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431
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1 open import Level
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2 open import Ordinals
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3 module Topology {n : Level } (O : Ordinals {n}) where
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4
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5 open import zf
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6 open import logic
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7 open _∧_
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8 open _∨_
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9 open Bool
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10
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11 import OD
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12 open import Relation.Nullary
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13 open import Data.Empty
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14 open import Relation.Binary.Core
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15 open import Relation.Binary.PropositionalEquality
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16 import BAlgbra
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17 open BAlgbra O
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18 open inOrdinal O
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19 open OD O
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20 open OD.OD
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21 open ODAxiom odAxiom
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22 import OrdUtil
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23 import ODUtil
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24 open Ordinals.Ordinals O
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25 open Ordinals.IsOrdinals isOrdinal
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26 open Ordinals.IsNext isNext
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27 open OrdUtil O
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28 open ODUtil O
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29
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30 import ODC
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31 open ODC O
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32
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33 open import filter
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34
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482
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35 record Topology ( L : HOD ) : Set (suc n) where
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431
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36 field
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37 OS : HOD
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38 OS⊆PL : OS ⊆ Power L
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39 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P
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40 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q)
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41
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482
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42 open Topology
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431
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43
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44 record _covers_ ( P q : HOD ) : Set (suc n) where
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45 field
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46 cover : {x : HOD} → q ∋ x → HOD
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47 P∋cover : {x : HOD} → {lt : q ∋ x} → P ∋ cover lt
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48 isCover : {x : HOD} → {lt : q ∋ x} → cover lt ∋ x
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49
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50 -- Base
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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
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52 -- W ⊆ U ∩ V and x ∈ W .
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53
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54 data genTop (P : HOD) : HOD → Set (suc n) where
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55 gi : {x : HOD} → P ∋ x → genTop P x
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56 g∩ : {x y : HOD} → genTop P x → genTop P y → genTop P (x ∩ y)
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57 g∪ : {Q x : HOD} → Q ⊆ P → genTop P (Union Q)
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58
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59 -- Limit point
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60
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482
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61 record LP ( L S x : HOD ) (top : Topology L) (S⊆PL : S ⊆ Power L ) ( S∋x : S ∋ x ) : Set (suc n) where
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431
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62 field
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63 neip : {y : HOD} → OS top ∋ y → y ∋ x → HOD
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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 )
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65
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66 -- Finite Intersection Property
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67
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68 data Finite-∩ (S : HOD) : HOD → Set (suc n) where
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69 fin-∩e : {x : HOD} → S ∋ x → Finite-∩ S x
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70 fin-∩ : {x y : HOD} → Finite-∩ S x → Finite-∩ S y → Finite-∩ S (x ∩ y)
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71
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72 record FIP ( L P : HOD ) : Set (suc n) where
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73 field
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74 fipS⊆PL : P ⊆ Power L
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75 fip≠φ : { x : HOD } → Finite-∩ P x → ¬ ( x ≡ od∅ )
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76
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77 -- Compact
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78
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79 data Finite-∪ (S : HOD) : HOD → Set (suc n) where
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80 fin-∪e : {x : HOD} → S ∋ x → Finite-∪ S x
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81 fin-∪ : {x y : HOD} → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (x ∪ y)
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82
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83 record Compact ( L P : HOD ) : Set (suc n) where
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84 field
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85 finCover : {X y : HOD} → X covers P → P ∋ y → HOD
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86 isFinCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → finCover xp P∋y ∋ y
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87 isFininiteCover : {X y : HOD} → (xp : X covers P ) → (P∋y : P ∋ y ) → Finite-∪ X (finCover xp P∋y )
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88
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89 -- FIP is Compact
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90
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482
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91 FIP→Compact : {L P : HOD} → Topology L → FIP L P → Compact L P
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92 FIP→Compact = {!!}
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431
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93
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482
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94 Compact→FIP : {L P : HOD} → Topology L → Compact L P → FIP L P
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95 Compact→FIP = {!!}
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431
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96
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97 -- Product Topology
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98
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482
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99 _Top⊗_ : {P Q : HOD} → Topology P → Topology Q → Topology {!!}
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100 _Top⊗_ = {!!}
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431
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101
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102 -- existence of Ultra Filter
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103
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104 -- Ultra Filter has limit point
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105
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106 -- FIP is UFL
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107
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108 -- Product of UFL has limit point (Tychonoff)
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109
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