formal/lean/mathlib/topology/compact_open.lean

/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton
-/
import tactic.tidy
import topology.continuous_function.basic
import topology.homeomorph
import topology.subset_properties
import topology.maps

/-!
# The compact-open topology

In this file, we define the compact-open topology on the set of continuous maps between two
topological spaces.

## Main definitions

* `compact_open` is the compact-open topology on `C(α, β)`. It is declared as an instance.
* `continuous_map.coev` is the coevaluation map `β → C(α, β × α)`. It is always continuous.
* `continuous_map.curry` is the currying map `C(α × β, γ) → C(α, C(β, γ))`. This map always exists
  and it is continuous as long as `α × β` is locally compact.
* `continuous_map.uncurry` is the uncurrying map `C(α, C(β, γ)) → C(α × β, γ)`. For this map to
  exist, we need `β` to be locally compact. If `α` is also locally compact, then this map is
  continuous.
* `homeomorph.curry` combines the currying and uncurrying operations into a homeomorphism
  `C(α × β, γ) ≃ₜ C(α, C(β, γ))`. This homeomorphism exists if `α` and `β` are locally compact.


## Tags

compact-open, curry, function space
-/

open set
open_locale topological_space

namespace continuous_map

section compact_open
variables {α : Type*} {β : Type*} {γ : Type*}
variables [topological_space α] [topological_space β] [topological_space γ]

/-- A generating set for the compact-open topology (when `s` is compact and `u` is open). -/
def compact_open.gen (s : set α) (u : set β) : set C(α,β) := {f | f '' s ⊆ u}

@[simp] lemma gen_empty (u : set β) : compact_open.gen (∅ : set α) u = set.univ :=
set.ext (λ f, iff_true_intro ((congr_arg (⊆ u) (image_empty f)).mpr u.empty_subset))

@[simp] lemma gen_univ (s : set α) : compact_open.gen s (set.univ : set β) = set.univ :=
set.ext (λ f, iff_true_intro (f '' s).subset_univ)

@[simp] lemma gen_inter (s : set α) (u v : set β) :
  compact_open.gen s (u ∩ v) = compact_open.gen s u ∩ compact_open.gen s v :=
set.ext (λ f, subset_inter_iff)

@[simp] lemma gen_union (s t : set α) (u : set β) :
  compact_open.gen (s ∪ t) u = compact_open.gen s u ∩ compact_open.gen t u :=
set.ext (λ f, (iff_of_eq (congr_arg (⊆ u) (image_union f s t))).trans union_subset_iff)

lemma gen_empty_right {s : set α} (h : s.nonempty) : compact_open.gen s (∅ : set β) = ∅ :=
eq_empty_of_forall_not_mem $ λ f, (h.image _).not_subset_empty

-- The compact-open topology on the space of continuous maps α → β.
instance compact_open : topological_space C(α, β) :=
topological_space.generate_from
  {m | ∃ (s : set α) (hs : is_compact s) (u : set β) (hu : is_open u), m = compact_open.gen s u}

protected lemma is_open_gen {s : set α} (hs : is_compact s) {u : set β} (hu : is_open u) :
  is_open (compact_open.gen s u) :=
topological_space.generate_open.basic _ (by dsimp [mem_set_of_eq]; tauto)

section functorial

variables (g : C(β, γ))

private lemma preimage_gen {s : set α} (hs : is_compact s) {u : set γ} (hu : is_open u) :
  continuous_map.comp g ⁻¹' (compact_open.gen s u) = compact_open.gen s (g ⁻¹' u) :=
begin
  ext ⟨f, _⟩,
  change g ∘ f '' s ⊆ u ↔ f '' s ⊆ g ⁻¹' u,
  rw [image_comp, image_subset_iff]
end

/-- C(α, -) is a functor. -/
lemma continuous_comp : continuous (continuous_map.comp g : C(α, β) → C(α, γ)) :=
continuous_generated_from $ assume m ⟨s, hs, u, hu, hm⟩,
  by rw [hm, preimage_gen g hs hu]; exact continuous_map.is_open_gen hs (hu.preimage g.2)

variable (f : C(α, β))

private lemma image_gen {s : set α} (hs : is_compact s) {u : set γ} (hu : is_open u) :
  (λ g : C(β, γ), g.comp f) ⁻¹' compact_open.gen s u = compact_open.gen (f '' s) u :=
begin
  ext ⟨g, _⟩,
  change g ∘ f '' s ⊆ u ↔ g '' (f '' s) ⊆ u,
  rw set.image_comp,
end

/-- C(-, γ) is a functor. -/
lemma continuous_comp_left : continuous (λ g, g.comp f : C(β, γ) → C(α, γ)) :=
continuous_generated_from $ assume m ⟨s, hs, u, hu, hm⟩,
  by { rw [hm, image_gen f hs hu], exact continuous_map.is_open_gen (hs.image f.2) hu }

end functorial

section ev

variables {α β}

/-- The evaluation map `C(α, β) × α → β` is continuous if `α` is locally compact.

