun.gen, Subtype.forall, UniformOnFun.gen, preimage_setOf_eq,
    Prod_map, Function.comp_apply, UniformFun.toFun_ofFun, restrict_apply]

protected theorem uniformity_eq : 𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 (UniformOnFun.gen 𝔖 s V) :=
  UniformOnFun.uniformity_eq_of_basis _ _ (𝓤 β).basis_sets

protected theorem gen_mem_uniformity (hs : s ∈ 𝔖) {V : Set (β × β)} (hV : V ∈ 𝓤 β) :
    UniformOnFun.gen 𝔖 s V ∈ 𝓤 (α →ᵤ[𝔖] β) := by
  rw [UniformOnFun.uniformity_eq]
  apply_rules [mem_iInf_of_mem, mem_principal_self]

/-- A version of `UniformOnFun.hasBasis_nhds_of_basis`
with weaker conclusion and weaker assumptions.

We make no assumptions about the set `𝔖`
but conclude only that the neighbourhoods filter is equal to some indexed infimum. -/
protected theorem nhds_eq_of_basis {ι : Sort*} {p : ι → Prop} {V : ι → Set (β × β)}
    (h : (𝓤 β).HasBasis p V) (f : α →ᵤ[𝔖] β) :
    𝓝 f = ⨅ s ∈ 𝔖, ⨅ (i) (_ : p i), 𝓟 {g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V i} := by
  simp_rw [nhds_eq_comap_uniformity, UniformOnFun.uniformity_eq_of_basis _ _ h, comap_iInf,
    comap_principal, UniformOnFun.gen, preimage_setOf_eq]

protected theorem nhds_eq (f : α →ᵤ[𝔖] β) :
    𝓝 f = ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 {g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V} :=
  UniformOnFun.nhds_eq_of_basis _ _ (𝓤 β).basis_sets f

protected theorem gen_mem_nhds (f : α →ᵤ[𝔖] β) (hs : s ∈ 𝔖) {V : Set (β × β)} (hV : V ∈ 𝓤 β) :
    {g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V} ∈ 𝓝 f := by
  rw [UniformOnFun.nhds_eq]
  apply_rules [mem_iInf_of_mem, mem_principal_self]

theorem uniformContinuous_ofUniformFun :
    UniformContinuous fun f : α →ᵤ β ↦ ofFun 𝔖 (UniformFun.toFun f) := by
  simp only [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf, tendsto_principal,
    (UniformFun.hasBasis_uniformity _ _).eventually_iff]
  exact fun _ _ U hU ↦ ⟨U, hU, fun f hf x _ ↦ hf x⟩

/-- The uniformity on `α →ᵤ[𝔖] β` is the same as the uniformity on `α →ᵤ β`,
provided that `Set.univ ∈ 𝔖`.

Here we formulate it as a `UniformEquiv`. -/
def uniformEquivUniformFun (h : univ ∈ 𝔖) : (α →ᵤ[𝔖] β) ≃ᵤ (α →ᵤ β) where
  toFun f := UniformFun.ofFun <| toFun _ f
  invFun f := ofFun _ <| UniformFun.toFun f
  left_inv _ := rfl
  right_inv _ := rfl
  uniformContinuous_toFun := by
    simp only [UniformContinuous, (UniformFun.hasBasis_uniformity _ _).tendsto_right_iff]
    intro U hU
    filter_upwards [UniformOnFun.gen_mem_uniformity _ _ h hU] with f hf x using hf x (mem_univ _)
  uniformContinuous_invFun := uniformContinuous_ofUniformFun _ _

/-- 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)`. -/
protected theorem mono ⦃u₁ u₂ : UniformSpace 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) := iInf_le_iInf_of_subset h𝔖
    _ ≤ 𝒱(α, γ, 𝔖₂, u₂) := iInf₂_mono fun _i _hi => UniformSpace.comap_mono <| UniformFun.mono hu
#align uniform_on_fun.mono UniformOnFun.mono

