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Coq-UniMath

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Definition pr1BaseOfFilter {X : UU} : BaseOfFilter X → BaseOfPreFilter X. Proof. intros base. exists (pr1 base). split. - apply (pr1 (pr2 base)). - apply (pr1 (pr2 (pr2 base))). Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

pr1BaseOfFilter

Lemma BaseOfPreFilter_and {X : UU} (base : BaseOfPreFilter X) : ∏ A B : X → hProp, base A → base B → ∃ C : X → hProp, base C × (∏ x, C x → A x ∧ B x). Proof. apply (pr1 (pr2 base)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

BaseOfPreFilter_and

Lemma BaseOfPreFilter_notempty {X : UU} (base : BaseOfPreFilter X) : ∃ A : X → hProp, base A. Proof. apply (pr2 (pr2 base)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

BaseOfPreFilter_notempty

Lemma BaseOfFilter_and {X : UU} (base : BaseOfFilter X) : ∏ A B : X → hProp, base A → base B → ∃ C : X → hProp, base C × (∏ x, C x → A x ∧ B x). Proof. apply (pr1 (pr2 base)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

BaseOfFilter_and

Lemma BaseOfFilter_notempty {X : UU} (base : BaseOfFilter X) : ∃ A : X → hProp, base A. Proof. apply (pr1 (pr2 (pr2 base))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

BaseOfFilter_notempty

Lemma BaseOfFilter_notfalse {X : UU} (base : BaseOfFilter X) : ∏ A, base A → ∃ x, A x. Proof. apply (pr2 (pr2 (pr2 base))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

BaseOfFilter_notfalse

Definition filterbase : (X → hProp) → hProp := λ A : X → hProp, (∃ B : X → hProp, base B × ∏ x, B x → A x).

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase

Lemma filterbase_imply : isfilter_imply filterbase. Proof. intros A B H. apply hinhfun. intros A'. exists (pr1 A'). split. - apply (pr1 (pr2 A')). - intros x Hx. apply H. now apply (pr2 (pr2 A')). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_imply

Lemma filterbase_htrue : isfilter_htrue filterbase. Proof. revert Hempty. apply hinhfun. intros A. exists (pr1 A). split. - apply (pr2 A). - intros. exact tt. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_htrue

Lemma filterbase_and : isfilter_and filterbase. Proof. intros A B. apply hinhuniv2. intros A' B'. refine (hinhfun _ _). 2: apply Hand. 2: (apply (pr1 (pr2 A'))). 2: (apply (pr1 (pr2 B'))). intros C'. exists (pr1 C'). split. - apply (pr1 (pr2 C')). - intros x Cx ; split. + apply (pr2 (pr2 A')). now refine (pr1 (pr2 (pr2 C') _ _)). + apply (pr2 (pr2 B')). now refine (pr2 (pr2 (pr2 C') _ _)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_and

Lemma filterbase_notempty : isfilter_notempty filterbase. Proof. intros A. apply hinhuniv. intros B. generalize (Hfalse _ (pr1 (pr2 B))). apply hinhfun. intros x. exists (pr1 x). apply (pr2 (pr2 B)), (pr2 x). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_notempty

Lemma base_finite_intersection : ∏ L : Sequence (X → hProp), (∏ n, base (L n)) → ∃ A, base A × (∏ x, A x → finite_intersection L x). Proof. intros L Hbase. apply (pr2 (finite_intersection_hProp (λ B, ∃ A : X → hProp, base A × (∏ x : X, A x → B x)))). - split. + revert Hempty. apply hinhfun. intros A. now exists (pr1 A), (pr2 A). + intros A B. apply hinhuniv2. intros A' B'. generalize (Hand _ _ (pr1 (pr2 A')) (pr1 (pr2 B'))). apply hinhfun. intros C. exists (pr1 C). split. * exact (pr1 (pr2 C)). * intros x Cx. split. ** apply (pr2 (pr2 A')), (pr1 (pr2 (pr2 C) _ Cx)). ** apply (pr2 (pr2 B')), (pr2 (pr2 (pr2 C) _ Cx)). - intros n. apply hinhpr. exists (L n). split. + now apply Hbase. + intros; assumption. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

base_finite_intersection

Lemma filterbase_genetated : filterbase = filtergenerated base. Proof. apply funextfun ; intros P. apply hPropUnivalence. - apply hinhfun. intros A. exists (singletonSequence (pr1 A)). split. + intros m ; simpl. exact (pr1 (pr2 A)). + intros x Ax. apply (pr2 (pr2 A)). simple refine (Ax _). now exists 0%nat. - apply hinhuniv. intros L. generalize (base_finite_intersection (pr1 L) (pr1 (pr2 L))). apply hinhfun. intros A. exists (pr1 A) ; split. + exact (pr1 (pr2 A)). + intros x Ax. now apply (pr2 (pr2 L)), (pr2 (pr2 A)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_genetated

