Module mathcomp.analysis.function_spaces

From HB Require Import structures.
From mathcomp Require Import all_ssreflect all_algebra finmap generic_quotient.
From mathcomp Require Import boolp classical_sets functions.
From mathcomp Require Import cardinality mathcomp_extra fsbigop.
Require Import reals signed topology.

# The topology of functions spaces Function spaces have no canonical topology. We develop the theory of several general-purpose function space topologies here. Contextually, there is often a natural choice, though. So we provide modules to assign topologies to arrow types locally. Topologies we consider are: - Topology of pointwise convergence - Topology of uniform convergence - Topology of uniform convergence on subspaces - The compact-open topology Available modules to assign a topology to `->`: - ArrowAsProduct assigns the product topology - ArrowAsUniformType assigns the uniform topology - ArrowAsCompactOpen assign the compact-open topology The major results are: - Compactness in the product topology via Tychonoff's - Compactness in the compact convergence topology via Ascoli's - Conditions when the supremum and weak topology commute in products - The compact-open topology is the topopology of compact convergence - Cartesian closedness for the category of locally compact topologies ## Function space notations ``` {uniform` A -> V} == the space U -> V, equipped with the topology of uniform convergence from a set A to V, where V is a uniformType {uniform U -> V} := {uniform` [set: U] -> V} {uniform A, F --> f} == F converges to f in {uniform A -> V} {uniform, F --> f} := {uniform setT, F --> f} prod_topology I T == the topology of pointwise convergence on the dependent space `forall (i:I), T i` {ptws U -> V} == prod_topology for the non-dependent product separate_points_from_closed f == for a closed set U and point x outside some member of the family f, it sends f_i(x) outside (closure (f_i @` U)) Used together with join_product. join_product f == the function (x => f ^~ x) When the family f separates points from closed {ptws, F --> f} == F converges to f in {ptws U -> V} {family fam, U -> V} == the supremum of {uniform A -> f} for each A in `fam` In particular, {family compact, U -> V} is the topology of compact convergence. {family fam, F --> f} == F converges to f in {family fam, U -> V} {compact_open, U -> V} == compact-open topology {compact_open, F --> f} == F converges to f in {compact_open, U -> V} ``` ## Ascoli's theorem notations ``` equicontinuous W x == the set (W : X -> Y) is equicontinuous at x pointwise_precompact W == for each (x : X), the set of images [f x | f in W] is precompact ```

Reserved Notation "{ 'uniform`' A -> V }"
  (at level 0, A at level 69, format "{ 'uniform`' A -> V }").
Reserved Notation "{ 'uniform' U -> V }"
  (at level 0, U at level 69, format "{ 'uniform' U -> V }").
Reserved Notation "{ 'uniform' A , F --> f }"
  (at level 0, A at level 69, F at level 69,
   format "{ 'uniform' A , F --> f }").
Reserved Notation "{ 'uniform' , F --> f }"
  (at level 0, F at level 69,
   format "{ 'uniform' , F --> f }").
Reserved Notation "{ 'ptws' U -> V }"
  (at level 0, U at level 69, format "{ 'ptws' U -> V }").
Reserved Notation "{ 'ptws' , F --> f }"
  (at level 0, F at level 69, format "{ 'ptws' , F --> f }").
Reserved Notation "{ 'family' fam , U -> V }"
  (at level 0, U at level 69, format "{ 'family' fam , U -> V }").
Reserved Notation "{ 'family' fam , F --> f }"
  (at level 0, F at level 69, format "{ 'family' fam , F --> f }").
Reserved Notation "{ 'compact-open' , U -> V }"
  (at level 0, U at level 69, format "{ 'compact-open' , U -> V }").
Reserved Notation "{ 'compact-open' , F --> f }"
  (at level 0, F at level 69, format "{ 'compact-open' , F --> f }").

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Obligation Tactic := idtac.

Import Order.TTheory GRing.Theory Num.Theory.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Product topology, also known as the topology of pointwise convergence

Section Product_Topology.

Definition prod_topology {I : Type} (T : I -> Type) := forall i, T i.

Variable I : Type.

Definition product_topology_def (T : I -> topologicalType) :=
  sup_topology (fun i => Topological.class
    (weak_topology (fun f : [the choiceType of forall i, T i] => f i))).

HB.instance Definition _ (T : I -> topologicalType) :=
  Topological.copy (prod_topology T) (product_topology_def T).

HB.instance Definition _ (T : I -> uniformType) :=
  Uniform.copy (prod_topology T)
    (sup_topology (fun i => Uniform.class
      [the uniformType of weak_topology (@proj _ T i)])).

HB.instance Definition _ (R : realType) (Ii : countType)
    (Tc : Ii -> pseudoMetricType R) := PseudoMetric.copy (prod_topology Tc)
  (sup_pseudometric (fun i => PseudoMetric.class
     [the pseudoMetricType R of weak_topology (@proj _ Tc i)]) (countableP _)).

End Product_Topology.

Notation "{ 'ptws' U -> V }" := (prod_topology (fun _ : U => V)) : type_scope.
Notation "{ 'ptws' , F --> f }" :=
  (cvg_to F (nbhs (f : {ptws _ -> _}))) : classical_set_scope.

Module ArrowAsProduct.
HB.instance Definition _ (U : Type) (T : U -> topologicalType) :=
  Topological.copy (forall x : U, T x) (prod_topology T).

HB.instance Definition _ (U : Type) (T : U -> uniformType) :=
  Uniform.copy (forall x : U, T x) (prod_topology T).

HB.instance Definition _ (U : Type) R (T : U -> pseudoMetricType R) :=
  Uniform.copy (forall x : U, T x) (prod_topology T).
End ArrowAsProduct.

Section product_spaces.
Local Import ArrowAsProduct.

Section projection_maps.
Context {I : eqType} {K : I -> topologicalType}.

Lemma proj_continuous i : continuous (@proj I K i).

Proof.

move=> f; have /cvg_sup/(_ i)/cvg_image : f --> f by apply: cvg_id.
move=> h; apply: cvg_trans (h _) => {h}.
by move=> Q /= [W nbdW <-]; apply: filterS nbdW; exact: preimage_image.
rewrite eqEsubset; split => y //; exists (dfwith f i y) => //.
by rewrite dfwithin.
Qed.

Lemma dfwith_continuous g (i : I) : continuous (@dfwith I K g i).

Proof.

move=> z U [] P [] [] Q QfinP <- [] V JV Vpz.
move/(@preimage_subset _ _ (dfwith g i))/filterS; apply.
apply: (@filterS _ _ _ ((dfwith g i) @^-1` V)); first by exists V.
have [L Lsub /[dup] VL <-] := QfinP _ JV; rewrite preimage_bigcap.
apply: filter_bigI => /= M /[dup] LM /Lsub /set_mem [] w _ [+] + /[dup] + <-.
have [->|wnx] := eqVneq w i => N oN NM.
  apply: (@filterS _ _ _ N); first by move=> ? ?; rewrite /= dfwithin.
  apply: open_nbhs_nbhs; split => //; move: Vpz.
  by rewrite -VL => /(_ _ LM); rewrite -NM /= dfwithin.
apply: nearW => y /=; move: Vpz.
by rewrite -VL => /(_ _ LM); rewrite -NM /= ?dfwithout // eq_sym.
Qed.

Lemma proj_open i (A : set (prod_topology K)) : open A -> open (proj i @` A).

Proof.

move=> oA; rewrite openE => z [f Af <-]; rewrite openE in oA.
have {oA} := oA _ Af; rewrite /interior => nAf.
apply: (@filterS _ _ _ ((dfwith f i) @^-1` A)).
by move=> w Apw; exists (dfwith f i w) => //; rewrite projK.
apply: dfwith_continuous => /=; move: nAf; congr (nbhs _ A).
by apply: functional_extensionality_dep => ?; case: dfwithP.
Qed.

Lemma hausdorff_product :
(forall x, hausdorff_space (K x)) -> hausdorff_space (forall x, K x).

Proof.

move=> hsdfK p q /= clstr; apply: functional_extensionality_dep => x.
apply: hsdfK; move: clstr; rewrite ?cluster_cvgE /= => -[G PG [GtoQ psubG]].
exists (proj x @ G); [exact: fmap_proper_filter|split].
apply: cvg_trans; last exact: (@proj_continuous x q).
by apply: cvg_app; exact: GtoQ.
move/(cvg_app (proj x)): psubG => /cvg_trans; apply.
exact: proj_continuous.
Qed.

End projection_maps.

Lemma tychonoff (I : eqType) (T : I -> topologicalType)
  (A : forall i, set (T i)) :
  (forall i, compact (A i)) ->
  compact [set f : forall i, T i | forall i, A i (f i)].

Proof.

case: (pselect ([set f : forall i, T i | forall i, A i (f i)] == set0)).
  move/eqP => -> _; exact: compact0.
case/negP/set0P=> a0 Aa0 Aco; rewrite compact_ultra => F FU FA.
set subst_coord := fun (i : I) (pi : T i) (f : forall x : I, T x) (j : I) =>
  if eqP is ReflectT e then ecast i (T i) (esym e) pi else f j.
have subst_coordT i pi f : subst_coord i pi f i = pi.
  rewrite /subst_coord; case: eqP => // e.
  by rewrite (eq_irrelevance e (erefl _)).
have subst_coordN i pi f j : i != j -> subst_coord i pi f j = f j.
  move=> inej; rewrite /subst_coord; case: eqP => // e.
  by move: inej; rewrite {1}e => /negP.
have pr_surj i : @^~ i @` [set: forall i, T i] = setT.
  rewrite predeqE => pi; split=> // _.
  by exists (subst_coord i pi a0) => //; rewrite subst_coordT.
pose pF i : set_system _ := [set @^~ i @` B | B in F].
have pFultra i : UltraFilter (pF i) by exact: ultra_image (pr_surj i).
have pFA i : pF i (A i).
  exists [set g | forall i, A i (g i)] => //.
  rewrite predeqE => pi; split; first by move=> [g Ag <-]; apply: Ag.
  move=> Aipi; have [f Af] := filter_ex FA.
  exists (subst_coord i pi f); last exact: subst_coordT.
  move=> j; have [<-{j}|] := eqVneq i j; first by rewrite subst_coordT.
  by move=> /subst_coordN ->; apply: Af.
have cvpFA i : A i `&` [set p | pF i --> p] !=set0.
  by rewrite -ultra_cvg_clusterE; apply: Aco.
exists (fun i => xget (a0 i) (A i `&` [set p | pF i --> p])).
split=> [i|]; first by have /(xgetPex (a0 i)) [] := cvpFA i.
apply/cvg_sup => i; apply/cvg_image=> //.
by have /(xgetPex (a0 i)) [] := cvpFA i.
Qed.

