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Commit 44897cb9 authored by François Clément's avatar François Clément
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Renaming + cleaning (of doubles and useless results).

Mv def of continuity (to Set_system_def_any).
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......@@ -16,12 +16,12 @@ COPYING file for more details.
(** General topology - bases and continuity (definitions and properties).
Generators of topologies are also called subbase, or prebase. *)
Generators of topologies are also called subbases, or prebases. *)
Require Import Set_system.
Section Basis_Facts3.
Section Basis_Facts1.
Context {U : Type}.
Variable T : set_system U.
......@@ -29,77 +29,36 @@ Variable PB : set_system U.
Hypothesis HT : is_Open T.
Hypothesis HPB : is_Basisp T PB.
Lemma Basisp_full_unionp_any : full (unionp_any PB).
Proof.
apply is_Basisp_wFull with T; try apply is_Open_equiv; easy.
Qed.
Lemma Basisp_Union_any_inter : Union_any_inter PB.
Proof.
apply is_Basisp_Inter with T; try apply is_Open_equiv; easy.
Qed.
Lemma Basisp_is_gen : T = Open PB.
Proof.
destruct HPB as [HPB1 HPB2]; rewrite <- HT; apply Open_ext.
(* *)
intros A HA; rewrite (HPB2 _ HA).
pose (PB_A := fun B => PB B /\ incl B A); fold PB_A.
destruct (empty_dec PB_A) as [H | H].
destruct (empty_dec (Subset A PB)) as [H | H].
(* . *)
rewrite empty_equiv in H; rewrite H, unionp_any_nullary.
apply Open_wEmpty.
(* . *)
apply Open_Union_any; try easy.
unfold PB_A; intros B HB; apply Open_Gen; easy.
intros B HB; induction HB; apply Open_Gen; easy.
(* *)
intros B HB; apply Open_Gen; auto.
Qed.
Lemma Basisp_fullset : unionp_any PB = fullset.
Proof.
rewrite <- full_equiv; apply is_Basisp_wFull with T; try easy.
apply is_Open_equiv; easy.
Qed.
End Basis_Facts1.
Lemma Basisp_Union_any_inter : Union_any_inter PB.
Proof.
apply is_Basisp_Inter with T; try easy.
apply is_Open_equiv; easy.
Qed.
End Basis_Facts3.
Section Basis_Facts4.
Context {U Idx : Type}.
Variable T : set_system U.
(* Note that the following results do not need T to be a topology! *)
Lemma all_is_Basisf : is_Basisf T (skolem T).
Proof.
split.
intros [x Hx]; easy.
intros A HA; apply set_ext_equiv; split; intros x.
intros; exists (exist _ _ HA); easy.
intros [[B HB1] [HB2 HB3]]; auto.
Qed.
Lemma all_is_Basisp : is_Basisp T T.
Proof.
split; try easy.
intros A HA; apply set_ext_equiv; split; intros x Hx.
exists A; easy.
destruct Hx as [B [[HB1 HB2] HB3]]; auto.
Qed.
Variable fB : Idx -> set U.
(* Useful? *)
Lemma Basisf_to_Basisp : is_Basisf T fB -> is_Basisp T (unskolem fB).
Proof.
apply (proj1 (is_Basisf_is_Basisp_equiv _ _)).
Qed.
Variable PB : set_system U.
(* Useful? *)
Lemma Basisf_of_Basisp : is_Basisp T PB -> is_Basisf T (skolem PB).
Proof.
apply (proj1 (is_Basisp_is_Basisf_equiv _ _)).
Qed.
Section Basis_Facts2.
Context {U1 U2 : Type}.
Variable T1 : set_system U1.
......@@ -115,10 +74,10 @@ apply Incl_trans with (Preimage f T2); try easy.
apply Preimage_monot; easy.
Qed.
End Basis_Facts4.
End Basis_Facts2.
Section Basis_Facts5.
Section Basis_Facts3.
Context {U1 U2 Idx1 Idx2 : Type}.
Variable T1 : set_system U1.
......@@ -162,27 +121,25 @@ intros [i [Hx1 Hx2]]; auto.
*)
Admitted.
End Basis_Facts5.
End Basis_Facts3.
Section Continuous_Def.
Section Continuous_fun_Facts1.
Context {U1 U2 : Type}. (* Universes. *)
Variable genU1 : set_system U1. (* Generator, or subbase. *)
Variable genU2 : set_system U2. (* Generator, or subbase. *)
Definition continuous_fun : set (U1 -> U2) :=
fun f => Incl (Preimage f (Open genU2)) (Open genU1).
Lemma continuous_fun_ext :
forall f g, same_fun f g -> continuous_fun f -> continuous_fun g.
forall f g, same_fun f g ->
continuous_fun genU1 genU2 f -> continuous_fun genU1 genU2 g.
Proof.
intros f g H Hf _ [A2 HA2].
rewrite <- (preimage_ext_fun f); try easy; apply Hf; easy.
Qed.
Lemma continuous_fun_equiv :
forall f, continuous_fun f <-> Incl (Preimage f genU2) (Open genU1).
forall f, continuous_fun genU1 genU2 f <-> Incl (Preimage f genU2) (Open genU1).
Proof.
intros f; split; intros Hf.
intros _ [A2 HA2]; apply Hf, Im, Open_Gen; easy.
......@@ -194,10 +151,10 @@ apply Open_Inter_finite; easy.
apply Open_Unionf_any; easy.
Qed.
End Continuous_Def.
End Continuous_fun_Facts1.
Section Continuous_Facts1.
Section Continuous_fun_Facts2.
Context {U1 U2 : Type}.
Variable genU1 : set_system U1.
......@@ -223,7 +180,7 @@ split; intros Hf A1 HA1; induction HA1 as [A2 HA2].
Admitted.
End Continuous_Facts1.
End Continuous_fun_Facts2.
(* TODO: add section(s) on UniformSpace.
......
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