/-- If `x : α` is in some `S ∈ 𝔖`, then evaluation at `x` is uniformly continuous on
`α →ᵤ[𝔖] β`. -/
theorem uniformContinuous_eval_of_mem {x : α} (hxs : x ∈ s) (hs : s ∈ 𝔖) :
    UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
  (UniformFun.uniformContinuous_eval β (⟨x, hxs⟩ : s)).comp
    (UniformOnFun.uniformContinuous_restrict α β 𝔖 hs)
#align uniform_on_fun.uniform_continuous_eval_of_mem UniformOnFun.uniformContinuous_eval_of_mem

theorem uniformContinuous_eval_of_mem_sUnion {x : α} (hx : x ∈ ⋃₀ 𝔖) :
    UniformContinuous ((Function.eval x : (α → β) → β) ∘ toFun 𝔖) :=
  let ⟨_s, hs, hxs⟩ := hx
  uniformContinuous_eval_of_mem _ _ hxs hs

variable {β} {𝔖}

/-- 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)`. -/
protected theorem iInf_eq {u : ι → UniformSpace γ} :
    𝒱(α, γ, 𝔖, ⨅ i, u i) = ⨅ i, 𝒱(α, γ, 𝔖, u i) := by
  simp_rw [UniformOnFun.uniformSpace, UniformFun.iInf_eq, UniformSpace.comap_iInf]
  rw [iInf_comm]
  exact iInf_congr fun s => iInf_comm
#align uniform_on_fun.infi_eq UniformOnFun.iInf_eq

/-- 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)`. -/
protected theorem inf_eq {u₁ u₂ : UniformSpace γ} :
    𝒱(α, γ, 𝔖, u₁ ⊓ u₂) = 𝒱(α, γ, 𝔖, u₁) ⊓ 𝒱(α, γ, 𝔖, u₂) := by
  rw [inf_eq_iInf, inf_eq_iInf, UniformOnFun.iInf_eq]
  refine' iInf_congr fun i => _
  cases i <;> rfl
#align uniform_on_fun.inf_eq UniformOnFun.inf_eq

/-- If `u` is a uniform structure on `β` and `f : γ → β`, then
`𝒱(α, γ, 𝔖, comap f u) = comap (fun g ↦ f ∘ g) 𝒱(α, γ, 𝔖, u₁)`. -/
protected theorem comap_eq {f : γ → β} :
    𝒱(α, γ, 𝔖, ‹UniformSpace β›.comap f) = 𝒱(α, β, 𝔖, _).comap (f ∘ ·) := by
  -- We reduce this to `UniformFun.comap_eq` using the fact that `comap` distributes
  -- on `iInf`.
  simp_rw [UniformOnFun.uniformSpace, UniformSpace.comap_iInf, UniformFun.comap_eq, ←
    UniformSpace.comap_comap]
  -- By definition, `∀ S ∈ 𝔖, (f ∘ —) ∘ S.restrict = S.restrict ∘ (f ∘ —)`.
  rfl
#align uniform_on_fun.comap_eq UniformOnFun.comap_eq

/-- 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
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous. -/
protected theorem postcomp_uniformContinuous [UniformSpace γ] {f : γ → β}
    (hf : UniformContinuous f) : UniformContinuous (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
  -- This is a direct consequence of `UniformOnFun.comap_eq`
  rw [uniformContinuous_iff]
  exact (UniformOnFun.mono (uniformContinuous_iff.mp hf) subset_rfl).trans_eq UniformOnFun.comap_eq
#align uniform_on_fun.postcomp_uniform_continuous UniformOnFun.postcomp_uniformContinuous

/-- 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
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing. -/
protected theorem postcomp_uniformInducing [UniformSpace γ] {f : γ → β} (hf : UniformInducing f) :
    UniformInducing (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
  -- This is a direct consequence of `UniformOnFun.comap_eq`
  constructor
  replace hf : (𝓤 β).comap (Prod.map f f) = _ := hf.comap_uniformity
  change comap (Prod.map (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖)) _ = _
  rw [← uniformity_comap] at hf ⊢
  congr
  rw [← UniformSpace.ext hf, UniformOnFun.comap_eq]
  rfl
#align uniform_on_fun.postcomp_uniform_inducing UniformOnFun.postcomp_uniformInducing

/-- Post-composition by a uniform embedding is a uniform embedding for the
uniform structures of `𝔖`-convergence.