Lemma filterbase_generated_hypothesis : ∏ L' : Sequence (X → hProp), (∏ m : stn (length L'), base (L' m)) → ∃ x : X, finite_intersection L' x. Proof. intros L Hbase. generalize (base_finite_intersection L Hbase). apply hinhuniv. intros A. generalize (Hfalse _ (pr1 (pr2 A))). apply hinhfun. intros x. exists (pr1 x). apply (pr2 (pr2 A)). exact (pr2 x). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_generated_hypothesis

Definition PreFilterBase {X : UU} (base : BaseOfPreFilter X) : PreFilter X. Proof. simple refine (make_PreFilter _ _ _ _). - apply (filterbase base). - apply filterbase_imply. - apply filterbase_htrue, BaseOfPreFilter_notempty. - apply filterbase_and. intros A B. now apply BaseOfPreFilter_and. Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

PreFilterBase

Definition FilterBase {X : UU} (base : BaseOfFilter X) : Filter X. Proof. intros. exists (pr1 (PreFilterBase base)). split. - simple refine (pr2 (PreFilterBase base)). - apply filterbase_notempty. intros A Ha. simple refine (BaseOfFilter_notfalse _ _ _). + apply base. + apply Ha. Defined.

Definition

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

FilterBase

Lemma PreFilterBase_Generated {X : UU} (base : BaseOfPreFilter X) : PreFilterBase base = PreFilterGenerated base. Proof. simple refine (subtypePath_prop (B := λ _, make_hProp _ _) _). - apply isapropdirprod. + apply isaprop_isfilter_imply. + apply isaprop_isfilter_finite_intersection. - simpl. apply filterbase_genetated. + intros A B. apply (BaseOfPreFilter_and base). + apply (BaseOfPreFilter_notempty base). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

PreFilterBase_Generated

Lemma FilterBase_Generated {X : UU} (base : BaseOfFilter X) Hbase : FilterBase base = FilterGenerated base Hbase. Proof. simple refine (subtypePath_prop (B := λ _, make_hProp _ _) _). - apply isapropdirprod. + apply isapropdirprod. * apply isaprop_isfilter_imply. * apply isaprop_isfilter_finite_intersection. + apply isaprop_isfilter_notempty. - simpl. apply filterbase_genetated. + intros A B. apply (BaseOfFilter_and base). + apply (BaseOfFilter_notempty base). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

FilterBase_Generated

Lemma FilterBase_Generated_hypothesis {X : UU} (base : BaseOfFilter X) : ∏ L' : Sequence (X → hProp), (∏ m : stn (length L'), base (L' m)) → ∃ x : X, finite_intersection L' x. Proof. apply filterbase_generated_hypothesis. - intros A B. apply (BaseOfFilter_and base). - apply (BaseOfFilter_notempty base). - intros A Ha. now apply (BaseOfFilter_notfalse base). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

FilterBase_Generated_hypothesis

Lemma filterbase_le {X : UU} (base base' : (X → hProp) → hProp) : (∏ P : X → hProp, base P → ∃ Q : X → hProp, base' Q × (∏ x, Q x → P x)) <-> (∏ P : X → hProp, filterbase base P → filterbase base' P). Proof. split. - intros Hbase P. apply hinhuniv. intros A. generalize (Hbase _ (pr1 (pr2 A))). apply hinhfun. intros B. exists (pr1 B). split. + apply (pr1 (pr2 B)). + intros x Bx. apply (pr2 (pr2 A)), (pr2 (pr2 B)), Bx. - intros Hbase P Hp. apply Hbase. apply hinhpr. now exists P. Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

filterbase_le

Lemma PreFilterBase_le {X : UU} (base base' : BaseOfPreFilter X) : (∏ P : X → hProp, base P → ∃ Q : X → hProp, base' Q × (∏ x, Q x → P x)) <-> filter_le (PreFilterBase base') (PreFilterBase base). Proof. split. - intros Hbase P. apply (pr1 (filterbase_le base base')), Hbase. - apply (pr2 (filterbase_le base base')). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

PreFilterBase_le

Lemma FilterBase_le {X : UU} (base base' : BaseOfFilter X) : (∏ P : X → hProp, base P → ∃ Q : X → hProp, base' Q × (∏ x, Q x → P x)) <-> filter_le (FilterBase base') (FilterBase base). Proof. split. - intros Hbase P. apply (pr1 (filterbase_le base base')), Hbase. - apply (pr2 (filterbase_le base base')). Qed.

Lemma

Topology

Require Import UniMath.Foundations.Preamble. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.PartA.

Topology\Filters.v

FilterBase_le

Lemma hinhuniv2' {P X Y : UU} : isaprop P → (X → Y → P) → (∥ X ∥ → ∥ Y ∥ → P). Proof. intros HP Hxy. apply (hinhuniv2 (P := make_hProp _ HP)). exact Hxy. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

hinhuniv2'

Lemma max_le_l : ∏ n m : nat, (n ≤ max n m)%nat. Proof. induction n as [ | n IHn] ; simpl max. - intros m ; reflexivity. - induction m as [ | m _]. + now apply isreflnatleh. + now apply IHn. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

max_le_l

Lemma max_le_r : ∏ n m : nat, (m ≤ max n m)%nat. Proof. induction n as [ | n IHn] ; simpl max. - intros m ; now apply isreflnatleh. - induction m as [ | m _]. + reflexivity. + now apply IHn. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

max_le_r

Definition singletonSequence {X} (A : X) : Sequence X := (1 ,, λ _, A).