Lemma perfect_prod {I : Type} (i : I) (K : I -> topologicalType) :
perfect_set [set: K i] -> perfect_set [set: forall i, K i].

Proof.

move=> /perfectTP KPo; apply/perfectTP => f oF; apply: (KPo (f i)).
rewrite (_ : [set f i] = proj i @` [set f]).
by apply: (@proj_open {classic I} _ i); exact: oF.
by rewrite eqEsubset; split => ? //; [move=> -> /=; exists f | case=> g ->].
Qed.

Lemma perfect_diagonal (K : nat -> topologicalType) :
(forall i, exists xy : K i * K i, xy.1 != xy.2) ->
perfect_set [set: forall i, K i].

Proof.

move=> npts; split; first exact: closedT.
rewrite eqEsubset; split => f // _.
pose distincts (i : nat) := projT1 (sigW (npts i)).
pose derange i (z : K i) :=
  if z == (distincts i).1 then (distincts i).2 else (distincts i).1.
pose g (N i : nat) := if (i < N)%N then f i else derange _ (f i).
have gcvg : g @ \oo --> f.
  apply/cvg_sup => N U [V] [[W] oW <-] WfN WU.
  by apply: (filterS WU); rewrite nbhs_simpl /g; exists N.+1 => // i /= ->.
move=> A /gcvg; rewrite nbhs_simpl => -[N _ An].
exists (g N); split => //; last by apply: An; rewrite /= leqnn.
apply/eqP => M; suff: g N N != f N by rewrite M; move/eqP.
rewrite /g ltnn /derange eq_sym; have [->|//] := eqVneq (f N) (distincts N).1.
exact: projT2 (sigW (npts N)).
Qed.

Lemma zero_dimension_prod (I : choiceType) (T : I -> topologicalType) :
(forall i, zero_dimensional (T i)) ->
zero_dimensional (forall i, T i).

Proof.

move=> dctTI x y /eqP xneqy.
have [i/eqP/dctTI [U [clU Ux nUy]]] : exists i, x i <> y i.
by apply/existsNP=> W; exact/xneqy/functional_extensionality_dep.
exists (proj i @^-1` U); split => //; apply: clopen_comp => //.
exact/proj_continuous.
Qed.

Lemma totally_disconnected_prod (I : choiceType)
  (T : I -> topologicalType) (A : forall i, set (T i)) :
  (forall i, totally_disconnected (A i)) ->
  @totally_disconnected (forall i, T i) (fun f => forall i, A i (f i)).

Proof.

move=> dsctAi x /= Aix; rewrite eqEsubset; split; last first.
by move=> ? ->; exact: connected_component_refl.
move=> f /= [C /= [Cx CA ctC Cf]]; apply/functional_extensionality_dep => i.
suff : proj i @` C `<=` [set x i] by apply; exists f.
rewrite -(dsctAi i) // => Ti ?; exists (proj i @` C) => //.
split; [by exists x | by move=> ? [r Cr <-]; exact: CA |].
apply/(connected_continuous_connected ctC)/continuous_subspaceT.
exact: proj_continuous.
Qed.

A handy technique for embedding a space `T` into a product. The key interface is `separate_points_from_closed`, which guarantees that the topologies - `T`'s native topology - `sup (weak f_i)`: the sup of all the weak topologies of `f_i` - `weak (x => (f_1 x, f_2 x, ...))`: the weak topology from the product space are equivalent (the last equivalence seems to require `accessible_space`).

Section product_embeddings.
Context {I : choiceType} {T : topologicalType} {U_ : I -> topologicalType}.
Variable (f_ : forall i, T -> U_ i).

Definition separate_points_from_closed := forall (U : set T) x,
  closed U -> ~ U x -> exists i, ~ (closure (f_ i @` U)) (f_ i x).

Hypothesis sepf : separate_points_from_closed.
Hypothesis ctsf : forall i, continuous (f_ i).

Let weakT := [the topologicalType of
  sup_topology (fun i => Topological.on (weak_topology (f_ i)))].

Let PU := [the topologicalType of prod_topology U_].

Local Notation sup_open := (@open weakT).
Local Notation "'weak_open' i" := (@open weakT) (at level 0).
Local Notation natural_open := (@open T).

Lemma weak_sep_cvg (F : set_system T) (x : T) :
  Filter F -> (F --> (x : T)) <-> (F --> (x : weakT)).

Proof.

move=> FF; split.
  move=> FTx; apply/cvg_sup => i U.
  have /= -> := @nbhsE (weak_topology (f_ i)) x.
  case=> B [[C oC <- ?]] /filterS; apply; apply: FTx; rewrite /= nbhsE.
  by exists (f_ i @^-1` C) => //; split => //; exact: open_comp.
move/cvg_sup => wiFx U; rewrite /= nbhs_simpl nbhsE => [[B [oB ?]]].
move/filterS; apply; have [//|i nclfix] := @sepf _ x (open_closedC oB).
apply: (wiFx i); have /= -> := @nbhsE (weak_topology (f_ i)) x.
exists (f_ i @^-1` (~` closure [set f_ i x | x in ~` B])); [split=>//|].
  apply: open_comp; last by rewrite ?openC//; exact: closed_closure.
  by move=> + _; exact: (@weak_continuous _ _ (f_ i)).
rewrite closureC preimage_bigcup => z [V [oV]] VnB => /VnB.
by move/forall2NP => /(_ z) [] // /contrapT.
Qed.

Lemma weak_sep_nbhsE x : @nbhs T T x = @nbhs T weakT x.

Proof.

rewrite predeqE => U; split; move: U.
by have P := weak_sep_cvg x (nbhs_filter (x : weakT)); exact/P.
by have P := weak_sep_cvg x (nbhs_filter (x : T)); exact/P.
Qed.

Lemma weak_sep_openE : @open T = @open weakT.

Proof.

rewrite predeqE => A; rewrite ?openE /interior.
by split => + z => /(_ z); rewrite weak_sep_nbhsE.
Qed.

Definition join_product (x : T) : PU := f_ ^~ x.

Lemma join_product_continuous : continuous join_product.

Proof.

suff : continuous (join_product : weakT -> PU).
by move=> cts x U => /cts; rewrite nbhs_simpl /= -weak_sep_nbhsE.
move=> x; apply/cvg_sup; first exact/fmap_filter/(nbhs_filter (x : weakT)).
move=> i; move: x; apply/(@continuousP _ (weak_topology (@^~ i))) => A [B ? E].
rewrite -E (_ : @^~ i = proj i) //.
have -> : join_product @^-1` (proj i @^-1` B) = f_ i @^-1` B by [].
apply: open_comp => // + _; rewrite /cvg_to => x U.
by rewrite nbhs_simpl /= -weak_sep_nbhsE; move: x U; exact: ctsf.
Qed.

Local Notation prod_open := (@open (subspace (range join_product))).

Lemma join_product_open (A : set T) : open A ->
open ((join_product @` A) : set (subspace (range join_product))).

Proof.

move=> oA; rewrite openE => y /= [x Ax] jxy.
have [// | i nAfiy] := @sepf (~` A) x (open_closedC oA).
pose B : set PU := proj i @^-1` (~` closure (f_ i @` ~` A)).
apply: (@filterS _ _ _ (range join_product `&` B)).
  move=> z [[w ?]] wzE Bz; exists w => //.
  move: Bz; rewrite /B -wzE closureC; case=> K [oK KsubA] /KsubA.
  have -> : proj i (join_product w) = f_ i w by [].
  by move=> /exists2P/forallNP/(_ w)/not_andP [] // /contrapT.
apply: open_nbhs_nbhs; split; last by rewrite -jxy.
apply: openI; first exact: open_subspaceT.
apply: open_subspaceW; apply: open_comp; last exact/closed_openC/closed_closure.
by move=> + _; exact: proj_continuous.
Qed.

Lemma join_product_inj : accessible_space T -> set_inj [set: T] join_product.

Proof.

move=> /accessible_closed_set1 cl1 x y; case: (eqVneq x y) => // xny _ _ jxjy.
have [] := @sepf [set y] x (cl1 y); first exact/eqP.
move=> i P; suff : join_product x i != join_product y i by rewrite jxjy => /eqP.
apply/negP; move: P; apply: contra_not => /eqP; rewrite /join_product => ->.
by apply: subset_closure; exists y.
Qed.

Lemma join_product_weak : set_inj [set: T] join_product ->
@open T = @open (weak_topology join_product).

Proof.

move=> inj; rewrite predeqE => U; split; first last.
by move=> [V ? <-]; apply: open_comp => // + _; exact: join_product_continuous.
move=> /join_product_open/open_subspaceP [V [oU VU]].
exists V => //; have := @f_equal _ _ (preimage join_product) _ _ VU.
rewrite !preimage_setI // !preimage_range !setIT => ->.
rewrite eqEsubset; split; last exact: preimage_image.
by move=> z [w Uw] /inj <- //; rewrite inE.
Qed.