More precisely, if `f : γ → β` is a uniform embedding, then
`(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform embedding. -/
protected theorem postcomp_uniformEmbedding [UniformSpace γ] {f : γ → β} (hf : UniformEmbedding f) :
    UniformEmbedding (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) where
  toUniformInducing := UniformOnFun.postcomp_uniformInducing hf.toUniformInducing
  inj _ _ H := funext fun _ ↦ hf.inj (congrFun H _)

/-- Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
by post-composing. -/
protected def congrRight [UniformSpace γ] (e : γ ≃ᵤ β) : (α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) :=
  { Equiv.piCongrRight fun _a => e.toEquiv with
    uniformContinuous_toFun := UniformOnFun.postcomp_uniformContinuous e.uniformContinuous
    uniformContinuous_invFun := UniformOnFun.postcomp_uniformContinuous e.symm.uniformContinuous }
#align uniform_on_fun.congr_right UniformOnFun.congrRight

/-- Let `f : γ → α`, `𝔖 : Set (Set α)`, `𝔗 : Set (Set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`.
Then, the function `(fun 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 prove that `𝒱(α, β, 𝔖, uβ) = 𝒱(α, β, 𝔖', uβ)` where `𝔖'` is the
***noncovering*** bornology generated by `𝔖`. -/
protected theorem precomp_uniformContinuous {𝔗 : Set (Set γ)} {f : γ → α}
    (hf : MapsTo (f '' ·) 𝔗 𝔖) :
    UniformContinuous fun g : α →ᵤ[𝔖] β => ofFun 𝔗 (toFun 𝔖 g ∘ f) := by
  -- This follows from the fact that `(· ∘ f) × (· ∘ f)` maps `gen (f '' t) V` to `gen t V`.
  simp_rw [UniformContinuous, UniformOnFun.uniformity_eq, tendsto_iInf]
  refine fun t ht V hV ↦ tendsto_iInf' (f '' t) <| tendsto_iInf' (hf ht) <|
    tendsto_iInf' V <| tendsto_iInf' hV ?_
  simpa only [tendsto_principal_principal, UniformOnFun.gen] using fun _ ↦ forall_mem_image.1
#align uniform_on_fun.precomp_uniform_continuous UniformOnFun.precomp_uniformContinuous

/-- 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. -/
protected def congrLeft {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) :=
  { Equiv.arrowCongr e (Equiv.refl _) with
    uniformContinuous_toFun := UniformOnFun.precomp_uniformContinuous fun s hs ↦ by
      change e.symm '' s ∈ 𝔗
      rw [← preimage_equiv_eq_image_symm]
      exact he' hs
    uniformContinuous_invFun := UniformOnFun.precomp_uniformContinuous he }
#align uniform_on_fun.congr_left UniformOnFun.congrLeft

/-- If `𝔖` covers `α`, then the topology of `𝔖`-convergence is T₂. -/
theorem t2Space_of_covering [T2Space β] (h : ⋃₀ 𝔖 = univ) : T2Space (α →ᵤ[𝔖] β) where
  t2 f g hfg := by
    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 (uniformContinuous_eval_of_mem β 𝔖 hxs hs).continuous hx
#align uniform_on_fun.t2_space_of_covering UniformOnFun.t2Space_of_covering

/-- The restriction map from `α →ᵤ[𝔖] β` to `⋃₀ 𝔖 → β` is uniformly continuous. -/
theorem uniformContinuous_restrict_toFun :
    UniformContinuous ((⋃₀ 𝔖).restrict ∘ toFun 𝔖 : (α →ᵤ[𝔖] β) → ⋃₀ 𝔖 → β) := by
  rw [uniformContinuous_pi]
  intro ⟨x, hx⟩
  obtain ⟨s : Set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := mem_sUnion.mpr hx
  exact uniformContinuous_eval_of_mem β 𝔖 hxs hs