Definition

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

singletonSequence

Definition pairSequence {X} (A B : X) : Sequence X. Proof. exists 2. intros m. induction m as [m _]. induction m as [ | _ _]. - exact A. - exact B. Defined.

Definition

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

pairSequence

Definition union {X : UU} (P : (X → hProp) → hProp) : X → hProp := λ x : X, ∃ A : X → hProp, P A × A x.

Definition

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

union

Lemma union_hfalse {X : UU} : union (λ _ : X → hProp, hfalse) = (λ _ : X, hfalse). Proof. apply funextfun ; intros x. apply hPropUnivalence. - apply hinhuniv. intros A. apply (pr1 (pr2 A)). - apply fromempty. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

union_hfalse

Lemma union_or {X : UU} : ∏ A B : X → hProp, union (λ C : X → hProp, C = A ∨ C = B) = (λ x : X, A x ∨ B x). Proof. intros A B. apply funextfun ; intro x. apply hPropUnivalence. - apply hinhuniv. intros C. generalize (pr1 (pr2 C)). apply hinhfun. apply sumofmaps. + intro e. rewrite <- e. left. apply (pr2 (pr2 C)). + intro e. rewrite <- e. right. apply (pr2 (pr2 C)). - apply hinhfun ; apply sumofmaps ; [ intros Ax | intros Bx]. + exists A. split. * apply hinhpr. now left. * exact Ax. + exists B. split. * apply hinhpr. now right. * exact Bx. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

union_or

Lemma union_hProp {X : UU} : ∏ (P : (X → hProp) → hProp), (∏ (L : (X → hProp) → hProp), (∏ A, L A → P A) → P (union L)) → (∏ A B, P A → P B → P (λ x : X, A x ∨ B x)). Proof. intros P Hp A B Pa Pb. rewrite <- union_or. apply Hp. intros C. apply hinhuniv. now apply sumofmaps ; intros ->. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

union_hProp

Lemma union_not_contained_in {X : UU} (U : (X -> hProp) -> hProp) (S : X -> hProp) : union U ⊈ S ⇔ (∃ T, U T ∧ T ⊈ S). Proof. unfold subtype_notContainedIn, union. use make_dirprod; intro H. - use (hinhuniv _ H); intro Hx. induction Hx as [x Hx]. induction Hx as [Hx HSx]. use (hinhfun _ Hx); intro HT. induction HT as [T HT]. exists T. use make_dirprod. + exact (dirprod_pr1 HT). + apply hinhpr. exists x. exact (make_dirprod (dirprod_pr2 HT) HSx). - use (hinhuniv _ H); intro HT. induction HT as [T HT]. use (hinhfun _ (dirprod_pr2 HT)); intro Hx. induction Hx as [x Hx]. exists x. use make_dirprod. + apply hinhpr. exists T. exact (make_dirprod (dirprod_pr1 HT) (dirprod_pr1 Hx)). + exact (dirprod_pr2 Hx). Defined.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

union_not_contained_in

Definition finite_intersection {X : UU} (P : Sequence (X → hProp)) : X → hProp. Proof. intros x. simple refine (make_hProp _ _). - apply (∏ n, P n x). - apply (impred_isaprop _ (λ _, propproperty _)). Defined.

Definition

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection

Lemma finite_intersection_htrue {X : UU} : finite_intersection nil = (λ _ : X, htrue). Proof. apply funextfun ; intros x. apply hPropUnivalence. - intros _. apply tt. - intros _ m. generalize (pr1 m) (pr2 m). intros m' Hm. apply fromempty. revert Hm. apply negnatlthn0. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_htrue

Lemma finite_intersection_1 {X : UU} : ∏ (A : X → hProp), finite_intersection (singletonSequence A) = A. Proof. intros A. apply funextfun ; intros x. apply hPropUnivalence. - intros H. simple refine (H _). now exists 0. - intros Lx m. exact Lx. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_1

Lemma finite_intersection_and {X : UU} : ∏ A B : X → hProp, finite_intersection (pairSequence A B) = (λ x : X, A x ∧ B x). Proof. intros A B. apply funextfun ; intro x. apply hPropUnivalence. - intros H. split. + simple refine (H (0,,_)). reflexivity. + simple refine (H (1,,_)). reflexivity. - intros H m ; simpl. change m with (pr1 m,,pr2 m). generalize (pr1 m) (pr2 m). clear m. intros m Hm. induction m as [ | m _]. + apply (pr1 H). + apply (pr2 H). Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_and