End product_embeddings.

Global Instance prod_topology_filter (U : Type) (T : U -> ptopologicalType) (f : prod_topology T) :
ProperFilter (nbhs f).

Proof.

exact: nbhs_pfilter.
Qed.

End product_spaces.

HB.instance Definition _ (U : Type) (T : U -> ptopologicalType) :=
Pointed.copy (forall x : U, T x) (prod_topology T).

the uniform topologies type

Section fct_Uniform.
Local Open Scope relation_scope.
Variables (T : choiceType) (U : uniformType).

Definition fct_ent := filter_from (@entourage U)
(fun P => [set fg | forall t : T, P (fg.1 t, fg.2 t)]).

Lemma fct_ent_filter : Filter fct_ent.

Proof.

apply: filter_from_filter; first by exists setT; apply: filterT.
move=> A B entA entB.
exists (A `&` B); first exact: filterI.
by move=> fg ABfg; split=> t; have [] := ABfg t.
Qed.

Lemma fct_ent_refl A : fct_ent A -> diagonal `<=` A.

Proof.

move=> [B entB sBA] fg feg; apply/sBA => t; rewrite feg.
exact: entourage_refl.
Qed.

Lemma fct_ent_inv A : fct_ent A -> fct_ent A^-1.

Proof.

move=> [B entB sBA]; exists B^-1; first exact: entourage_inv.
by move=> fg Bgf; exact/sBA.
Qed.

Lemma fct_ent_split A : fct_ent A -> exists2 B, fct_ent B & B \; B `<=` A.

Proof.

move=> [B entB sBA].
exists [set fg | forall t, split_ent B (fg.1 t, fg.2 t)].
by exists (split_ent B).
move=> fg [h spBfh spBhg].
by apply: sBA => t; apply: entourage_split (spBfh t) (spBhg t).
Qed.

Definition arrow_uniform_type : Type := T -> U.

#[export] HB.instance Definition _ := Choice.on arrow_uniform_type.
#[export] HB.instance Definition _ := isUniform.Build arrow_uniform_type
  fct_ent_filter fct_ent_refl fct_ent_inv fct_ent_split.

End fct_Uniform.

#[export] HB.instance Definition _ {T : choiceType} {U : puniformType} :=
  Pointed.on (arrow_uniform_type T U).

Lemma cvg_fct_entourageP (T : choiceType) (U : uniformType)
    (F : set_system (arrow_uniform_type T U)) (FF : Filter F)
    (f : arrow_uniform_type T U) :
  F --> f <-> forall A, entourage A ->
              \forall g \near F, forall t, A (f t, g t).

Proof.

split => [/cvg_entourageP Ff A entA|Ff].
by apply: (Ff [set fg | forall t : T, A (fg.1 t, fg.2 t)]); exists A.
apply/cvg_entourageP => A [P entP sPA].
by near=> g do apply: sPA; apply: Ff.
Unshelve. all: by end_near. Qed.

Section fun_Complete.
Context {T : choiceType} {U : completeType}.

Lemma fun_complete (F : set_system (arrow_uniform_type T U))
{FF : ProperFilter F} : cauchy F -> cvg F.

Proof.

move=> Fc.
have /(_ _) /cauchy_cvg /cvg_app_entourageP cvF : cauchy (@^~_ @ F).
move=> t A /= entA; rewrite near_simpl -near2E near_map2.
by apply: Fc; exists A.
apply/cvg_ex; exists (fun t => lim (@^~t @ F)).
apply/cvg_fct_entourageP => A entA; near=> f => t; near F => g.
apply: (entourage_split (g t)) => //; first by near: g; apply: cvF.
move: (t); near: g; near: f; apply: nearP_dep; apply: Fc.
by exists (split_ent A)^-1%relation => /=.
Unshelve. all: by end_near. Qed.

HB.instance Definition _ := Uniform_isComplete.Build
(arrow_uniform_type T U) fun_complete.

HB.instance Definition _ (R : numFieldType) :=
Uniform_isComplete.Build (arrow_uniform_type T U) cauchy_cvg.

End fun_Complete.

Functional metric spaces

Section fct_PseudoMetric.
Variable (T : choiceType) (R : numFieldType) (U : pseudoMetricType R).
Definition fct_ball (x : arrow_uniform_type T U) (eps : R)
(y : arrow_uniform_type T U) := forall t : T, ball (x t) eps (y t).
Lemma fct_ball_center (x : T -> U) (e : R) : 0 < e -> fct_ball x e x.

Proof.

by move=> /posnumP[{}e] ?. Qed.

Lemma fct_ball_sym (x y : T -> U) (e : R) : fct_ball x e y -> fct_ball y e x.

Proof.

by move=> P t; apply: ball_sym. Qed.

Lemma fct_ball_triangle (x y z : T -> U) (e1 e2 : R) :
fct_ball x e1 y -> fct_ball y e2 z -> fct_ball x (e1 + e2) z.

Proof.

by move=> xy yz t; apply: (@ball_triangle _ _ (y t)). Qed.

Lemma fct_entourage : entourage = entourage_ fct_ball.

Proof.

rewrite predeqE => A; split; last first.
by move=> [_/posnumP[e] sbeA]; exists [set xy | ball xy.1 e%:num xy.2].
move=> [P]; rewrite -entourage_ballE => -[_/posnumP[e] sbeP] sPA.
by exists e%:num => //= fg fg_e; apply: sPA => t; apply: sbeP; apply: fg_e.
Qed.

HB.instance Definition _ := Uniform_isPseudoMetric.Build R
  (arrow_uniform_type T U) fct_ball_center fct_ball_sym
  fct_ball_triangle fct_entourage.
End fct_PseudoMetric.

Module ArrowAsUniformType.
HB.instance Definition _ (U : choiceType) (V : uniformType) :=
  Uniform.copy (U -> V) (arrow_uniform_type U V).

HB.instance Definition _ (U : choiceType) (R : numFieldType)
    (V : pseudoMetricType R) :=
  PseudoMetric.copy (U -> V) (arrow_uniform_type U V).
End ArrowAsUniformType.

Limit switching

Section Cvg_switch.
Context {T1 T2 : choiceType}.
Local Import ArrowAsUniformType.

Lemma cvg_switch_1 {U : uniformType}
  F1 {FF1 : ProperFilter F1} F2 {FF2 : Filter F2}
  (f : T1 -> T2 -> U) (g : T2 -> U) (h : T1 -> U) (l : U) :
  f @ F1 --> g -> (forall x1, f x1 @ F2 --> h x1) -> h @ F1 --> l ->
  g @ F2 --> l.

Proof.

move=> fg fh hl; apply/cvg_app_entourageP => A entA.
near F1 => x1; near=> x2; apply: (entourage_split (h x1)) => //.
by apply/xsectionP; near: x1; exact: hl.
apply: (entourage_split (f x1 x2)) => //.
by apply/xsectionP; near: x2; exact: fh.
move: (x2); near: x1; have /cvg_fct_entourageP /(_ _^-1%relation):= fg; apply.
exact: entourage_inv.
Unshelve. all: by end_near. Qed.

Lemma cvg_switch_2 {U : completeType}
  F1 {FF1 : ProperFilter F1} F2 {FF2 : ProperFilter F2}
  (f : T1 -> T2 -> U) (g : T2 -> U) (h : T1 -> U) :
  f @ F1 --> g -> (forall x, f x @ F2 --> h x) ->
  [cvg h @ F1 in U].

Proof.

move=> fg fh; apply: cauchy_cvg => A entA.
rewrite !near_simpl -near2_pair near_map2; near=> x1 y1 => /=; near F2 => x2.
apply: (entourage_split (f x1 x2)) => //.
  by apply/xsectionP; near: x2; exact: fh.
apply: (entourage_split (f y1 x2)) => //; last first.
  apply/xsectionP; near: x2; apply/(fh _ (xsection _^-1%relation _)).
  exact: nbhs_entourage (entourage_inv _).
apply: (entourage_split (g x2)) => //; move: (x2); [near: x1|near: y1].
  have /cvg_fct_entourageP /(_ _^-1%relation) := fg; apply.
  exact: entourage_inv.
by have /cvg_fct_entourageP := fg; apply.
Unshelve. all: by end_near. Qed.

Lemma cvg_switch {U : completeType}
  F1 (FF1 : ProperFilter F1) F2 (FF2 : ProperFilter F2)
  (f : T1 -> T2 -> U) (g : T2 -> U) (h : T1 -> U) :
  f @ F1 --> g -> (forall x1, f x1 @ F2 --> h x1) ->
  exists l : U, h @ F1 --> l /\ g @ F2 --> l.

Proof.

move=> Hfg Hfh; have hcv := !! cvg_switch_2 Hfg Hfh.
by exists (lim (h @ F1)); split=> //; apply: cvg_switch_1 Hfg Hfh hcv.
Qed.

End Cvg_switch.

Definition uniform_fun {U : Type} (A : set U) (V : Type) : Type := U -> V.

Notation "{ 'uniform`' A -> V }" := (@uniform_fun _ A V) : type_scope.
Notation "{ 'uniform' U -> V }" := ({uniform` [set: U] -> V}) : type_scope.
Notation "{ 'uniform' A , F --> f }" :=
  (cvg_to F (nbhs (f : {uniform` A -> _}))) : classical_set_scope.
Notation "{ 'uniform' , F --> f }" :=
  (cvg_to F (nbhs (f : {uniform _ -> _}))) : classical_set_scope.

Definition sigL_arrow {U : choiceType} (A : set U) (V : uniformType) :
  (U -> V) -> arrow_uniform_type A V := @sigL _ V A.