/-- If `𝔖` covers `α`, the natural map `UniformOnFun.toFun` 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. -/
protected theorem uniformContinuous_toFun (h : ⋃₀ 𝔖 = univ) :
    UniformContinuous (toFun 𝔖 : (α →ᵤ[𝔖] β) → α → β) := by
  rw [uniformContinuous_pi]
  intro x
  obtain ⟨s : Set α, hs : s ∈ 𝔖, hxs : x ∈ s⟩ := sUnion_eq_univ_iff.mp h x
  exact uniformContinuous_eval_of_mem β 𝔖 hxs hs
#align uniform_on_fun.uniform_continuous_to_fun UniformOnFun.uniformContinuous_toFun

/-- Convergence in the topology of `𝔖`-convergence means uniform convergence on `S` (in the sense
of `TendstoUniformlyOn`) for all `S ∈ 𝔖`. -/
protected theorem tendsto_iff_tendstoUniformlyOn {F : ι → α →ᵤ[𝔖] β} {f : α →ᵤ[𝔖] β} :
    Tendsto F p (𝓝 f) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (toFun 𝔖 ∘ F) (toFun 𝔖 f) p s := by
  simp only [UniformOnFun.nhds_eq, tendsto_iInf, tendsto_principal, TendstoUniformlyOn,
    Function.comp_apply, mem_setOf]
#align uniform_on_fun.tendsto_iff_tendsto_uniformly_on UniformOnFun.tendsto_iff_tendstoUniformlyOn

protected lemma continuous_rng_iff {X : Type*} [TopologicalSpace X] {f : X → (α →ᵤ[𝔖] β)} :
    Continuous f ↔ ∀ s ∈ 𝔖,
      Continuous (UniformFun.ofFun ∘ s.restrict ∘ UniformOnFun.toFun 𝔖 ∘ f) := by
  simp only [continuous_iff_continuousAt, ContinuousAt,
    UniformOnFun.tendsto_iff_tendstoUniformlyOn, UniformFun.tendsto_iff_tendstoUniformly,
    tendstoUniformlyOn_iff_tendstoUniformly_comp_coe, @forall_swap X, Function.comp_apply,
    Function.comp_def, restrict_eq, UniformFun.toFun_ofFun]

instance [CompleteSpace β] : CompleteSpace (α →ᵤ[𝔖] β) := by
  rcases isEmpty_or_nonempty β
  · infer_instance
  · refine ⟨fun {F} hF ↦ ?_⟩
    have := hF.1
    have : ∀ x ∈ ⋃₀ 𝔖, ∃ y : β, Tendsto (toFun 𝔖 · x) F (𝓝 y) := fun x hx ↦
      CompleteSpace.complete (hF.map (uniformContinuous_eval_of_mem_sUnion _ _ hx))
    choose! g hg using this
    use ofFun 𝔖 g
    simp_rw [UniformOnFun.nhds_eq_of_basis _ _ uniformity_hasBasis_closed, le_iInf₂_iff,
      le_principal_iff]
    intro s hs U ⟨hU, hUc⟩
    rcases cauchy_iff.mp hF |>.2 _ <| UniformOnFun.gen_mem_uniformity _ _ hs hU
      with ⟨V, hV, hVU⟩
    filter_upwards [hV] with f hf x hx
    refine hUc.mem_of_tendsto ((hg x ⟨s, hs, hx⟩).prod_mk_nhds tendsto_const_nhds) ?_
    filter_upwards [hV] with g' hg' using hVU (mk_mem_prod hg' hf) _ hx