Lemma finite_intersection_case {X : UU} : ∏ (L : Sequence (X → hProp)), finite_intersection L = sumofmaps (λ _ _, htrue) (λ (AB : (X → hProp) × Sequence (X → hProp)) (x : X), pr1 AB x ∧ finite_intersection (pr2 AB) x) (disassembleSequence L). Proof. intros L. apply funextfun ; intros x. apply hPropUnivalence. - intros Hx. induction L as [n L]. induction n as [ | n _] ; simpl. + apply tt. + split. * simple refine (Hx _). * intros m. simple refine (Hx _). - induction L as [n L]. induction n as [ | n _] ; simpl. + intros _ n. apply fromempty. induction (negnatlthn0 _ (pr2 n)). + intros Hx m. change m with (pr1 m,,pr2 m). generalize (pr1 m) (pr2 m) ; clear m ; intros m Hm. induction (natlehchoice _ _ (natlthsntoleh _ _ Hm)) as [Hm' | ->]. * generalize (pr2 Hx (m,,Hm')). unfold dni_lastelement ; simpl. assert (H : Hm = natlthtolths m n Hm' ). { apply (pr2 (natlth m (S n))). } now rewrite H. * assert (H : lastelement = (n,, Hm)). { now apply subtypePath_prop. } rewrite <- H. exact (pr1 Hx). Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_case

Lemma finite_intersection_append {X : UU} : ∏ (A : X → hProp) (L : Sequence (X → hProp)), finite_intersection (append L A) = (λ x : X, A x ∧ finite_intersection L x). Proof. intros A L. rewrite finite_intersection_case. simpl. rewrite append_vec_compute_2. apply funextfun ; intro x. apply maponpaths. apply map_on_two_paths. - induction L as [n L] ; simpl. apply maponpaths. apply funextfun ; intro m. simpl. rewrite <- replace_dni_last. apply append_vec_compute_1. - reflexivity. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_append

Lemma finite_intersection_hProp {X : UU} : ∏ (P : (X → hProp) → hProp), (∏ (L : Sequence (X → hProp)), (∏ n, P (L n)) → P (finite_intersection L)) <-> (P (λ _, htrue) × (∏ A B, P A → P B → P (λ x : X, A x ∧ B x))). Proof. intros P. split. - split. + rewrite <- finite_intersection_htrue. apply X0. intros n. induction (negnatlthn0 _ (pr2 n)). + intros A B Pa Pb. rewrite <- finite_intersection_and. apply X0. intros n. change n with (pr1 n,,pr2 n). generalize (pr1 n) (pr2 n) ; clear n ; intros n Hn. now induction n as [ | n _] ; simpl. - intros Hp. apply (Sequence_rect (P := λ L : Sequence (X → hProp), (∏ n : stn (length L), P (L n)) → P (finite_intersection L))). + intros _. rewrite finite_intersection_htrue. exact (pr1 Hp). + intros L A IHl Hl. rewrite finite_intersection_append. apply (pr2 Hp). * rewrite <- (append_vec_compute_2 L A). now apply Hl. * apply IHl. intros n. rewrite <- (append_vec_compute_1 L A). apply Hl. Qed.

Lemma

Topology

Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Subtypes.

Topology\Prelim.v

finite_intersection_hProp

Definition isSetOfOpen_union := ∏ P : (X → hProp) → hProp, (∏ A : X → hProp, P A → O A) → O (union P).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_union

Lemma isaprop_isSetOfOpen_union : isaprop isSetOfOpen_union. Proof. apply (impred_isaprop _ (λ _, isapropimpl _ _ (propproperty _))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isaprop_isSetOfOpen_union

Definition isSetOfOpen_finite_intersection := ∏ (P : Sequence (X → hProp)), (∏ m, O (P m)) → O (finite_intersection P).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_finite_intersection

Lemma isaprop_isSetOfOpen_finite_intersection : isaprop isSetOfOpen_finite_intersection. Proof. apply (impred_isaprop _ (λ _, isapropimpl _ _ (propproperty _))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isaprop_isSetOfOpen_finite_intersection

Definition isSetOfOpen_htrue := O (λ _, htrue).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_htrue

Definition isSetOfOpen_and := ∏ A B, O A → O B → O (λ x, A x ∧ B x).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_and

Lemma isaprop_isSetOfOpen_and : isaprop isSetOfOpen_and. Proof. apply impred_isaprop ; intros A. apply impred_isaprop ; intros B. apply isapropimpl, isapropimpl. now apply propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isaprop_isSetOfOpen_and

Lemma isSetOfOpen_hfalse : isSetOfOpen_union → O (λ _ : X, hfalse). Proof. intros H0. rewrite <- union_hfalse. apply H0. intro. apply fromempty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_hfalse

Lemma isSetOfOpen_finite_intersection_htrue : isSetOfOpen_finite_intersection → isSetOfOpen_htrue. Proof. intro H0. unfold isSetOfOpen_htrue. rewrite <- finite_intersection_htrue. apply H0. intros m. induction (nopathsfalsetotrue (pr2 m)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_finite_intersection_htrue

Lemma isSetOfOpen_finite_intersection_and : isSetOfOpen_finite_intersection → isSetOfOpen_and. Proof. intros H0 A B Ha Hb. rewrite <- finite_intersection_and. apply H0. intros m ; simpl. induction m as [m Hm]. now induction m. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_finite_intersection_and