HB.instance Definition _ (U : choiceType) (A : set U) (V : uniformType) :=
  Uniform.copy {uniform` A -> V} (weak_topology (@sigL_arrow _ A V)).

Section RestrictedUniformTopology.
Context {U : choiceType} (A : set U) {V : uniformType} .

Lemma uniform_nbhs (f : {uniform` A -> V}) P:
  nbhs f P <-> (exists E, entourage E /\
    [set h | forall y, A y -> E(f y, h y)] `<=` P).

Proof.

split=> [[Q [[/= W oW <- /=] Wf subP]]|[E [entE subP]]].
  rewrite openE /= /interior in oW.
  case: (oW _ Wf) => ? [ /= E entE] Esub subW.
  exists E; split=> // h Eh; apply/subP/subW/xsectionP/Esub => /= [[u Au]].
  by apply: Eh => /=; rewrite -inE.
case : (pselect (exists (u : U), True)); first last.
  move=> nU; apply: (filterS subP); apply: (@filterS _ _ _ setT).
  by move=> t _ /= y; move: nU; apply: absurd; exists y.
  exact: filterT.
case=> u0 _; near=> g; apply: subP => y /mem_set Ay; rewrite -!(sigLE A).
move: (SigSub _); near: g.
have := (@cvg_image _ _ (@sigL_arrow _ A V) _ f (nbhs_filter f)
  (image_sigL (f u0))).1 cvg_id [set h | forall y, E (sigL A f y, h y)].
case.
  exists [set fg | forall y, E (fg.1 y, fg.2 y)] => //; first by exists E.
  by move=> g /xsectionP.
move=> B nbhsB rBrE; apply: (filterS _ nbhsB) => g Bg [y yA].
by move: rBrE; rewrite eqEsubset; case => [+ _]; apply; exists g.
Unshelve. all: by end_near. Qed.

Lemma uniform_entourage :
  @entourage [the uniformType of {uniform` A -> V}] =
  filter_from
    (@entourage V)
    (fun P => [set fg | forall t : U, A t -> P (fg.1 t, fg.2 t)]).

Proof.

rewrite eqEsubset; split => P /=.
  case=> /= E [F entF FsubE EsubP]; exists F => //; case=> f g Ffg.
  by apply/EsubP/FsubE=> [[x p]] /=; apply: Ffg; move/set_mem: (p).
case=> E entE EsubP; exists [set fg | forall t, E (fg.1 t, fg.2 t)].
  by exists E.
case=> f g Efg; apply: EsubP => t /mem_set At.
by move: Efg => /= /(_ (@exist _ (fun x => in_mem x (mem A)) _ At)).
Qed.

End RestrictedUniformTopology.

Lemma restricted_cvgE {U : choiceType} {V : uniformType}
(F : set_system (U -> V)) A (f : U -> V) :
{uniform A, F --> f} = (F --> (f : {uniform` A -> V})).

Proof.

by []. Qed.

Lemma pointwise_cvgE {U : Type} {V : topologicalType}
(F : set_system (U -> V)) (A : set U) (f : U -> V) :
{ptws, F --> f} = (F --> (f : {ptws U -> V})).

Proof.

by []. Qed.

We use this function to help Coq identify the correct notation to use when printing. Otherwise you get goals like `F --> f -> F --> f`.

Definition uniform_fun_family {U} V (fam : set U -> Prop) := U -> V.

Notation "{ 'family' fam , U -> V }" := (@uniform_fun_family U V fam).
Notation "{ 'family' fam , F --> f }" :=
  (cvg_to F (@nbhs _ {family fam, _ -> _} f)) : type_scope.

HB.instance Definition _ {U : choiceType} {V : uniformType}
    (fam : set U -> Prop) :=
  Uniform.copy {family fam, U -> V} (sup_topology (fun k : sigT fam =>
       Uniform.class [the uniformType of {uniform` projT1 k -> V}])).

Section UniformCvgLemmas.
Context {U : choiceType} {V : uniformType}.

Lemma uniform_set1 F (f : U -> V) (x : U) :
  Filter F -> {uniform [set x], F --> f} = (g x @[g --> F] --> f x).

Proof.

move=> FF; rewrite propeqE; split.
  move=> + W => /(_ [set t | W (t x)]) +; rewrite -nbhs_entourageE.
  rewrite uniform_nbhs => + [Q entQ subW].
  by apply; exists Q; split => // h Qf; exact/subW/xsectionP/Qf.
move=> Ff W; rewrite uniform_nbhs => [[E] [entE subW]].
apply: (filterS subW); move/(nbhs_entourage (f x))/Ff: entE => //=; near_simpl.
by apply: filter_app; apply: nearW=> ? /xsectionP ? ? ->.
Qed.

Lemma uniform_subset_nbhs (f : U -> V) (A B : set U) :
B `<=` A -> nbhs (f : {uniform` A -> V}) `=>` nbhs (f : {uniform` B -> V}).

Proof.

move => BsubA P /uniform_nbhs [E [entE EsubP]].
apply: (filterS EsubP); apply/uniform_nbhs; exists E; split => //.
by move=> h Eh y /BsubA Ay; exact: Eh.
Qed.

Lemma uniform_subset_cvg (f : U -> V) (A B : set U) F :
Filter F -> B `<=` A -> {uniform A, F --> f} -> {uniform B, F --> f}.

Proof.

move => FF /uniform_subset_nbhs => /(_ f).
by move=> nbhsF Acvg; apply: cvg_trans; [exact: Acvg|exact: nbhsF].
Qed.

Lemma pointwise_uniform_cvg (f : U -> V) (F : set_system (U -> V)) :
Filter F -> {uniform, F --> f} -> {ptws, F --> f}.

Proof.

move=> FF; rewrite cvg_sup => + i; have isubT : [set i] `<=` setT by move=> ?.
move=> /(uniform_subset_cvg _ isubT); rewrite uniform_set1.
rewrite cvg_image; last by rewrite eqEsubset; split=> v // _; exists (cst v).
apply: cvg_trans => W /=; rewrite nbhs_simpl; exists (@^~ i @^-1` W) => //.
by rewrite image_preimage // eqEsubset; split=> // j _; exists (fun _ => j).
Qed.

Lemma cvg_sigL (A : set U) (f : U -> V) (F : set_system (U -> V)) :
    Filter F ->
  {uniform A, F --> f} <->
  {uniform, sigL A @ F --> sigL A f}.

Proof.

move=> FF; split.
- move=> cvgF P' /uniform_nbhs [E [entE EsubP]].
  apply: (filterS EsubP); apply: cvgF => /=.
  apply: (filterS (P := [set h | forall y, A y -> E (f y, h y)])).
    + by move=> h/= Eh [y ?] _; apply Eh; rewrite -inE.
    + by (apply/uniform_nbhs; eexists; split; eauto).
- move=> cvgF P' /= /uniform_nbhs [ E [/= entE EsubP]].
  apply: (filterS EsubP).
  move: (cvgF [set h | (forall y , E (sigL A f y, h y))]) => /=.
  set Q := (x in (_ -> x) -> _); move=> W.
  have: Q by apply W, uniform_nbhs; exists E; split => // h + ?; apply.
  rewrite {}/W {}/Q; near_simpl => /= R; apply: (filterS _ R) => h /=.
  by rewrite forall_sig /sigL /=.
Qed.

Lemma eq_in_close (A : set U) (f g : {uniform` A -> V}) :
{in A, f =1 g} -> close f g.

Proof.

rewrite entourage_close => /eq_sigLP eqfg ? [E entE]; apply=> /=.
by rewrite /map_pair/sigL_arrow eqfg; exact: entourage_refl.
Qed.

Lemma hausdorrf_close_eq_in (A : set U) (f g : {uniform` A -> V}) :
hausdorff_space V -> close f g = {in A, f =1 g}.

Proof.

move=> hV.
rewrite propeqE; split; last exact: eq_in_close.
rewrite entourage_close => C u; rewrite inE => uA; apply: hV.
rewrite /cluster -nbhs_entourageE /= => X Y [X' eX X'X] [Y' eY Y'Y].
exists (g u); split; [apply: X'X| apply: Y'Y]; apply/xsectionP; last first.
exact: entourage_refl.
apply: (C [set fg | forall y, A y -> X' (fg.1 y, fg.2 y)]) => //=.
by rewrite uniform_entourage; exists X'.
Qed.

Lemma uniform_nbhsT (f : U -> V) :
(nbhs (f : {uniform U -> V})) = nbhs (f : arrow_uniform_type U V).

Proof.

rewrite eqEsubset; split=> A.
  case/uniform_nbhs => E [entE] /filterS; apply.
  exists [set fh | forall y, E (fh.1 y, fh.2 y)]; first by exists E.
  by move=> ? /xsectionP /=.
case => J [E entE EJ] /filterS; apply; apply/uniform_nbhs; exists E.
by split => // z /= Efz; apply/xsectionP/EJ => t /=; exact: Efz.
Qed.

Lemma cvg_uniformU (f : U -> V) (F : set_system (U -> V)) A B : Filter F ->
{uniform A, F --> f} -> {uniform B, F --> f} ->
{uniform (A `|` B), F --> f}.

Proof.

move=> FF AFf BFf Q /=/uniform_nbhs [E [entE EsubQ]].
apply: (filterS EsubQ).
rewrite (_: [set h | (forall y : U, (A `|` B) y -> E (f y, h y))] =
    [set h | forall y, A y -> E (f y, h y)] `&`
    [set h | forall y, B y -> E (f y, h y)]).
- apply: filterI; [apply: AFf| apply: BFf].
  + by apply/uniform_nbhs; exists E; split.
  + by apply/uniform_nbhs; exists E; split.
- rewrite eqEsubset; split=> h.
  + by move=> R; split=> t ?; apply: R;[left| right].
  + by move=> [R1 R2] y [? | ?]; [apply: R1| apply: R2].
Qed.

