Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff src/Topology.agda @ 431:a5f8084b8368
reorganiztion for apkg
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 21 Dec 2020 10:23:37 +0900 |
parents | |
children | ce4f3f180b8e |
line wrap: on
line diff
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/Topology.agda Mon Dec 21 10:23:37 2020 +0900 @@ -0,0 +1,109 @@ +open import Level +open import Ordinals +module Topology {n : Level } (O : Ordinals {n}) where + +open import zf +open import logic +open _∧_ +open _∨_ +open Bool + +import OD +open import Relation.Nullary +open import Data.Empty +open import Relation.Binary.Core +open import Relation.Binary.PropositionalEquality +import BAlgbra +open BAlgbra O +open inOrdinal O +open OD O +open OD.OD +open ODAxiom odAxiom +import OrdUtil +import ODUtil +open Ordinals.Ordinals O +open Ordinals.IsOrdinals isOrdinal +open Ordinals.IsNext isNext +open OrdUtil O +open ODUtil O + +import ODC +open ODC O + +open import filter + +record Toplogy ( L : HOD ) : Set (suc n) where + field + OS : HOD + OS⊆PL : OS ⊆ Power L + o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P + o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) + +open Toplogy + +record _covers_ ( P q : HOD ) : Set (suc n) where + field + cover : {x : HOD} → q ∋ x → HOD + P∋cover : {x : HOD} → {lt : q ∋ x} → P ∋ cover lt + isCover : {x : HOD} → {lt : q ∋ x} → cover lt ∋ x + +-- Base +-- The elements of B cover X ; For any U , V ∈ B and any point x ∈ U ∩ V there is a W ∈ B such that +-- W ⊆ U ∩ V and x ∈ W . + +data genTop (P : HOD) : HOD → Set (suc n) where + gi : {x : HOD} → P ∋ x → genTop P x + g∩ : {x y : HOD} → genTop P x → genTop P y → genTop P (x ∩ y) + g∪ : {Q x : HOD} → Q ⊆ P → genTop P (Union Q) + +-- Limit point + +record LP ( L S x : HOD ) (top : Toplogy L) (S⊆PL : S ⊆ Power L ) ( S∋x : S ∋ x ) : Set (suc n) where + field + neip : {y : HOD} → OS top ∋ y → y ∋ x → HOD + isNeip : {y : HOD} → (o∋y : OS top ∋ y ) → (y∋x : y ∋ x ) → ¬ ( x ≡ neip o∋y y∋x) ∧ ( y ∋ neip o∋y y∋x ) + +-- Finite Intersection Property + +data Finite-∩ (S : HOD) : HOD → Set (suc n) where + fin-∩e : {x : HOD} → S ∋ x → Finite-∩ S x + fin-∩ : {x y : HOD} → Finite-∩ S x → Finite-∩ S y → Finite-∩ S (x ∩ y) + +record FIP ( L P : HOD ) : Set (suc n) where + field + fipS⊆PL : P ⊆ Power L + fip≠φ : { x : HOD } → Finite-∩ P x → ¬ ( x ≡ od∅ ) + +-- Compact + +data Finite-∪ (S : HOD) : HOD → Set (suc n) where + fin-∪e : {x : HOD} → S ∋ x → Finite-∪ S x + fin-∪ : {x y : HOD} → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (x ∪ y) + +record Compact ( L P : HOD ) : Set (suc n) where + field + finCover : {X y : HOD} → X covers P → P ∋ y → HOD + isFinCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → finCover xp P∋y ∋ y + isFininiteCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → Finite-∪ X (finCover xp P∋y ) + +-- FIP is Compact + +FIP→Compact : {L P : HOD} → Tolopogy L → FIP L P → Compact L P +FIP→Compact = ? + +Compact→FIP : {L P : HOD} → Tolopogy L → Compact L P → FIP L P +Compact→FIP = ? + +-- Product Topology + +_Top⊗_ : {P Q : HOD} → Topology P → Tolopogy Q → Topology ( P ⊗ Q ) +_Top⊗_ = ? + +-- existence of Ultra Filter + +-- Ultra Filter has limit point + +-- FIP is UFL + +-- Product of UFL has limit point (Tychonoff) +