Lemma isSetOfOpen_finite_intersection_carac : isSetOfOpen_htrue → isSetOfOpen_and → isSetOfOpen_finite_intersection. Proof. intros Htrue Hpair L. apply (pr2 (finite_intersection_hProp O)). split. - exact Htrue. - exact Hpair. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen_finite_intersection_carac

Definition isSetOfOpen := isSetOfOpen_union × isSetOfOpen_finite_intersection.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isSetOfOpen

Definition isTopologicalSpace (X : hSet) := ∑ O : (X → hProp) → hProp, isSetOfOpen O.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isTopologicalSpace

Definition TopologicalSpace := ∑ X : hSet, isTopologicalSpace X.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologicalSpace

Definition make_TopologicalSpace (X : hSet) (O : (X → hProp) → hProp) (is : isSetOfOpen_union O) (is0 : isSetOfOpen_htrue O) (is1 : isSetOfOpen_and O) : TopologicalSpace := (X,,O,,is,,(isSetOfOpen_finite_intersection_carac _ is0 is1)).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

make_TopologicalSpace

Definition pr1TopologicatSpace : TopologicalSpace → hSet := pr1.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

pr1TopologicatSpace

Definition isOpen {T : TopologicalSpace} : (T → hProp) → hProp := pr1 (pr2 T).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen

Definition Open {T : TopologicalSpace} := ∑ O : T → hProp, isOpen O.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

Open

Definition pr1Open {T : TopologicalSpace} : Open → (T → hProp) := pr1.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

pr1Open

Lemma isOpen_union : ∏ P : (T → hProp) → hProp, (∏ A : T → hProp, P A → isOpen A) → isOpen (union P). Proof. apply (pr1 (pr2 (pr2 T))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_union

Lemma isOpen_finite_intersection : ∏ (P : Sequence (T → hProp)), (∏ m , isOpen (P m)) → isOpen (finite_intersection P). Proof. apply (pr2 (pr2 (pr2 T))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_finite_intersection

Lemma isOpen_hfalse : isOpen (λ _ : T, hfalse). Proof. apply isSetOfOpen_hfalse. intros P H. now apply isOpen_union. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_hfalse

Lemma isOpen_htrue : isOpen (λ _ : T, htrue). Proof. rewrite <- finite_intersection_htrue. apply isOpen_finite_intersection. intros m. induction (nopathsfalsetotrue (pr2 m)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_htrue

Lemma isOpen_and : ∏ A B : T → hProp, isOpen A → isOpen B → isOpen (λ x : T, A x ∧ B x). Proof. apply isSetOfOpen_finite_intersection_and. intros P Hp. apply isOpen_finite_intersection. apply Hp. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_and

Lemma isOpen_or : ∏ A B : T → hProp, isOpen A → isOpen B → isOpen (λ x : T, A x ∨ B x). Proof. intros A B Ha Hb. rewrite <- union_or. apply isOpen_union. intros C. apply hinhuniv. apply sumofmaps ; intros ->. - exact Ha. - exact Hb. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isOpen_or

Definition neighborhood (x : T) : (T → hProp) → hProp := λ P : T → hProp, ∃ O : Open, O x × (∏ y : T, O y → P y).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood

Lemma neighborhood_isOpen (P : T → hProp) : (∏ x, P x → neighborhood x P) <-> isOpen P. Proof. split. - intros Hp. assert (H : ∏ A : T → hProp, isaprop (∏ y : T, A y → P y)). { intros A. apply impred_isaprop. intro y. apply isapropimpl. apply propproperty. } set (Q := λ A : T → hProp, isOpen A ∧ (make_hProp (∏ y : T, A y → P y) (H A))). assert (X : P = (union Q)). { apply funextfun. intros x. apply hPropUnivalence. - intros Px. generalize (Hp _ Px). apply hinhfun. intros A. exists (pr1 A) ; split. + split. * apply (pr2 (pr1 A)). * exact (pr2 (pr2 A)). + exact (pr1 (pr2 A)). - apply hinhuniv. intros A. apply (pr2 (pr1 (pr2 A))). exact (pr2 (pr2 A)). } rewrite X. apply isOpen_union. intros A Ha. apply (pr1 Ha). - intros Hp x Px. apply hinhpr. exists (P,,Hp). split. + exact Px. + intros y Py. exact Py. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_isOpen

Lemma neighborhood_imply : ∏ (x : T) (P Q : T → hProp), (∏ y : T, P y → Q y) → neighborhood x P → neighborhood x Q. Proof. intros x P Q H. apply hinhfun. intros O. exists (pr1 O). split. - apply (pr1 (pr2 O)). - intros y Hy. apply H. apply (pr2 (pr2 O)). exact Hy. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_imply

Lemma neighborhood_forall : ∏ (x : T) (P : T → hProp), (∏ y, P y) → neighborhood x P. Proof. intros x P H. apply hinhpr. exists ((λ _ : T, htrue),,isOpen_htrue). split. - reflexivity. - intros y _. now apply H. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_forall