Lemma cvg_uniform_set0 (F : set_system (U -> V)) (f : U -> V) : Filter F ->
{uniform set0, F --> f}.

Proof.

move=> FF P /= /uniform_nbhs [E [? R]].
suff -> : P = setT by exact: filterT.
rewrite eqEsubset; split => //=.
by apply: subset_trans R => g _ ?.
Qed.

Lemma fam_cvgP (fam : set U -> Prop) (F : set_system (U -> V)) (f : U -> V) :
Filter F -> {family fam, F --> f} <->
(forall A : set U, fam A -> {uniform A, F --> f }).

Proof.

split; first by move=> /cvg_sup + A FA; move/(_ (existT _ _ FA)).
by move=> famFf /=; apply/cvg_sup => [[? ?] FA]; apply: famFf.
Qed.

Lemma family_cvg_subset (famA famB : set U -> Prop) (F : set_system (U -> V))
(f : U -> V) : Filter F ->
famA `<=` famB -> {family famB, F --> f} -> {family famA, F --> f}.

Proof.

by move=> FF S /fam_cvgP famBFf; apply/fam_cvgP => A ?; apply/famBFf/S.
Qed.

Lemma family_cvg_finite_covers (famA famB : set U -> Prop)
  (F : set_system (U -> V)) (f : U -> V) : Filter F ->
  (forall P, famA P ->
    exists (I : choiceType) f,
      (forall i, famB (f i)) /\ finite_subset_cover [set: I] f P) ->
  {family famB, F --> f} -> {family famA, F --> f}.

Proof.

move=> FF ex_finCover /fam_cvgP rFf; apply/fam_cvgP => A famAA.
move: ex_finCover => /(_ _ famAA) [R [g [g_famB [D _]]]].
move/uniform_subset_cvg; apply.
elim/finSet_rect: D => X IHX.
have [->|/set0P[x xX]] := eqVneq [set` X] set0.
by rewrite coverE bigcup_set0; apply: cvg_uniform_set0.
rewrite coverE (bigcup_fsetD1 x)//; apply: cvg_uniformU.
exact/rFf/g_famB.
exact/IHX/fproperD1.
Qed.

End UniformCvgLemmas.

Lemma uniform_restrict_cvg {U : choiceType} {V : puniformType}
(F : set_system (U -> V)) (f : U -> V) A : Filter F ->
{uniform A, F --> f} <-> {uniform, restrict A @ F --> restrict A f}.

Proof.

move=> FF; rewrite cvg_sigL; split.
- rewrite -sigLK; move/(cvg_app valL) => D.
  apply: cvg_trans; first exact: D.
  move=> P /uniform_nbhs [E [/=entE EsubP]]; apply: (filterS EsubP).
  apply/uniform_nbhs; exists E; split=> //= h /=.
  rewrite /sigL => R u _; rewrite oinv_set_val.
  by case: insubP=> /= *; [apply: R|apply: entourage_refl].
- move/(@cvg_app _ _ _ _ (sigL A)).
  rewrite -fmap_comp sigL_restrict => D.
  apply: cvg_trans; first exact: D.
  move=> P /uniform_nbhs [E [/=entE EsubP]]; apply: (filterS EsubP).
  apply/uniform_nbhs; exists E; split=> //= h /=.
  rewrite /sigL => R [u Au] _ /=.
  by have := R u I; rewrite /patch Au.
Qed.

Section FamilyConvergence.

Lemma fam_cvgE {U : choiceType} {V : uniformType} (F : set_system (U -> V))
(f : U -> V) fam :
{family fam, F --> f} = (F --> (f : {family fam, U -> V})).

Proof.

by []. Qed.

Lemma fam_nbhs {U : choiceType} {V : uniformType} (fam : set U -> Prop)
(A : set U) (E : set (V * V)) (f : {family fam, U -> V}) :
entourage E -> fam A -> nbhs f [set g | forall y, A y -> E (f y, g y)].

Proof.

move=> entE famA; have /fam_cvgP /(_ A) : (nbhs f --> f) by []; apply => //.
by apply uniform_nbhs; exists E; split.
Qed.

Lemma fam_compact_nbhs {U : topologicalType} {V : uniformType}
    (A : set U) (O : set V) (f : {family compact, U -> V}) :
  open O -> f @` A `<=` O -> compact A -> continuous f ->
  nbhs (f : {family compact, U -> V}) [set g | forall y, A y -> O (g y)].

Proof.

move=> oO fAO /[dup] cA /compact_near_coveringP/near_covering_withinP cfA ctsf.
near=> z => /=; (suff: A `<=` [set y | O (z y)] by exact); near: z.
apply: cfA => x Ax; have : O (f x) by exact: fAO.
move: (oO); rewrite openE /= => /[apply] /[dup] /ctsf Ofx /=.
rewrite /interior -nbhs_entourageE => -[E entE EfO].
exists (f @^-1` xsection (split_ent E) (f x),
    [set g | forall w, A w -> split_ent E (f w, g w)]).
  split => //=; last exact: fam_nbhs.
  by apply: ctsf; rewrite /= -nbhs_entourageE; exists (split_ent E).
case=> y g [/= /xsectionP Efxy] AEg Ay; apply/EfO/xsectionP.
by apply: subset_split_ent => //; exists (f y) => //=; exact: AEg.
Unshelve. all: by end_near. Qed.

End FamilyConvergence.

It turns out `{family compact, U -> V}` can be generalized to only assume `topologicalType` on `V`. This topology is called the compact-open topology. This topology is special because it is the _only_ topology that will allow `curry`/`uncurry` to be continuous.

Section compact_open.
Context {T U : topologicalType}.

Definition compact_open : Type := T -> U.

Section compact_open_setwise.
Context {K : set T}.

Definition compact_openK := let _ := K in compact_open.

Definition compact_openK_nbhs (f : compact_openK) :=
  filter_from
    [set O | f @` K `<=` O /\ open O]
    (fun O => [set g | g @` K `<=` O]).

Global Instance compact_openK_nbhs_filter (f : compact_openK) :
  ProperFilter (compact_openK_nbhs f).

Proof.

split; first by case=> g [gKO oO] /(_ f); apply.
apply: filter_from_filter; first by exists setT; split => //; exact: openT.
move=> P Q [fKP oP] [fKQ oQ]; exists (P `&` Q); first split.
- by move=> ? [z Kz M-]; split; [apply: fKP | apply: fKQ]; exists z.
- exact: openI.
by move=> g /= gPQ; split; exact: (subset_trans gPQ).
Qed.

HB.instance Definition _ := Choice.on compact_openK.

HB.instance Definition _ := hasNbhs.Build compact_openK compact_openK_nbhs.

Definition compact_open_of_nbhs := [set A : set compact_openK | A `<=` nbhs^~ A].

Lemma compact_openK_nbhsE_subproof (p : compact_openK) :
  compact_openK_nbhs p =
    [set A | exists B : set compact_openK,
      [/\ compact_open_of_nbhs B, B p & B `<=` A]].

Proof.

rewrite eqEsubset; split => A /=.
case=> B /= [fKB oB gKBA]; exists [set g | g @` K `<=` B]; split => //.
by move=> h /= hKB; exists B.
by case=> B [oB Bf /filterS]; apply; exact: oB.
Qed.

Lemma compact_openK_openE_subproof :
compact_open_of_nbhs = [set A | A `<=` compact_openK_nbhs^~ A].

Proof.

by []. Qed.

HB.instance Definition _ :=
  Nbhs_isTopological.Build compact_openK compact_openK_nbhs_filter
  compact_openK_nbhsE_subproof compact_openK_openE_subproof.

End compact_open_setwise.

Definition compact_open_def :=
  sup_topology (fun i : sigT (@compact T) =>
    Topological.class (@compact_openK (projT1 i))).

HB.instance Definition _ := Nbhs.copy compact_open compact_open_def.

HB.instance Definition _ : Nbhs_isTopological compact_open :=
  Topological.copy compact_open compact_open_def.

Lemma compact_open_cvgP (F : set_system compact_open)
    (f : compact_open) :
  Filter F ->
  F --> f <-> forall K O, @compact T K -> @open U O -> f @` K `<=` O ->
    F [set g | g @` K `<=` O].

Proof.

move=> FF; split.
by move/cvg_sup => + K O cptK ? ? => /(_ (existT _ _ cptK)); apply; exists O.
move=> fko; apply/cvg_sup => -[A cptK] O /= [C /= [fAC oC]].
by move/filterS; apply; exact: fko.
Qed.

Lemma compact_open_open (K : set T) (O : set U) :
compact K -> open O -> open ([set g | g @` K `<=` O] : set compact_open).

Proof.

pose C := [set g | g @` K `<=` O]; move=> cptK oO.
exists [set C]; last by rewrite bigcup_set1.
move=> _ ->; exists (fset1 C) => //; last by rewrite set_fset1 bigcap_set1.
by move=> _ /[!inE] ->; exists (existT _ _ cptK) => // z Cz; exists O.
Qed.

End compact_open.

HB.instance Definition _ {U : topologicalType} {V : ptopologicalType} K :=
    Pointed.on (@compact_openK U V K).

HB.instance Definition _ {U : topologicalType} {V : ptopologicalType} :=
  Pointed.on (@compact_open U V).

Notation "{ 'compact-open' , U -> V }" := (@compact_open U V).
Notation "{ 'compact-open' , F --> f }" :=
  (F --> (f : @compact_open _ _)).

Section compact_open_uniform.
Context {U : topologicalType} {V : puniformType}.

Let small_ent_sub := @small_set_sub _ (@entourage V).

Lemma compact_open_fam_compactP (f : U -> V) (F : set_system (U -> V)) :
  continuous f -> Filter F ->
  {compact-open, F --> f} <-> {family compact, F --> f}.