/-- 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)`. -/
protected def uniformEquivProdArrow [UniformSpace 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
  -- `UniformOnFun.inf_eq` and `UniformOnFun.comap_eq`,
  -- which leaves us to check that some square commutes.
  -- We could also deduce this from `UniformFun.uniformEquivProdArrow`,
  -- but it turns out to be more annoying.
  ((UniformOnFun.ofFun 𝔖).symm.trans <|
    (Equiv.arrowProdEquivProdArrow _ _ _).trans <|
      (UniformOnFun.ofFun 𝔖).prodCongr (UniformOnFun.ofFun 𝔖)).toUniformEquivOfUniformInducing <| by
      constructor
      rw [uniformity_prod, comap_inf, comap_comap, comap_comap]
      have H := @UniformOnFun.inf_eq α (β × γ) 𝔖
        (UniformSpace.comap Prod.fst ‹_›) (UniformSpace.comap Prod.snd ‹_›)
      apply_fun (fun u ↦ @uniformity (α →ᵤ[𝔖] β × γ) u) at H
      convert H.symm using 1
      rw [UniformOnFun.comap_eq, UniformOnFun.comap_eq]
      erw [inf_uniformity]
      rw [uniformity_comap, uniformity_comap]
      rfl
#align uniform_on_fun.uniform_equiv_prod_arrow UniformOnFun.uniformEquivProdArrow

-- the relevant diagram commutes by definition
variable (𝔖) (δ : ι → Type*) [∀ i, UniformSpace (δ i)]

/-- 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`. -/
protected def uniformEquivPiComm : (α →ᵤ[𝔖] ((i:ι) → δ i)) ≃ᵤ ((i:ι) → 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
  -- `UniformOnFun.iInf_eq` and `UniformOnFun.comap_eq`,
  -- which leaves us to check that some square commutes.
  -- We could also deduce this from `UniformFun.uniformEquivPiComm`, but it turns out
  -- to be more annoying.
  @Equiv.toUniformEquivOfUniformInducing (α →ᵤ[𝔖] ((i:ι) → δ i)) ((i:ι) → α →ᵤ[𝔖] δ i)
      _ _ (Equiv.piComm _) <| by
    constructor
    change comap (Prod.map Function.swap Function.swap) _ = _
    erw [← uniformity_comap]
    congr
    rw [Pi.uniformSpace, UniformSpace.ofCoreEq_toCore, Pi.uniformSpace,
      UniformSpace.ofCoreEq_toCore, UniformSpace.comap_iInf, UniformOnFun.iInf_eq]
    refine' iInf_congr fun i => _
    rw [← UniformSpace.comap_comap, UniformOnFun.comap_eq]
    rfl
#align uniform_on_fun.uniform_equiv_Pi_comm UniformOnFun.uniformEquivPiComm

-- Like in the previous lemma, the diagram actually commutes by definition

/-- Suppose that the topology on `α` is defined by its restrictions to the sets of `𝔖`.

Then the set of continuous functions is closed
in the topology of uniform convergence on the sets of `𝔖`. -/
theorem isClosed_setOf_continuous_of_le [t : TopologicalSpace α]
    (h : t ≤ ⨆ s ∈ 𝔖, .coinduced (Subtype.val : s → α) inferInstance) :
    IsClosed {f : α →ᵤ[𝔖] β | Continuous (toFun 𝔖 f)} := by
  refine isClosed_iff_forall_filter.2 fun f u _ hu huf ↦ ?_
  rw [← tendsto_id', UniformOnFun.tendsto_iff_tendstoUniformlyOn] at huf
  have hcont : ∀ s ∈ 𝔖, ContinuousOn f s := fun s hs ↦
    (huf s hs).continuousOn <| hu fun _ ↦ Continuous.continuousOn
  refine continuous_le_dom h ?_
  simpa only [continuous_iSup_dom, continuous_coinduced_dom] using fun s hs ↦ (hcont s hs).restrict

end UniformOnFun

namespace UniformFun

instance {α β : Type*} [UniformSpace β] [CompleteSpace β] : CompleteSpace (α →ᵤ β) :=
  (UniformOnFun.uniformEquivUniformFun β {univ} (mem_singleton _)).completeSpace_iff.1 inferInstance

end UniformFun