Lemma neighborhood_and : ∏ (x : T) (A B : T → hProp), neighborhood x A → neighborhood x B → neighborhood x (λ y, A y ∧ B y). Proof. intros x A B. apply hinhfun2. intros Oa Ob. exists ((λ x, pr1 Oa x ∧ pr1 Ob x) ,, isOpen_and _ _ (pr2 (pr1 Oa)) (pr2 (pr1 Ob))). simpl. split. - split. + apply (pr1 (pr2 Oa)). + apply (pr1 (pr2 Ob)). - intros y Hy. split. + apply (pr2 (pr2 Oa)). apply (pr1 Hy). + apply (pr2 (pr2 Ob)). apply (pr2 Hy). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_and

Lemma neighborhood_point : ∏ (x : T) (P : T → hProp), neighborhood x P → P x. Proof. intros x P. apply hinhuniv. intros O. apply (pr2 (pr2 O)). apply (pr1 (pr2 O)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_point

Lemma neighborhood_neighborhood : ∏ (x : T) (P : T → hProp), neighborhood x P → ∃ Q : T → hProp, neighborhood x Q × ∏ y : T, Q y → neighborhood y P. Proof. intros x P. apply hinhfun. intros Q. exists (pr1 Q). split. - apply (pr2 (neighborhood_isOpen _)). + apply (pr2 (pr1 Q)). + apply (pr1 (pr2 Q)). - intros y Qy. apply hinhpr. exists (pr1 Q). split. + exact Qy. + exact (pr2 (pr2 Q)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_neighborhood

Definition locally {T : TopologicalSpace} (x : T) : Filter T. Proof. simple refine (make_Filter _ _ _ _ _). - apply (neighborhood x). - abstract (intros A B ; apply neighborhood_imply). - abstract (apply (pr2 (neighborhood_isOpen _)) ; [ apply isOpen_htrue | apply tt]). - abstract (intros A B ; apply neighborhood_and). - abstract (intros A Ha ; apply hinhpr ; exists x ; now apply neighborhood_point in Ha). Defined.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

locally

Definition is_base_of_neighborhood {T : TopologicalSpace} (x : T) (B : (T → hProp) → hProp) := (∏ P : T → hProp, B P → neighborhood x P) × (∏ P : T → hProp, neighborhood x P → ∃ Q : T → hProp, B Q × (∏ t : T, Q t → P t)).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

is_base_of_neighborhood

Definition base_of_neighborhood {T : TopologicalSpace} (x : T) := ∑ (B : (T → hProp) → hProp), is_base_of_neighborhood x B.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

base_of_neighborhood

Definition pr1base_of_neighborhood {T : TopologicalSpace} (x : T) : base_of_neighborhood x → ((T → hProp) → hProp) := pr1.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

pr1base_of_neighborhood

Definition base_default : (T → hProp) → hProp := λ P : T → hProp, isOpen P ∧ P x.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

base_default

Lemma base_default_1 : ∏ P : T → hProp, base_default P → neighborhood x P. Proof. intros P Hp. apply hinhpr. exists (P,,(pr1 Hp)) ; split. - exact (pr2 Hp). - intros. assumption. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

base_default_1

Lemma base_default_2 : ∏ P : T → hProp, neighborhood x P → ∃ Q : T → hProp, base_default Q × (∏ t : T, Q t → P t). Proof. intros P. apply hinhfun. intros Q. exists (pr1 Q). repeat split. - exact (pr2 (pr1 Q)). - exact (pr1 (pr2 Q)). - exact (pr2 (pr2 Q)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

base_default_2

Definition base_of_neighborhood_default {T : TopologicalSpace} (x : T) : base_of_neighborhood x. Proof. exists (base_default x). split. - now apply base_default_1. - now apply base_default_2. Defined.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

base_of_neighborhood_default

Definition neighborhood' {T : TopologicalSpace} (x : T) (B : base_of_neighborhood x) : (T → hProp) → hProp := λ P : T → hProp, ∃ O : T → hProp, B O × (∏ t : T, O t → P t).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood'

Lemma neighborhood_equiv {T : TopologicalSpace} (x : T) (B : base_of_neighborhood x) : ∏ P, neighborhood' x B P <-> neighborhood x P. Proof. split. - apply hinhuniv. intros O. generalize ((pr1 (pr2 B)) (pr1 O) (pr1 (pr2 O))). apply neighborhood_imply. exact (pr2 (pr2 O)). - intros Hp. generalize (pr2 (pr2 B) P Hp). apply hinhfun. intros O. exists (pr1 O). exact (pr2 O). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

neighborhood_equiv

Definition isNeighborhood {X : UU} (B : X → (X → hProp) → hProp) := (∏ x, isfilter_imply (B x)) × (∏ x, isfilter_htrue (B x)) × (∏ x, isfilter_and (B x)) × (∏ x P, B x P → P x) × (∏ x P, B x P → ∃ Q, B x Q × ∏ y, Q y → B y P).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isNeighborhood