Proof.

move=> ctsf FF; split; first last.
  move=> cptF; apply/compact_open_cvgP => K O cptK oO fKO.
  apply: cptF; have := fam_compact_nbhs oO fKO cptK ctsf; apply: filter_app.
  by near=> g => /= gKO ? [z Kx <-]; exact: gKO.
move/compact_open_cvgP=> cptOF; apply/cvg_sup => -[K cptK R].
case=> D [[E oE <-] Ekf] /filterS; apply.
move: oE; rewrite openE => /(_ _ Ekf); case => A [J entJ] EKR KfE.
near=> z; apply/KfE/xsectionP/EKR => -[u Kp]; rewrite /sigL_arrow /= /set_val /= /eqincl.
(have Ku : K u by rewrite inE in Kp); move: u Ku {D Kp}; near: z.
move/compact_near_coveringP/near_covering_withinP : (cptK); apply.
move=> u Ku; near (powerset_filter_from (@entourage V)) => E'.
have entE' : entourage E' by exact: (near (near_small_set _)).
pose C := f @^-1` xsection E' (f u).
pose B := \bigcup_(z in K `&` closure C) interior (xsection E' (f z)).
have oB : open B by apply: bigcup_open => ? ?; exact: open_interior.
have fKB : f @` (K `&` closure C) `<=` B.
  move=> _ [z KCz <-]; exists z => //; rewrite /interior.
  by rewrite -nbhs_entourageE; exists E'.
have cptKC : compact (K `&` closure C).
  by apply: compact_closedI => //; exact: closed_closure.
have := cptOF (K `&` closure C) B cptKC oB fKB.
exists (C, [set g | [set g x | x in K `&` closure C] `<=` B]).
  split; last exact: cptOF.
  by apply: (ctsf) => //; rewrite /filter_of -nbhs_entourageE; exists E'.
case=> z h /= [Cz KB Kz].
case: (KB (h z)); first by exists z; split => //; exact: subset_closure.
move=> w [Kw Cw /interior_subset Jfwhz]; apply: subset_split_ent => //.
exists (f w); last first.
  apply: (near (small_ent_sub _) E') => //.
  exact/xsectionP.
apply: subset_split_ent => //; exists (f u).
  apply/entourage_sym; apply: (near (small_ent_sub _) E') => //.
  exact/xsectionP.
have [] := Cw (f @^-1` xsection E' (f w)).
  by apply: ctsf; rewrite /= -nbhs_entourageE; exists E'.
move=> r [Cr /= Ewr]; apply: subset_split_ent => //; exists (f r).
  apply: (near (small_ent_sub _) E') => //.
  exact/xsectionP.
apply/entourage_sym; apply: (near (small_ent_sub _) E') => //.
exact/xsectionP.
Unshelve. all: by end_near. Qed.

End compact_open_uniform.

Module ArrowAsCompactOpen.
HB.instance Definition _ (U : topologicalType) (V : topologicalType) :=
  Topological.copy (U -> V) {compact-open, U -> V}.
End ArrowAsCompactOpen.

Definition compactly_in {U : topologicalType} (A : set U) :=
  [set B | B `<=` A /\ compact B].

Lemma compact_cvg_within_compact {U : topologicalType} {V : uniformType}
    (C : set U) (F : set_system (U -> V)) (f : U -> V) :
  Filter F -> compact C ->
  {uniform C, F --> f} <-> {family compactly_in C, F --> f}.

Proof.

move=> FF CC.
apply: (iff_trans _ (iff_sym (fam_cvgP _ _ FF))); split.
- by move=> CFf D [/uniform_subset_cvg + _]; apply.
- by apply; split.
Qed.

Section UniformContinuousLimits.

Lemma uniform_limit_continuous {U : topologicalType} {V : uniformType}
    (F : set_system (U -> V)) (f : U -> V) :
  ProperFilter F -> (\forall g \near F, continuous (g : U -> V)) ->
  {uniform, F --> f} -> continuous f.

Proof.

move=> PF ctsF Ff x; apply/cvg_app_entourageP => A entA; near F => g; near=> y.
apply: (entourage_split (g x)) => //.
by near: g; apply/Ff/uniform_nbhs; exists (split_ent A); split => // ?; exact.
apply: (entourage_split (g y)) => //; near: y; near: g.
by apply: (filterS _ ctsF) => g /(_ x) /cvg_app_entourageP; exact.
apply/Ff/uniform_nbhs; exists (split_ent (split_ent A))^-1%relation.
by split; [exact: entourage_inv | move=> g fg; near_simpl; near=> z; exact: fg].
Unshelve. all: end_near. Qed.

Lemma uniform_limit_continuous_subspace {U : topologicalType} {V : puniformType}
    (K : set U) (F : set_system (U -> V)) (f : subspace K -> V) :
  ProperFilter F -> (\forall g \near F, continuous (g : subspace K -> V)) ->
  {uniform K, F --> f} -> {within K, continuous f}.

Proof.

move=> PF ctsF Ff; apply: (@subspace_eq_continuous _ _ _ (restrict K f)).
  by rewrite /restrict => ? ->.
apply: (@uniform_limit_continuous (subspace K) _ (restrict K @ F) _).
  apply: (filterS _ ctsF) => g; apply: subspace_eq_continuous.
  by rewrite /restrict => ? ->.
by apply (@uniform_restrict_cvg _ _ F ) => //; exact: PF.
Qed.

End UniformContinuousLimits.

Section UniformPointwise.
Context {U : topologicalType} {V : uniformType}.

Definition singletons {T : Type} := [set [set x] | x in [set: T]].

Lemma pointwise_cvg_family_singleton F (f: U -> V):
Filter F -> {ptws, F --> f} = {family @singletons U, F --> f}.

Proof.

move=> FF; apply/propext.
rewrite (@fam_cvgP _ _ singletons). (* BUG: slowdown if no arguments *)
rewrite cvg_sup; split.
  move=> + A [x _ <-] => /(_ x); rewrite uniform_set1.
  rewrite cvg_image; last by rewrite eqEsubset; split=> v // _; exists (cst v).
  apply: cvg_trans => W /=; rewrite ?nbhs_simpl /fmap /= => [[W' + <-]].
  by apply: filterS => g W'g /=; exists g.
move=> + i; have /[swap] /[apply] : singletons [set i] by exists i.
rewrite uniform_set1.
rewrite cvg_image; last by rewrite eqEsubset; split=> v // _; exists (cst v).
move=> + W //=; rewrite ?nbhs_simpl => Q => /Q Q'; exists (@^~ i @^-1` W) => //.
by rewrite eqEsubset; split => [j [? + <-//]|j Wj]; exists (fun _ => j).
Qed.

Lemma pointwise_cvg_compact_family F (f : U -> V) :
Filter F -> {family compact, F --> f} -> {ptws, F --> f}.

Proof.

move=> PF; rewrite pointwise_cvg_family_singleton; apply: family_cvg_subset.
by move=> A [x _ <-]; exact: compact_set1.
Qed.

Lemma pointwise_cvgP F (f: U -> V):
Filter F -> {ptws, F --> f} <-> forall (t : U), (fun g => g t) @ F --> f t.

Proof.

move=> Ff; rewrite pointwise_cvg_family_singleton; split.
move/fam_cvgP => + t A At => /(_ [set t]); rewrite uniform_set1; apply => //.
by exists t.
by move=> pf; apply/fam_cvgP => ? [t _ <-]; rewrite uniform_set1; exact: pf.
Qed.

End UniformPointwise.

Section ArzelaAscoli.
Context {X : topologicalType} {Y : puniformType} {hsdf : hausdorff_space Y}.
Implicit Types (I : Type).

The key condition in Arzela-Ascoli that, like uniform continuity, moves a quantifier around so all functions have the same "deltas":

Definition equicontinuous {I} (W : set I) (d : I -> (X -> Y)) :=
  forall x (E : set (Y * Y)), entourage E ->
    \forall y \near x, forall i, W i -> E (d i x, d i y).

Lemma equicontinuous_subset {I J} (W : set I) (V : set J)
    {fW : I -> X -> Y} {fV : J -> X -> Y} :
  fW @`W `<=` fV @` V -> equicontinuous V fV -> equicontinuous W fW.

Proof.

move=> WsubV + x E entE => /(_ x E entE); apply: filterS => y VE i Wi.
by case: (WsubV (fW i)); [exists i | move=> j Vj <-; exact: VE].
Qed.

Lemma equicontinuous_subset_id (W V : set (X -> Y)) :
W `<=` V -> equicontinuous V id -> equicontinuous W id.

Proof.

move=> WsubV; apply: equicontinuous_subset => ? [y ? <- /=]; exists y => //.
exact: WsubV.
Qed.

Lemma equicontinuous_continuous_for {I} (W : set I) (fW : I -> X -> Y) i x :
{for x, equicontinuous W fW} -> W i -> {for x, continuous (fW i)}.

Proof.

move=> ectsW Wf; apply/cvg_entourageP => E entE; near_simpl.
by near=> y; apply: (near (ectsW _ entE) y).
Unshelve. end_near. Qed.

Lemma equicontinuous_continuous {I} (W : set I) (fW : I -> (X -> Y)) (i : I) :
equicontinuous W fW -> W i -> continuous (fW i).

Proof.

move=> ectsW Wf x; apply: equicontinuous_continuous_for; last exact: Wf.
by move=> ?; exact: ectsW.
Qed.

A convenient notion that is in between compactness in `{family compact, X -> y}` and compactness in `{ptws X -> y}`:

Definition pointwise_precompact {I} (W : set I) (d : I -> X -> Y) :=
  forall x, precompact [set d i x | i in W].

Lemma pointwise_precompact_subset {I J} (W : set I) (V : set J)
    {fW : I -> X -> Y} {fV : J -> X -> Y} :
  fW @` W `<=` fV @` V -> pointwise_precompact V fV ->
  pointwise_precompact W fW.

