Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison Topology.agda @ 428:94392feeebc5
add Topology
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 19 Dec 2020 21:43:21 +0900 |
parents | |
children | 8d8149bcd4d1 |
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427:9b0630f03c4b | 428:94392feeebc5 |
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1 open import Level | |
2 open import Ordinals | |
3 module Topology {n : Level } (O : Ordinals {n}) where | |
4 | |
5 open import zf | |
6 open import logic | |
7 open _∧_ | |
8 open _∨_ | |
9 open Bool | |
10 | |
11 import OD | |
12 open import Relation.Nullary | |
13 open import Data.Empty | |
14 open import Relation.Binary.Core | |
15 open import Relation.Binary.PropositionalEquality | |
16 import BAlgbra | |
17 open BAlgbra O | |
18 open inOrdinal O | |
19 open OD O | |
20 open OD.OD | |
21 open ODAxiom odAxiom | |
22 import OrdUtil | |
23 import ODUtil | |
24 open Ordinals.Ordinals O | |
25 open Ordinals.IsOrdinals isOrdinal | |
26 open Ordinals.IsNext isNext | |
27 open OrdUtil O | |
28 open ODUtil O | |
29 | |
30 import ODC | |
31 open ODC O | |
32 | |
33 open import filter | |
34 | |
35 record Toplogy ( L : HOD ) : Set (suc n) where | |
36 field | |
37 OS : HOD | |
38 OS⊆PL : OS ⊆ Power L | |
39 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P | |
40 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) | |
41 | |
42 open Toplogy | |
43 | |
44 record _covers_ ( P q : HOD ) : Set (suc n) where | |
45 field | |
46 cover : {x : HOD} → q ∋ x → HOD | |
47 P∋cover : {x : HOD} → {lt : q ∋ x} → P ∋ cover lt | |
48 isCover : {x : HOD} → {lt : q ∋ x} → cover lt ∋ x | |
49 | |
50 -- Base | |
51 -- The elements of B cover X ; For any U , V ∈ B and any point x ∈ U ∩ V there is a W ∈ B such that | |
52 -- W ⊆ U ∩ V and x ∈ W . | |
53 | |
54 data genTop (P : HOD) : HOD → Set (suc n) where | |
55 gi : {x : HOD} → P ∋ x → genTop P x | |
56 g∩ : {x y : HOD} → genTop P x → genTop P y → genTop P (x ∩ y) | |
57 g∪ : {Q x : HOD} → Q ⊆ P → genTop P (Union Q) | |
58 | |
59 -- Limit point | |
60 | |
61 record LP ( L S x : HOD ) (top : Toplogy L) (S⊆PL : S ⊆ Power L ) ( S∋x : S ∋ x ) : Set (suc n) where | |
62 field | |
63 neip : {y : HOD} → OS top ∋ y → y ∋ x → HOD | |
64 isNeip : {y : HOD} → (o∋y : OS top ∋ y ) → (y∋x : y ∋ x ) → ¬ ( x ≡ neip o∋y y∋x) ∧ ( y ∋ neip o∋y y∋x ) | |
65 | |
66 -- Finite Intersection Property | |
67 | |
68 data Finite-∩ (S : HOD) : HOD → Set (suc n) where | |
69 fin-∩e : {x : HOD} → S ∋ x → Finite-∩ S x | |
70 fin-∩ : {x y : HOD} → Finite-∩ S x → Finite-∩ S y → Finite-∩ S (x ∩ y) | |
71 | |
72 data Finite-∪ (S : HOD) : HOD → Set (suc n) where | |
73 fin-∪e : {x : HOD} → S ∋ x → Finite-∪ S x | |
74 fin-∪ : {x y : HOD} → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (x ∪ y) | |
75 | |
76 record FIP ( L : HOD ) : Set (suc n) where | |
77 field | |
78 fipS : HOD | |
79 fipS⊆PL : fipS ⊆ Power L | |
80 fip≠φ : { x : HOD } → Finite-∩ fipS x → ¬ ( x ≡ od∅ ) | |
81 | |
82 -- Compact | |
83 | |
84 record Compact ( L P : HOD ) : Set (suc n) where | |
85 field | |
86 finCover : {X y : HOD} → X covers P → P ∋ y → HOD | |
87 isFinCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → finCover xp P∋y ∋ y | |
88 isFininiteCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → Finite-∪ X (finCover xp P∋y ) | |
89 | |
90 | |
91 -- Product Topology |