See also `continuous_map.continuous_eval` -/
lemma continuous_eval' [locally_compact_space α] : continuous (λ p : C(α, β) × α, p.1 p.2) :=
continuous_iff_continuous_at.mpr $ assume ⟨f, x⟩ n hn,
  let ⟨v, vn, vo, fxv⟩ := mem_nhds_iff.mp hn in
  have v ∈ 𝓝 (f x), from is_open.mem_nhds vo fxv,
  let ⟨s, hs, sv, sc⟩ :=
    locally_compact_space.local_compact_nhds x (f ⁻¹' v)
      (f.continuous.tendsto x this) in
  let ⟨u, us, uo, xu⟩ := mem_nhds_iff.mp hs in
  show (λ p : C(α, β) × α, p.1 p.2) ⁻¹' n ∈ 𝓝 (f, x), from
  let w := compact_open.gen s v ×ˢ u in
  have w ⊆ (λ p : C(α, β) × α, p.1 p.2) ⁻¹' n, from assume ⟨f', x'⟩ ⟨hf', hx'⟩, calc
    f' x' ∈ f' '' s  : mem_image_of_mem f' (us hx')
    ...       ⊆ v            : hf'
    ...       ⊆ n            : vn,
  have is_open w, from (continuous_map.is_open_gen sc vo).prod uo,
  have (f, x) ∈ w, from ⟨image_subset_iff.mpr sv, xu⟩,
  mem_nhds_iff.mpr ⟨w, by assumption, by assumption, by assumption⟩

/-- See also `continuous_map.continuous_eval_const` -/
lemma continuous_eval_const' [locally_compact_space α] (a : α) : continuous (λ f : C(α, β), f a) :=
continuous_eval'.comp (continuous_id.prod_mk continuous_const)

/-- See also `continuous_map.continuous_coe` -/
lemma continuous_coe' [locally_compact_space α] : @continuous (C(α, β)) (α → β) _ _ coe_fn :=
continuous_pi continuous_eval_const'

instance [t2_space β] : t2_space C(α, β) :=
⟨ begin
    intros f₁ f₂ h,
    obtain ⟨x, hx⟩ := not_forall.mp (mt (fun_like.ext f₁ f₂) h),
    obtain ⟨u, v, hu, hv, hxu, hxv, huv⟩ := t2_separation hx,
    refine ⟨compact_open.gen {x} u, compact_open.gen {x} v, continuous_map.is_open_gen
      is_compact_singleton hu, continuous_map.is_open_gen is_compact_singleton hv, _, _, _⟩,
    { rwa [compact_open.gen, mem_set_of_eq, image_singleton, singleton_subset_iff] },
    { rwa [compact_open.gen, mem_set_of_eq, image_singleton, singleton_subset_iff] },
    { rw [disjoint_iff_inter_eq_empty, ←gen_inter, huv.inter_eq,
        gen_empty_right (singleton_nonempty _)] }
  end ⟩

end ev

section Inf_induced

lemma compact_open_le_induced (s : set α) :
  (continuous_map.compact_open : topological_space C(α, β))
  ≤ topological_space.induced (continuous_map.restrict s) continuous_map.compact_open :=
begin
  simp only [induced_generate_from_eq, continuous_map.compact_open],
  apply generate_from_mono,
  rintros b ⟨a, ⟨c, hc, u, hu, rfl⟩, rfl⟩,
  refine ⟨coe '' c, hc.image continuous_subtype_coe, u, hu, _⟩,
  ext f,
  simp only [compact_open.gen, mem_set_of_eq, mem_preimage, continuous_map.coe_restrict],
  rw image_comp f (coe : s → α),
end