Proof.

move=> WsubV + x => /(_ x) pcptV; apply: precompact_subset pcptV => y [i Wi <-].
by case: (WsubV (fW i)); [exists i | move=> j Vj <-; exists j].
Qed.

Lemma pointwise_precompact_precompact {I} (W : set I) (fW : I -> (X -> Y)) :
pointwise_precompact W fW -> precompact ((fW @` W) : set {ptws X -> Y}).

Proof.

rewrite precompactE => ptwsPreW.
pose K := fun x => closure [set fW i x | i in W].
set R := [set f : {ptws X -> Y} | forall x : X, K x (f x)].
have C : compact R.
  by apply: tychonoff => x; rewrite -precompactE; move: ptwsPreW; exact.
apply: (subclosed_compact _ C); first exact: closed_closure.
have WsubR : (fW @` W) `<=` R.
  move=> f Wf x; rewrite /R /K closure_limit_point; left.
  by case: Wf => i ? <-; exists i.
rewrite closureE; apply: smallest_sub (compact_closed _ C) WsubR.
exact: hausdorff_product.
Qed.

Lemma uniform_pointwise_compact (W : set (X -> Y)) :
compact (W : set (@uniform_fun_family X Y compact)) ->
compact (W : set {ptws X -> Y}).

Proof.

rewrite [x in x _ -> _]compact_ultra [x in _ -> x _]compact_ultra.
move=> + F UF FW => /(_ F UF FW) [h [Wh Fh]]; exists h; split => //.
by move=> Q Fq; apply: (pointwise_cvg_compact_family _ Fh).
Qed.

Lemma precompact_pointwise_precompact (W : set {family compact, X -> Y}) :
precompact W -> pointwise_precompact W id.

Proof.

move=> + x; rewrite ?precompactE => pcptW.
have : compact (proj x @` (closure W)).
  apply: continuous_compact => //; apply: continuous_subspaceT=> g.
  move=> E nbhsE; have := (@proj_continuous _ _ x g E nbhsE).
  exact: (@pointwise_cvg_compact_family _ _ (nbhs g)).
move=> /[dup]/(compact_closed hsdf)/closure_id -> /subclosed_compact.
apply; first exact: closed_closure.
by apply/closure_subset/image_subset; exact: (@subset_closure _ W).
Qed.

Lemma pointwise_cvg_entourage (x : X) (f : {ptws X -> Y}) E :
entourage E -> \forall g \near f, E (f x, g x).

Proof.

move=> entE; have : ({ptws, nbhs f --> f}) by [].
have ? : Filter (nbhs f) by exact: nbhs_pfilter. (* NB: This Filter (nbhs f) used to infer correctly. *)
rewrite pointwise_cvg_family_singleton => /fam_cvgP /(_ [set x]).
rewrite uniform_set1 => /(_ _ [set y | E (f x, y)]); apply; first by exists x.
by move: E entE; exact/cvg_entourageP.
Qed.

Lemma equicontinuous_closure (W : set {ptws X -> Y}) :
equicontinuous W id -> equicontinuous (closure W) id.

Proof.

move=> ectsW x E entE; near=> y => f clWf.
have ? : ProperFilter (within W (nbhs (f : {ptws X -> Y}))).
exact: within_nbhs_proper. (* TODO: This ProperFilter _ also used to infer correctly. *)
near (within W (nbhs (f : {ptws X -> Y}))) => g.
near: g; rewrite near_withinE; near_simpl; near=> g => Wg.
apply: (@entourage_split _ (g x)) => //.
exact: (near (pointwise_cvg_entourage _ _ _)).
apply: (@entourage_split _ (g y)) => //; first exact: (near (@ectsW x _ _)).
by apply/entourage_sym; exact: (near (pointwise_cvg_entourage _ _ _)).
Unshelve. all: by end_near. Qed.

Definition small_ent_sub := @small_set_sub _ (@entourage Y).

Lemma pointwise_compact_cvg (F : set_system {ptws X -> Y}) (f : {ptws X -> Y}) :
  ProperFilter F ->
  (\forall W \near powerset_filter_from F, equicontinuous W id) ->
  {ptws, F --> f} <-> {family compact, F --> f}.

Proof.

move=> PF /near_powerset_filter_fromP; case.
  exact: equicontinuous_subset_id.
move=> W; wlog Wf : f W / W f.
  move=> + FW /equicontinuous_closure => /(_ f (closure (W : set {ptws X -> Y}))) Q.
  split => Ff; last by apply: pointwise_cvg_compact_family.
  apply/Q => //.
    by rewrite closureEcvg; exists F; [|split] => // ? /= /filterS; apply.
  by apply: (filterS _ FW) => z Wz; apply: subset_closure.
move=> FW ectsW; split=> [ptwsF|]; last exact: pointwise_cvg_compact_family.
apply/fam_cvgP => K ? U /=; rewrite uniform_nbhs => [[E [eE EsubU]]].
suff : \forall g \near within W (nbhs (f : {ptws X -> Y})),
    forall y, K y -> E (f y, g y).
  rewrite near_withinE; near_simpl => N; apply: (filter_app _ _ FW).
  by apply: ptwsF; near=> g => ?; apply: EsubU; apply: (near N g).
near (powerset_filter_from (@entourage Y)) => E'.
have entE' : entourage E' by exact: (near (near_small_set _)).
pose Q := fun (h : X -> Y) x => E' (f x, h x).
apply: (iffLR (compact_near_coveringP K)) => // x Kx.
near=> y g => /=.
apply: (entourage_split (f x) eE).
  apply entourage_sym; apply: (near (small_ent_sub _) E') => //.
  exact: (near (ectsW x E' entE') y).
apply: (@entourage_split _ (g x)) => //.
  apply: (near (small_ent_sub _) E') => //.
  near: g; near_simpl; apply: (@cvg_within _ (nbhs (f : {ptws X -> Y}))).
  exact: pointwise_cvg_entourage.
apply: (near (small_ent_sub _) E') => //.
apply: (near (ectsW x E' entE')) => //.
exact: (near (withinT _ (nbhs_filter (f : {ptws X -> Y})))).
Unshelve. all: end_near. Qed.

Lemma pointwise_compact_closure (W : set (X -> Y)) :
  equicontinuous W id ->
  closure (W : set {family compact, X -> Y}) =
  closure (W : set {ptws X -> Y}).

Proof.

rewrite ?closureEcvg // predeqE => ? ?.
split; move=> [F PF [Fx WF]]; (exists F; last split) => //.
  apply/@pointwise_compact_cvg => //; apply/near_powerset_filter_fromP.
    exact: equicontinuous_subset_id.
  by exists W => //; exact: WF.
apply/@pointwise_compact_cvg => //; apply/near_powerset_filter_fromP.
  exact: equicontinuous_subset_id.
by exists W => //; exact: WF.
Qed.

Lemma pointwise_precompact_equicontinuous (W : set (X -> Y)) :
  pointwise_precompact W id ->
  equicontinuous W id ->
  precompact (W : set {family compact, X -> Y }).

Proof.

move=> /pointwise_precompact_precompact + ectsW.
rewrite ?precompactE compact_ultra compact_ultra pointwise_compact_closure //.
move=> /= + F UF FcW => /(_ F UF); rewrite image_id => /(_ FcW)[p [cWp Fp]].
exists p; split => //; apply/pointwise_compact_cvg => //.
apply/near_powerset_filter_fromP; first exact: equicontinuous_subset_id.
exists (closure (W : set {ptws X -> Y })) => //.
exact: equicontinuous_closure.
Qed.

Section precompact_equicontinuous.
Hypothesis lcptX : locally_compact [set: X].

Lemma compact_equicontinuous (W : set {family compact, X -> Y}) :
  (forall f, W f -> continuous f) ->
  compact (W : set {family compact, X -> Y}) ->
  equicontinuous W id.

Proof.

move=> ctsW cptW x E entE.
have [//|U UWx [cptU clU]] := @lcptX x; rewrite withinET in UWx.
near (powerset_filter_from (@entourage Y)) => E'.
have entE' : entourage E' by exact: (near (near_small_set _)).
pose Q := fun (y : X) (f : {family compact, X -> Y}) => E' (f x, f y).
apply: (iffLR (compact_near_coveringP W)) => // f Wf; near=> g y => /=.
apply: (entourage_split (f x) entE).
  apply/entourage_sym; apply: (near (small_ent_sub _) E') => //.
  exact: (near (fam_nbhs _ entE' (@compact_set1 _ x)) g).
apply: (entourage_split (f y) (entourage_split_ent entE)).
  apply: (near (small_ent_sub _) E') => //.
  by apply/xsectionP; near: y; apply: (@ctsW f Wf x); exact: nbhs_entourage.
apply: (near (small_ent_sub _) E') => //.
by apply: (near (fam_nbhs _ entE' cptU) g) => //; exact: (near UWx y).
Unshelve. all: end_near. Qed.

Lemma precompact_equicontinuous (W : set {family compact, X -> Y}) :
  (forall f, W f -> continuous f) ->
  precompact (W : set {family compact, X -> Y}) ->
  equicontinuous W id.

Proof.

move=> pcptW ctsW; apply: (equicontinuous_subset_id (@subset_closure _ W)).
apply: compact_equicontinuous; last by rewrite -precompactE.
move=> f; rewrite closureEcvg => [[G PG [Gf GW]]] x B /=.
rewrite -nbhs_entourageE => -[E entE] /filterS; apply; near_simpl.
suff ctsf : continuous f.
  near=> x0; apply/xsectionP; near: x0.
  by move: E entE; apply/cvg_app_entourageP; exact: ctsf.
apply/continuous_localP => x'; apply/near_powerset_filter_fromP.
  by move=> ? ?; exact: continuous_subspaceW.
case: (@lcptX x') => // U; rewrite withinET => nbhsU [cptU _].
exists U => //; apply: (uniform_limit_continuous_subspace PG _ _).
  by near=> g; apply: continuous_subspaceT; near: g; exact: GW.
by move/fam_cvgP/(_ _ cptU) : Gf.
Unshelve. all: end_near. Qed.

