Module mathcomp.analysis.topology_theory.initial_topology

From HB Require Import structures.
From mathcomp Require Import all_ssreflect_compat all_algebra all_classical.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import interval_inference reals topology_structure.
From mathcomp Require Import uniform_structure order_topology.
From mathcomp Require Import pseudometric_structure.

# Initial topology This file defines the initial topology for $S$ by a function $f$ whose domain is $S$. This topology is also known as initial topology on $S$ with respect to $f$. NB: Before version 1.16.0, the initial topology was called the weak topology. Though in some literature (e.g., Wilansky) it can be called that way, we reserve "weak topology" for the topology induced on a topological vector space by its dual. ``` initial_topology f == initial topology by a function f : S -> T on S S must be a choiceType and T a topologicalType. ``` `initial_topology` is equipped with the structures of: - uniform space - pseudometric space (the metric space for initial topologies)

Import Order.TTheory GRing.Theory Num.Theory.

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

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Definition initial_topology {S : Type} {T : Type}
(f : S -> T) : Type := S.

Section Initial_Topology.
Variable (S : choiceType) (T : topologicalType) (f : S -> T).
Local Notation W := (initial_topology f).

Definition wopen := [set f @^-1` A | A in open].

Local Lemma wopT : wopen [set: W].

Proof.

by exists setT => //; apply: openT. Qed.

Local Lemma wopI : setI_closed wopen.

Proof.

by move=> ? ? [C Cop <-] [D Dop <-]; exists (C `&` D) => //; exact: openI.
Qed.

Local Lemma wop_bigU (I : Type) (g : I -> set W) :
(forall i, wopen (g i)) -> wopen (\bigcup_i g i).

Proof.

move=> gop.
set opi := fun i => [set Ui | open Ui /\ g i = f @^-1` Ui].
exists (\bigcup_i get (opi i)).
  apply: bigcup_open => i.
  by have /getPex [] : exists U, opi i U by have [U] := gop i; exists U.
have g_preim i : g i = f @^-1` (get (opi i)).
  by have /getPex [] : exists U, opi i U by have [U] := gop i; exists U.
rewrite predeqE => s; split=> [[i _]|[i _]]; last by rewrite g_preim; exists i.
by rewrite -[_ _]/((f @^-1` _) _) -g_preim; exists i.
Qed.

HB.instance Definition _ := Choice.on W.
HB.instance Definition _ :=
isOpenTopological.Build W wopT wopI wop_bigU.

Lemma initial_continuous : continuous (f : W -> T).

Proof.

by apply/continuousP => A ?; exists A. Qed.

Lemma cvg_image (F : set_system S) (s : S) :
Filter F -> f @` setT = setT ->
F --> (s : W) <-> ([set f @` A | A in F] : set_system _) --> f s.

Proof.

move=> FF fsurj; split=> [cvFs|cvfFfs].
  move=> A /initial_continuous [B [Bop Bs sBAf]].
  have /cvFs FB : nbhs (s : W) B by apply: open_nbhs_nbhs.
  rewrite nbhs_simpl; exists (f @^-1` A); first exact: filterS FB.
  exact: image_preimage.
move=> A /= [_ [[B Bop <-] Bfs sBfA]].
have /cvfFfs [C FC fCeB] : nbhs (f s) B by rewrite nbhsE; exists B.
rewrite nbhs_filterE; apply: filterS FC.
by apply: subset_trans sBfA; rewrite -fCeB; apply: preimage_image.
Qed.

End Initial_Topology.

HB.instance Definition _ (S : pointedType) (T : topologicalType) (f : S -> T) :=
Pointed.on (initial_topology f).

Section initial_uniform.
Local Open Scope relation_scope.
Variable (pS : choiceType) (U : uniformType) (f : pS -> U).

Let S := initial_topology f.

Definition initial_ent : set_system (S * S) :=
filter_from (@entourage U) (fun V => (map_pair f)@^-1` V).

Let initial_ent_filter : Filter initial_ent.

Proof.

apply: filter_from_filter; first by exists setT; exact: entourageT.
by move=> P Q ??; (exists (P `&` Q); first exact: filterI) => ?.
Qed.

Let initial_ent_refl A : initial_ent A -> diagonal `<=` A.

Proof.

by move=> [B ? sBA] [x y]/diagonalP ->; apply/sBA; exact: entourage_refl.
Qed.

Let initial_ent_inv A : initial_ent A -> initial_ent A^-1.

Proof.

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

Let initial_ent_split A : initial_ent A -> exists2 B, initial_ent B & B \; B `<=` A.

Proof.

move=> [B entB sBA]; have : exists C, entourage C /\ C \; C `<=` B.
exact/exists2P/entourage_split_ex.
case=> C [entC CsubB]; exists ((map_pair f)@^-1` C); first by exists C.
by case=> x y [a ? ?]; apply/sBA/CsubB; exists (f a).
Qed.

Let initial_ent_nbhs : nbhs = nbhs_ initial_ent.

Proof.

rewrite predeq2E => x V; split.
  case=> [? [[B ? <-] ? BsubV]]; have: nbhs (f x) B by apply: open_nbhs_nbhs.
  move=> /nbhsP [W ? WsubB]; exists ((map_pair f) @^-1` W); first by exists W.
  by move=> ? ?; exact/BsubV/WsubB.
case=> W [V' entV' V'subW] /filterS; apply.
have : nbhs (f x) (xsection V' (f x)) by apply/nbhsP; exists V'.
rewrite (@nbhsE U) => [[O [openU Ofx Osub]]].
(exists (f @^-1` O); repeat split => //); first by exists O => //.
by move=> w ?; apply/mem_set; apply: V'subW; apply/set_mem; exact: Osub.
Qed.