/-- The compact-open topology on `C(α, β)` is equal to the infimum of the compact-open topologies
on `C(s, β)` for `s` a compact subset of `α`.  The key point of the proof is that the union of the
compact subsets of `α` is equal to the union of compact subsets of the compact subsets of `α`. -/
lemma compact_open_eq_Inf_induced :
  (continuous_map.compact_open : topological_space C(α, β))
  = ⨅ (s : set α) (hs : is_compact s),
    topological_space.induced (continuous_map.restrict s) continuous_map.compact_open :=
begin
  refine le_antisymm _ _,
  { refine le_infi₂ _,
    exact λ s hs, compact_open_le_induced s },
  simp only [← generate_from_Union, induced_generate_from_eq, continuous_map.compact_open],
  apply generate_from_mono,
  rintros _ ⟨s, hs, u, hu, rfl⟩,
  rw mem_Union₂,
  refine ⟨s, hs, _, ⟨univ, is_compact_iff_is_compact_univ.mp hs, u, hu, rfl⟩, _⟩,
  ext f,
  simp only [compact_open.gen, mem_set_of_eq, mem_preimage, continuous_map.coe_restrict],
  rw image_comp f (coe : s → α),
  simp
end

/-- For any subset `s` of `α`, the restriction of continuous functions to `s` is continuous as a
function from `C(α, β)` to `C(s, β)` with their respective compact-open topologies. -/
lemma continuous_restrict (s : set α) : continuous (λ F : C(α, β), F.restrict s) :=
by { rw continuous_iff_le_induced, exact compact_open_le_induced s }

lemma nhds_compact_open_eq_Inf_nhds_induced (f : C(α, β)) :
  𝓝 f = ⨅ s (hs : is_compact s), (𝓝 (f.restrict s)).comap (continuous_map.restrict s) :=
by { rw [compact_open_eq_Inf_induced], simp [nhds_infi, nhds_induced] }

lemma tendsto_compact_open_restrict {ι : Type*} {l : filter ι} {F : ι → C(α, β)} {f : C(α, β)}
  (hFf : filter.tendsto F l (𝓝 f)) (s : set α) :
  filter.tendsto (λ i, (F i).restrict s) l (𝓝 (f.restrict s)) :=
(continuous_restrict s).continuous_at.tendsto.comp hFf

lemma tendsto_compact_open_iff_forall {ι : Type*} {l : filter ι} (F : ι → C(α, β)) (f : C(α, β)) :
  filter.tendsto F l (𝓝 f)
  ↔ ∀ s (hs : is_compact s), filter.tendsto (λ i, (F i).restrict s) l (𝓝 (f.restrict s)) :=
by { rw [compact_open_eq_Inf_induced], simp [nhds_infi, nhds_induced, filter.tendsto_comap_iff] }

/-- A family `F` of functions in `C(α, β)` converges in the compact-open topology, if and only if
it converges in the compact-open topology on each compact subset of `α`. -/
lemma exists_tendsto_compact_open_iff_forall [locally_compact_space α] [t2_space α] [t2_space β]
  {ι : Type*} {l : filter ι} [filter.ne_bot l] (F : ι → C(α, β)) :
  (∃ f, filter.tendsto F l (𝓝 f))
  ↔ ∀ (s : set α) (hs : is_compact s), ∃ f, filter.tendsto (λ i, (F i).restrict s) l (𝓝 f) :=
begin
  split,
  { rintros ⟨f, hf⟩ s hs,
    exact ⟨f.restrict s, tendsto_compact_open_restrict hf s⟩ },
  { intros h,
    choose f hf using h,
    -- By uniqueness of limits in a `t2_space`, since `λ i, F i x` tends to both `f s₁ hs₁ x` and
    -- `f s₂ hs₂ x`, we have `f s₁ hs₁ x = f s₂ hs₂ x`
    have h : ∀ s₁ (hs₁ : is_compact s₁) s₂ (hs₂ : is_compact s₂) (x : α) (hxs₁ : x ∈ s₁)
      (hxs₂ : x ∈ s₂), f s₁ hs₁ ⟨x, hxs₁⟩ = f s₂ hs₂ ⟨x, hxs₂⟩,
    { rintros s₁ hs₁ s₂ hs₂ x hxs₁ hxs₂,
      haveI := is_compact_iff_compact_space.mp hs₁,
      haveI := is_compact_iff_compact_space.mp hs₂,
      have h₁ := (continuous_eval_const' (⟨x, hxs₁⟩ : s₁)).continuous_at.tendsto.comp (hf s₁ hs₁),
      have h₂ := (continuous_eval_const' (⟨x, hxs₂⟩ : s₂)).continuous_at.tendsto.comp (hf s₂ hs₂),
      exact tendsto_nhds_unique h₁ h₂ },
    -- So glue the `f s hs` together and prove that this glued function `f₀` is a limit on each
    -- compact set `s`
    have hs : ∀ x : α, ∃ s (hs : is_compact s), s ∈ 𝓝 x,
    { intros x,
      obtain ⟨s, hs, hs'⟩ := exists_compact_mem_nhds x,
      exact ⟨s, hs, hs'⟩ },
    refine ⟨lift_cover' _ _ h hs, _⟩,
    rw tendsto_compact_open_iff_forall,
    intros s hs,
    rw lift_cover_restrict',
    exact hf s hs }
end