End precompact_equicontinuous.

Theorem Ascoli (W : set {family compact, X -> Y}) :
    locally_compact [set: X] ->
  pointwise_precompact W id /\ equicontinuous W id <->
    (forall f, W f -> continuous f) /\
    precompact (W : set {family compact, X -> Y}).

Proof.

move=> lcpt; split => [[Wid ectsW]|[fWf]pcptW].
split=> [?|]; first exact: equicontinuous_continuous.
exact: pointwise_precompact_equicontinuous.
split; last exact: precompact_equicontinuous.
exact: precompact_pointwise_precompact.
Qed.

End ArzelaAscoli.

Section currying.
Local Import ArrowAsCompactOpen.

Section cartesian_closed.
Context {U V W : topologicalType}.

In this section, we consider under what conditions \ `[f in U ~> V ~> W | continuous f /\ forall u, continuous (f u)]` \ and \ `[f in U * V ~> W | continuous f]` \ are homeomorphic. - Always: \ `curry` sends continuous functions to continuous functions. - `V` locally_compact + regular or Hausdorff: \ `uncurry` sends continuous functions to continuous functions. - `U` regular or Hausdorff: \ `curry` itself is a continuous map. - `U` regular or Hausdorff AND `V` locally_compact + regular or Hausdorff \ `uncurry` itself is a continuous map. \ Therefore `curry`/`uncurry` are homeomorphisms. So the category of locally compact regular spaces is cartesian closed.

Lemma continuous_curry (f : U * V -> W) :
continuous f ->
continuous (curry f) /\ forall u, continuous (curry f u).

Proof.

move=> ctsf; split; first last.
  move=> u z; apply: continuous_comp; last exact: ctsf.
  by apply: cvg_pair => //=; exact: cvg_cst.
move=> x; apply/compact_open_cvgP => K O /= cptK oO fKO.
near=> z => w /= [+ + <-]; near: z.
move/compact_near_coveringP/near_covering_withinP : cptK; apply.
move=> v Kv; have [[P Q] [Px Qv] PQfO] : nbhs (x, v) (f @^-1` O).
  by apply: ctsf; move: oO; rewrite openE; apply; apply: fKO; exists v.
by exists (Q, P) => // -[b a] /= [Qb Pa] Kb; exact: PQfO.
Unshelve. all: by end_near. Qed.

Lemma continuous_uncurry_regular (f : U -> V -> W) :
locally_compact [set: V] -> @regular_space V -> continuous f ->
(forall u, continuous (f u)) -> continuous (uncurry f).

Proof.

move=> lcV reg cf cfp /= [u v] D; rewrite /= nbhsE => -[O [oO Ofuv]] /filterS.
apply; have [B] := @lcV v I; rewrite withinET => Bv [cptB clB].
have [R Rv RO] : exists2 R, nbhs v R & forall z, closure R z -> O (f u z).
  have [] := reg v (f u @^-1` O); first by apply: cfp; exact: open_nbhs_nbhs.
  by move=> R ? ?; exists R.
exists (f @^-1` [set g | g @` (B `&` closure R) `<=` O], B `&` closure R).
  split; [apply/cf/open_nbhs_nbhs; split | apply: filterI] => //.
  - apply: compact_open_open => //; apply: compact_closedI => //.
    exact: closed_closure.
  - by move=> ? [x [? + <-]]; apply: RO.
  - by apply: filterS; first exact: subset_closure.
by case=> a r /= [fBMO [Br] cmR]; apply: fBMO; exists r.
Qed.

Lemma continuous_uncurry (f : U -> V -> W) :
locally_compact [set: V] -> hausdorff_space V -> continuous f ->
(forall u, continuous (f u)) -> continuous (uncurry f).

Proof.

move=> lcV hsdf ctsf cf; apply: continuous_uncurry_regular => //.
move=> v; have [B] := @lcV v I; rewrite withinET => Bv [cptB clB].
by move=> z; exact: (@compact_regular V hsdf v B).
Qed.

Lemma curry_continuous (f : (U * V)%type -> W) : continuous f -> @regular_space U ->
{for f, continuous curry}.

Proof.

move=> ctsf regU; apply/compact_open_cvgP.
  by apply: fmap_filter; exact: nbhs_filter.
move=> K ? cptK [D OfinIo <-] fKD /=; near=> z => w [+ + <-]; near: z.
move/compact_near_coveringP/near_covering_withinP : (cptK); apply => u Ku.
have [] := fKD (curry f u); first by exists u.
move=> E /[dup] /[swap] /OfinIo [N Asub <- DIN INf].
suff : \forall x' \near u & i \near nbhs f, K x' ->
    (\bigcap_(i in [set` N]) i) (curry i x').
  apply: filter_app; near=> a b => /[apply] ?.
  by exists (\bigcap_(i in [set` N]) i).
apply: filter_bigI_within => R RN; have /set_mem [[M cptM _]] := Asub _ RN.
have Rfu : R (curry f u) by exact: INf.
move/(_ _ Rfu) => [O [fMO oO] MOR]; near=> p => /= Ki; apply: MOR => + [+ + <-].
move=> _ v Mv; move: v Mv Ki; near: p.
have umb : \forall y \near u, (forall b, M b -> nbhs (y, b) (f @^-1` O)).
  move/compact_near_coveringP/near_covering_withinP : (cptM); apply => v Mv.
  have [[P Q] [Pu Qv] PQO] : nbhs (u, v) (f @^-1` O).
    by apply: ctsf; apply: open_nbhs_nbhs; split => //; apply: fMO; exists v.
  exists (Q, P); [by []| move=> [b a [/= Qb Pa Mb]]].
  by apply: ctsf; apply: open_nbhs_nbhs; split => //; exact: PQO.
move/compact_near_coveringP/near_covering_withinP : (cptM); apply => v Mv.
have [P' P'u cPO] := regU u _ umb.
pose L := [set h | h @` ((K `&` closure P') `*` M) `<=` O].
exists (setT, P' `*` L).
  split => //; [exact: filterT|]; exists (P', L) => //; split => //.
  apply: open_nbhs_nbhs; split; first apply: compact_open_open => //.
    apply: compact_setX => //; apply: compact_closedI => //.
    exact: closed_closure.
  by move=> ? [[a b] [[Ka /cPO +] Mb <-]] => /(_ _ Mb)/nbhs_singleton.
move=> [b [a h]] [/= _ [Pa] +] Ma Ka; apply.
by exists (a, b); split => //; split => //; exact/subset_closure.
Unshelve. all: by end_near. Qed.

Lemma uncurry_continuous (f : U -> V -> W) :
  locally_compact [set: V] -> @regular_space V -> @regular_space U ->
  continuous f -> (forall u, continuous (f u)) ->
  {for f, continuous uncurry}.

Proof.

move=> lcV regV regU ctsf ctsfp; apply/compact_open_cvgP.
  by apply: fmap_filter; exact:nbhs_filter.
move=> /= K O cptK oO fKO; near=> h => ? [+ + <-]; near: h.
move/compact_near_coveringP/near_covering_withinP: (cptK); apply.
case=> u v Kuv.
have : exists P Q, [/\ closed P, compact Q, nbhs u P,
    nbhs v Q & P `*` Q `<=` uncurry f @^-1` O].
  have : continuous (uncurry f) by exact: continuous_uncurry_regular.
  move/continuousP/(_ _ oO); rewrite openE => /(_ (u, v))[].
    by apply: fKO; exists (u, v).
  case=> /= P' Q' [P'u Q'v] PQO.
  have [B] := @lcV v I; rewrite withinET; move=> Bv [cptB clB].
  have [P Pu cPP'] := regU u P' P'u; have [Q Qv cQQ'] := regV v Q' Q'v.
  exists (closure P), (B `&` closure Q); split.
  - exact: closed_closure.
  - by apply: compact_closedI => //; exact: closed_closure.
  - by apply: filterS; first exact: subset_closure.
  - by apply: filterI=> //; apply: filterS; first exact: subset_closure.
  - by case => a b [/cPP' ?] [_ /cQQ' ?]; exact: PQO.
case=> P [Q [clP cptQ Pu Qv PQfO]]; pose R := [set g : V -> W | g @` Q `<=` O].
(have oR : open R by exact: compact_open_open); pose P' := f @^-1` R.
pose L := [set h : U -> V -> W | h @` (fst @` K `&` P) `<=` R].
exists ((P `&` P') `*` Q, L); first split => /=.
- exists (P `&` P', Q) => //; split => //=; apply: filterI => //.
  apply: ctsf; apply: open_nbhs_nbhs; split => // _ [b Qb <-].
  by apply: (PQfO (u, b)); split => //; exact: nbhs_singleton.
- rewrite nbhs_simpl /=; apply: open_nbhs_nbhs; split.
    apply: compact_open_open => //; apply: compact_closedI => //.
    apply: continuous_compact => //; apply: continuous_subspaceT => x.
    exact: cvg_fst.
  move=> /= _ [a [Kxa Pa] <-] _ [b Qb <-].
  by apply: (PQfO (a, b)); split => //; exact: nbhs_singleton.
move=> [[a b h]] [/= [[Pa P'a] Qb Lh] Kab].
apply: (Lh (h a)); first by exists a => //; split => //; exists (a, b).
by exists b.
Unshelve. all: by end_near. Qed.

End cartesian_closed.

End currying.

Files

Module mathcomp.analysis.function_spaces