HB.instance Definition _ := @Nbhs_isUniform.Build (initial_topology f)
  initial_ent initial_ent_filter initial_ent_refl initial_ent_inv
  initial_ent_split initial_ent_nbhs.

End initial_uniform.

HB.instance Definition _ (pS : pointedType) (U : uniformType) (f : pS -> U) :=
  Pointed.on (initial_topology f).

Section initial_pseudoMetric.
Context {R : realType} (pS : choiceType) (U : pseudoMetricType R) .
Variable (f : pS -> U).

Notation S := (initial_topology f).

Definition initial_ball (x : S) (r : R) (y : S) := ball (f x) r (f y).

Let initial_pseudo_metric_ball_center (x : S) (e : R) : 0 < e ->
  initial_ball x e x.

Proof.

by move=> /posnumP[{}e]; exact: ball_center. Qed.

Let initial_pseudo_metric_entourageE : entourage = entourage_ initial_ball.

Proof.

rewrite /entourage /= /initial_ent -entourage_ballE /entourage_.
have -> : (fun e => [set xy | ball (f xy.1) e (f xy.2)]) =
   (preimage (map_pair f) \o fun e => [set xy | ball xy.1 e xy.2])%FUN.
  by [].
rewrite eqEsubset; split; apply/filter_fromP.
- apply: filter_from_filter; first by exists 1 => /=.
  move=> e1 e2 e1pos e2pos; wlog e1lee2 : e1 e2 e1pos e2pos / e1 <= e2.
    by have [?|/ltW ?] := lerP e1 e2; [exact | rewrite setIC; exact].
  exists e1 => //; rewrite -preimage_setI; apply: preimage_subset.
  by move=> ? ?; split => //; apply: le_ball; first exact: e1lee2.
- by move=> E [e ?] heE; exists e => //; apply: preimage_subset.
- apply: filter_from_filter.
    by exists [set xy | ball xy.1 1 xy.2]; exists 1 => /=.
  move=> E1 E2 [e1 e1pos he1E1] [e2 e2pos he2E2].
  wlog ? : E1 E2 e1 e2 e1pos e2pos he1E1 he2E2 / e1 <= e2.
    have [? /(_ _ _ e1 e2)|/ltW ? ] := lerP e1 e2; first exact.
    by rewrite setIC => /(_ _ _ e2 e1); exact.
  exists (E1 `&` E2) => //; exists e1 => // xy /= B; split; first exact: he1E1.
  by apply/he2E2/le_ball; last exact: B.
- by move=> e ?; exists [set xy | ball xy.1 e xy.2] => //; exists e => /=.
Qed.

HB.instance Definition _ := Uniform_isPseudoMetric.Build R S
  initial_pseudo_metric_ball_center (fun _ _ _ => @ball_sym _ _ _ _ _)
  (fun _ _ _ _ _ => @ball_triangle _ _ _ _ _ _ _)
  initial_pseudo_metric_entourageE.

Lemma initial_ballE (e : R) (x : S) : f @^-1` (ball (f x) e) = ball x e.

Proof.

by []. Qed.

End initial_pseudoMetric.

for an orderedTopologicalType T, and subtype U (order_topology (sub_type U)) `<=` (initial_topology (sub_type U)) but generally the topologies are not equal! Consider `

0,1[ | {2}` as a subset of `[0,3]` for an example

Section initial_order_refine.
Context {d} {X : orderTopologicalType d} {Y : subType X}.

Let OrdU : orderTopologicalType d := order_topology (sub_type Y).
Let InitialU : topologicalType := @initial_topology (sub_type Y) X val.

Lemma open_order_initial (U : set Y) : @open OrdU U -> @open InitialU U.

Proof.

rewrite ?openE /= /interior => + x Ux => /(_ x Ux); rewrite itv_nbhsE /=.
move=> [][][[]l|[]] [[]r|[]][]//= _ xlr /filterS; apply.
- exists `]l, r[%classic; split => //=; exists `]\val l, \val r[%classic.
    exact: itv_open.
  by rewrite eqEsubset; split => z; rewrite preimage_itv.
- exists `]l, +oo[%classic; split => //=; exists `]\val l, +oo[%classic.
    exact: rray_open.
  by rewrite eqEsubset; split => z; rewrite preimage_itv.
- exists `]-oo, r[%classic; split => //=; exists `]-oo, \val r[%classic.
    exact: lray_open.
  by rewrite eqEsubset; split => z; rewrite preimage_itv.
- by rewrite set_itvE; exact: filterT.
Qed.

End initial_order_refine.

Lemma continuous_comp_initial {Y : choiceType} {X Z : topologicalType}
(w : Y -> Z) (f : X -> initial_topology w) :
continuous (w \o f) -> continuous f.

Proof.

move=> cf z U [?/= [[W oW <-]]] /= Wsfz /filterS; apply; apply: cf.
exact: open_nbhs_nbhs.
Qed.

Files

Module mathcomp.analysis.topology_theory.initial_topology