end Inf_induced

section coev

variables (α β)

/-- The coevaluation map `β → C(α, β × α)` sending a point `x : β` to the continuous function
on `α` sending `y` to `(x, y)`. -/
def coev (b : β) : C(α, β × α) := ⟨prod.mk b, continuous_const.prod_mk continuous_id⟩

variables {α β}
lemma image_coev {y : β} (s : set α) : (coev α β y) '' s = ({y} : set β) ×ˢ s := by tidy

-- The coevaluation map β → C(α, β × α) is continuous (always).
lemma continuous_coev : continuous (coev α β) :=
continuous_generated_from $ begin
  rintros _ ⟨s, sc, u, uo, rfl⟩,
  rw is_open_iff_forall_mem_open,
  intros y hy,
  change (coev α β y) '' s ⊆ u at hy,
  rw image_coev s at hy,
  rcases generalized_tube_lemma is_compact_singleton sc uo hy
    with ⟨v, w, vo, wo, yv, sw, vwu⟩,
  refine ⟨v, _, vo, singleton_subset_iff.mp yv⟩,
  intros y' hy',
  change (coev α β y') '' s ⊆ u,
  rw image_coev s,
  exact subset.trans (prod_mono (singleton_subset_iff.mpr hy') sw) vwu
end

end coev

section curry

/-- Auxiliary definition, see `continuous_map.curry` and `homeomorph.curry`. -/
def curry' (f : C(α × β, γ)) (a : α) : C(β, γ) := ⟨function.curry f a⟩

/-- If a map `α × β → γ` is continuous, then its curried form `α → C(β, γ)` is continuous. -/
lemma continuous_curry' (f : C(α × β, γ)) : continuous (curry' f) :=
have hf : curry' f = continuous_map.comp f ∘ coev _ _, by { ext, refl },
hf ▸ continuous.comp (continuous_comp f) continuous_coev

/-- To show continuity of a map `α → C(β, γ)`, it suffices to show that its uncurried form
    `α × β → γ` is continuous. -/
lemma continuous_of_continuous_uncurry (f : α → C(β, γ))
  (h : continuous (function.uncurry (λ x y, f x y))) : continuous f :=
by { convert continuous_curry' ⟨_, h⟩, ext, refl }

/-- The curried form of a continuous map `α × β → γ` as a continuous map `α → C(β, γ)`.
    If `a × β` is locally compact, this is continuous. If `α` and `β` are both locally
    compact, then this is a homeomorphism, see `homeomorph.curry`. -/
def curry (f : C(α × β, γ)) : C(α, C(β, γ)) :=
⟨_, continuous_curry' f⟩

/-- The currying process is a continuous map between function spaces. -/
lemma continuous_curry [locally_compact_space (α × β)] :
  continuous (curry : C(α × β, γ) → C(α, C(β, γ))) :=
begin
  apply continuous_of_continuous_uncurry,
  apply continuous_of_continuous_uncurry,
  rw ←homeomorph.comp_continuous_iff' (homeomorph.prod_assoc _ _ _).symm,
  convert continuous_eval';
  tidy
end

@[simp]
lemma curry_apply (f : C(α × β, γ)) (a : α) (b : β) : f.curry a b = f (a, b) := rfl

/-- The uncurried form of a continuous map `α → C(β, γ)` is a continuous map `α × β → γ`. -/
lemma continuous_uncurry_of_continuous [locally_compact_space β] (f : C(α, C(β, γ))) :
  continuous (function.uncurry (λ x y, f x y)) :=
continuous_eval'.comp $ f.continuous.prod_map continuous_id