Lemma isNeighborhood_neighborhood {T : TopologicalSpace} : isNeighborhood (neighborhood (T := T)). Proof. repeat split. - intros x A B. apply (neighborhood_imply x). - intros x. apply (pr2 (neighborhood_isOpen _)). + exact (isOpen_htrue (T := T)). + apply tt. - intros A B. apply neighborhood_and. - intros x P. apply neighborhood_point. - intros x P. apply neighborhood_neighborhood. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isNeighborhood_neighborhood

Definition topologyfromneighborhood (A : X → hProp) := ∏ x : X, A x → N x A.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologyfromneighborhood

Lemma isaprop_topologyfromneighborhood : ∏ A, isaprop (topologyfromneighborhood A). Proof. intros A. apply impred_isaprop ; intros x ; apply isapropimpl, propproperty. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isaprop_topologyfromneighborhood

Lemma topologyfromneighborhood_open : isSetOfOpen_union (λ A : X → hProp, make_hProp (topologyfromneighborhood A) (isaprop_topologyfromneighborhood A)). Proof. intros L Hl x. apply hinhuniv. intros A. apply Himpl with (pr1 A). - intros y Hy. apply hinhpr. now exists (pr1 A), (pr1 (pr2 A)). - apply Hl. + exact (pr1 (pr2 A)). + exact (pr2 (pr2 A)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologyfromneighborhood_open

Definition TopologyFromNeighborhood {X : hSet} (N : X → (X → hProp) → hProp) (H : isNeighborhood N) : TopologicalSpace. Proof. use make_TopologicalSpace. - apply X. - intros A. simple refine (make_hProp _ _). + apply (topologyfromneighborhood N A). + apply isaprop_topologyfromneighborhood. - apply topologyfromneighborhood_open. apply (pr1 H). - intros x _. apply (pr1 (pr2 H)). - intros A B Ha Hb x Hx. apply (pr1 (pr2 (pr2 H))). + now apply Ha, (pr1 Hx). + now apply Hb, (pr2 Hx). Defined.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologyFromNeighborhood

Lemma TopologyFromNeighborhood_correct {X : hSet} (N : X → (X → hProp) → hProp) (H : isNeighborhood N) : ∏ (x : X) (P : X → hProp), N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. Proof. split. - intros Hx. apply hinhpr. simple refine (tpair _ _ _). + simple refine (tpair _ _ _). * intros y. apply (N y P). * simpl ; intros y Hy. generalize (pr2 (pr2 (pr2 (pr2 H))) _ _ Hy). apply hinhuniv. intros Q. apply (pr1 H) with (2 := pr1 (pr2 Q)). exact (pr2 (pr2 Q)). + split ; simpl. * exact Hx. * intros y. now apply (pr1 (pr2 (pr2 (pr2 H)))). - apply hinhuniv. intros O. apply (pr1 H) with (pr1 (pr1 O)). + apply (pr2 (pr2 O)). + simple refine (pr2 (pr1 O) _ _). exact (pr1 (pr2 O)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologyFromNeighborhood_correct

Lemma isNeighborhood_isPreFilter {X : hSet} N : isNeighborhood N -> ∏ x : X, isPreFilter (N x). Proof. intros Hn x. split. - apply (pr1 Hn). - apply isfilter_finite_intersection_carac. + apply (pr1 (pr2 Hn)). + apply (pr1 (pr2 (pr2 Hn))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isNeighborhood_isPreFilter

Lemma isNeighborhood_isFilter {X : hSet} N : isNeighborhood N -> ∏ x : X, isFilter (N x). Proof. intros Hn x. split. - apply isNeighborhood_isPreFilter, Hn. - intros A Fa. apply hinhpr. exists x. apply ((pr1 (pr2 (pr2 (pr2 Hn)))) _ _ Fa). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

isNeighborhood_isFilter

Definition topologygenerated := λ (x : X) (A : X → hProp), (∃ L : Sequence (X → hProp), (∏ n, O (L n)) × (finite_intersection L x) × (∏ y, finite_intersection L y → A y)).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated

Lemma topologygenerated_imply : ∏ x : X, isfilter_imply (topologygenerated x). Proof. intros x A B H. apply hinhfun. intros L. exists (pr1 L). repeat split. - exact (pr1 (pr2 L)). - exact (pr1 (pr2 (pr2 L))). - intros y Hy. apply H, (pr2 (pr2 (pr2 L))), Hy. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated_imply

Lemma topologygenerated_htrue : ∏ x : X, isfilter_htrue (topologygenerated x). Proof. intros x. apply hinhpr. exists nil. repeat split; intros n; induction (nopathsfalsetotrue (pr2 n)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated_htrue