/-- The uncurried form of a continuous map `α → C(β, γ)` as a continuous map `α × β → γ` (if `β` is
    locally compact). If `α` is also locally compact, then this is a homeomorphism between the two
    function spaces, see `homeomorph.curry`. -/
@[simps] def uncurry [locally_compact_space β] (f : C(α, C(β, γ))) : C(α × β, γ) :=
⟨_, continuous_uncurry_of_continuous f⟩

/-- The uncurrying process is a continuous map between function spaces. -/
lemma continuous_uncurry [locally_compact_space α] [locally_compact_space β] :
  continuous (uncurry : C(α, C(β, γ)) → C(α × β, γ)) :=
begin
  apply continuous_of_continuous_uncurry,
  rw ←homeomorph.comp_continuous_iff' (homeomorph.prod_assoc _ _ _),
  apply continuous.comp continuous_eval' (continuous.prod_map continuous_eval' continuous_id);
  apply_instance
end

/-- The family of constant maps: `β → C(α, β)` as a continuous map. -/
def const' : C(β, C(α, β)) := curry ⟨prod.fst, continuous_fst⟩

@[simp] lemma coe_const' : (const' : β → C(α, β)) = const α := rfl

lemma continuous_const' : continuous (const α : β → C(α, β)) := const'.continuous

end curry

end compact_open

end continuous_map

open continuous_map

namespace homeomorph
variables {α : Type*} {β : Type*} {γ : Type*}
variables [topological_space α] [topological_space β] [topological_space γ]

/-- Currying as a homeomorphism between the function spaces `C(α × β, γ)` and `C(α, C(β, γ))`. -/
def curry [locally_compact_space α] [locally_compact_space β] : C(α × β, γ) ≃ₜ C(α, C(β, γ)) :=
⟨⟨curry, uncurry, by tidy, by tidy⟩, continuous_curry, continuous_uncurry⟩

/-- If `α` has a single element, then `β` is homeomorphic to `C(α, β)`. -/
def continuous_map_of_unique [unique α] : β ≃ₜ C(α, β) :=
{ to_fun := const α,
  inv_fun := λ f, f default,
  left_inv := λ a, rfl,
  right_inv := λ f, by { ext, rw unique.eq_default a, refl },
  continuous_to_fun := continuous_const',
  continuous_inv_fun := continuous_eval'.comp (continuous_id.prod_mk continuous_const) }

@[simp] lemma continuous_map_of_unique_apply [unique α] (b : β) (a : α) :
  continuous_map_of_unique b a = b :=
rfl

@[simp] lemma continuous_map_of_unique_symm_apply [unique α] (f : C(α, β)) :
  continuous_map_of_unique.symm f = f default :=
rfl

end homeomorph

section quotient_map

variables {X₀ X Y Z : Type*} [topological_space X₀] [topological_space X]
  [topological_space Y] [topological_space Z] [locally_compact_space Y] {f : X₀ → X}

lemma quotient_map.continuous_lift_prod_left (hf : quotient_map f) {g : X × Y → Z}
  (hg : continuous (λ p : X₀ × Y, g (f p.1, p.2))) : continuous g :=
begin
  let Gf : C(X₀, C(Y, Z)) := continuous_map.curry ⟨_, hg⟩,
  have h : ∀ x : X, continuous (λ y, g (x, y)),
  { intros x,
    obtain ⟨x₀, rfl⟩ := hf.surjective x,
    exact (Gf x₀).continuous },
  let G : X → C(Y, Z) := λ x, ⟨_, h x⟩,
  have : continuous G,
  { rw hf.continuous_iff,
    exact Gf.continuous },
  convert continuous_map.continuous_uncurry_of_continuous ⟨G, this⟩,
  ext x,
  cases x,
  refl,
end

lemma quotient_map.continuous_lift_prod_right (hf : quotient_map f) {g : Y × X → Z}
  (hg : continuous (λ p : Y × X₀, g (p.1, f p.2))) : continuous g :=
begin
  have : continuous (λ p : X₀ × Y, g ((prod.swap p).1, f (prod.swap p).2)),
  { exact hg.comp continuous_swap },
  have : continuous (λ p : X₀ × Y, (g ∘ prod.swap) (f p.1, p.2)) := this,
  convert (hf.continuous_lift_prod_left this).comp continuous_swap,
  ext x,
  simp,
end

end quotient_map