Lemma topologygenerated_and : ∏ x : X, isfilter_and (topologygenerated x). Proof. intros x A B. apply hinhfun2. intros La Lb. exists (concatenate (pr1 La) (pr1 Lb)). repeat split. - simpl ; intro. apply (coprod_rect (λ x, O (coprod_rect _ _ _ x))) ; intros m. + rewrite coprod_rect_compute_1. exact (pr1 (pr2 La) _). + rewrite coprod_rect_compute_2. exact (pr1 (pr2 Lb) _). - simpl ; intro. apply (coprod_rect (λ y, (coprod_rect (λ _ : stn (length (pr1 La)) ⨿ stn (length (pr1 Lb)), X → hProp) (λ j : stn (length (pr1 La)), (pr1 La) j) (λ k : stn (length (pr1 Lb)), (pr1 Lb) k) y) x)) ; intros m. + rewrite coprod_rect_compute_1. exact (pr1 (pr2 (pr2 La)) _). + rewrite coprod_rect_compute_2. exact (pr1 (pr2 (pr2 Lb)) _). - apply (pr2 (pr2 (pr2 La))). intros n. simpl in X0. unfold concatenate' in X0. specialize (X0 (weqfromcoprodofstn_map (length (pr1 La)) (length (pr1 Lb)) (ii1 n))). now rewrite (weqfromcoprodofstn_eq1 _ _) , coprod_rect_compute_1 in X0. - apply (pr2 (pr2 (pr2 Lb))). intros n. simpl in X0. unfold concatenate' in X0. specialize (X0 (weqfromcoprodofstn_map (length (pr1 La)) (length (pr1 Lb)) (ii2 n))). now rewrite (weqfromcoprodofstn_eq1 _ _), coprod_rect_compute_2 in X0. Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated_and

Lemma topologygenerated_point : ∏ (x : X) (P : X → hProp), topologygenerated x P → P x. Proof. intros x P. apply hinhuniv. intros L. apply (pr2 (pr2 (pr2 L))). exact (pr1 (pr2 (pr2 L))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated_point

Lemma topologygenerated_neighborhood : ∏ (x : X) (P : X → hProp), topologygenerated x P → ∃ Q : X → hProp, topologygenerated x Q × (∏ y : X, Q y → topologygenerated y P). Proof. intros x P. apply hinhfun. intros L. exists (finite_intersection (pr1 L)). split. - apply hinhpr. exists (pr1 L). repeat split. + exact (pr1 (pr2 L)). + exact (pr1 (pr2 (pr2 L))). + intros. assumption. - intros y Hy. apply hinhpr. exists (pr1 L). repeat split. + exact (pr1 (pr2 L)). + exact Hy. + exact (pr2 (pr2 (pr2 L))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologygenerated_neighborhood

Definition TopologyGenerated {X : hSet} (O : (X → hProp) → hProp) : TopologicalSpace. Proof. simple refine (TopologyFromNeighborhood _ _). - apply X. - apply topologygenerated, O. - repeat split. + apply topologygenerated_imply. + apply topologygenerated_htrue. + apply topologygenerated_and. + apply topologygenerated_point. + apply topologygenerated_neighborhood. Defined.

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologyGenerated

Lemma TopologyGenerated_included {X : hSet} : ∏ (O : (X → hProp) → hProp) (P : X → hProp), O P → isOpen (T := TopologyGenerated O) P. Proof. intros O P Op. apply neighborhood_isOpen. intros x Hx. apply TopologyFromNeighborhood_correct. apply hinhpr. exists (singletonSequence P). repeat split. - induction n as [n Hn]. exact Op. - intros n ; induction n as [n Hn]. exact Hx. - intros y Hy. now apply (Hy (0%nat,,paths_refl _)). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologyGenerated_included

Lemma TopologyGenerated_smallest {X : hSet} : ∏ (O : (X → hProp) → hProp) (T : isTopologicalSpace X), (∏ P : X → hProp, O P → pr1 T P) → ∏ P : X → hProp, isOpen (T := TopologyGenerated O) P → pr1 T P. Proof. intros O T Ht P Hp. apply (neighborhood_isOpen (T := (X,,T))). intros x Px. generalize (Hp x Px) ; clear Hp. apply hinhfun. intros L. simple refine (tpair _ _ _). - simple refine (tpair _ _ _). + apply (finite_intersection (pr1 L)). + apply (isOpen_finite_intersection (T := X,,T)). intros m. apply Ht. apply (pr1 (pr2 L)). - split. + exact (pr1 (pr2 (pr2 L))). + exact (pr2 (pr2 (pr2 L))). Qed.

Lemma

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

TopologyGenerated_smallest

Definition topologydirprod := λ (z : U × V) (A : U × V → hProp), (∃ (Ax : U → hProp) (Ay : V → hProp), (Ax (pr1 z) × isOpen Ax) × (Ay (pr2 z) × isOpen Ay) × (∏ x y, Ax x → Ay y → A (x,,y))).

Definition

Topology

Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.DivisionRig. Require Import UniMath.Algebra.ConstructiveStructures. Section TopologyFromNeighborhood. End TopologyFromNeighborhood. Definition TopologyFromNeighborhood {X : hSet} Lemma TopologyFromNeighborhood_correct {X : hSet} N x P <-> neighborhood (T := TopologyFromNeighborhood N H) x P. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _). apply TopologyFromNeighborhood_correct. simple refine (TopologyFromNeighborhood _ _).

Topology\Topology.v

topologydirprod