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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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1102
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33 open import filter O
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1101
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34 open import OPair O
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35
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431
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36
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482
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37 record Topology ( L : HOD ) : Set (suc n) where
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431
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38 field
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39 OS : HOD
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40 OS⊆PL : OS ⊆ Power L
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41 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P
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42 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q)
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1101
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43 -- closed Set
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44 CS : HOD
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45 CS = record { od = record { def = λ x → odef OS (& ( L ＼ (* x ))) } ; odmax = & L ; <odmax = tp02 } where
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46 tp02 : {y : Ordinal } → odef OS (& (L ＼ * y)) → y o< & L
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47 tp02 {y} nop = ?
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1108
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48 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L
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49 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy )
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1101
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50 -- ∈∅< ( proj1 nop )
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431
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51
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482
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52 open Topology
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431
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53
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1107
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54 -- Base
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55 -- 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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56 -- W ⊆ U ∩ V and x ∈ W .
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57
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58 data Subbase (P : HOD) : Ordinal → Set n where
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59 gi : {x : Ordinal } → odef P x → Subbase P x
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60 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y))
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61
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1108
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62 Subbases : (P : HOD) → HOD
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63 Subbases P = record { od = record { def = λ x → Subbase P x } ; odmax = ? ; <odmax = ? }
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1107
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64
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65 record Base (P : HOD) (x : Ordinal) : Set n where
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66 field
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67 b : Ordinal
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1108
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68 pb : odef (Power (Subbases P) ) b
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69 bx : odef (* b) x
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1107
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70
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71 GeneratedTop : (P : HOD) → HOD
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72 GeneratedTop P = record { od = record { def = λ x → Base P x } ; odmax = & P ; <odmax = ? }
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73
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74 -- covers
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1101
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75
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431
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76 record _covers_ ( P q : HOD ) : Set (suc n) where
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77 field
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78 cover : {x : HOD} → q ∋ x → HOD
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79 P∋cover : {x : HOD} → {lt : q ∋ x} → P ∋ cover lt
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80 isCover : {x : HOD} → {lt : q ∋ x} → cover lt ∋ x
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81
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82 -- Finite Intersection Property
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83
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84 data Finite-∩ (S : HOD) : HOD → Set (suc n) where
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1101
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85 fin-e : {x : HOD} → S ∋ x → Finite-∩ S x
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86 fin-∩ : {x y : HOD} → Finite-∩ S x → Finite-∩ S y → Finite-∩ S (x ∩ y)
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431
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87
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1102
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88 record FIP {L : HOD} (top : Topology L) : Set (suc n) where
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431
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89 field
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1102
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90 fipS⊆PL : L ⊆ CS top
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91 fip≠φ : { x : HOD } → Finite-∩ L x → ¬ ( x ≡ od∅ )
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431
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92
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93 -- Compact
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94
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95 data Finite-∪ (S : HOD) : HOD → Set (suc n) where
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1101
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96 fin-e : {x : HOD} → S ∋ x → Finite-∪ S x
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431
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97 fin-∪ : {x y : HOD} → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (x ∪ y)
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98
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1102
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99 record Compact {L : HOD} (top : Topology L) : Set (suc n) where
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431
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100 field
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1102
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101 finCover : {X : HOD} → X ⊆ OS top → X covers L → HOD
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102 isCover : {X : HOD} → (xo : X ⊆ OS top) → (xcp : X covers L ) → (finCover xo xcp ) covers L
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103 isFinite : {X : HOD} → (xo : X ⊆ OS top) → (xcp : X covers L ) → Finite-∪ X (finCover xo xcp )
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431
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104
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105 -- FIP is Compact
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106
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1102
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107 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top
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108 FIP→Compact {L} TL fip = record { finCover = ? ; isCover = ? ; isFinite = ? }
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431
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109
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1102
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110 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top
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482
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111 Compact→FIP = {!!}
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431
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112
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113 -- Product Topology
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114
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1101
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115 open ZFProduct
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116
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1102
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117 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where
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118 field
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1106
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119 p q : Ordinal
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1102
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120 op : odef (OS TP) p
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1106
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121 prod : x ≡ & (ZFP (* p) Q )
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1102
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122
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123 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where
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124 field
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1106
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125 p q : Ordinal
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1102
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126 oq : odef (OS TQ) q
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1106
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127 prod : x ≡ & (ZFP P (* q ))
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128
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1107
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129 -- box : HOD
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130 -- box = ZFP (OS TP) (OS TQ)
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131
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1106
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132 base : {P Q : HOD} → Topology P → Topology Q → HOD
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133 base {P} {Q} TP TQ = record { od = record { def = λ x → BaseP TP Q x ∨ BaseQ P TQ x } ; odmax = & (ZFP P Q) ; <odmax = ? }
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134
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135 POS : {P Q : HOD} → Topology P → Topology Q → HOD
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1107
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136 POS {P} {Q} TP TQ = GeneratedTop (base TP TQ)
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1102
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137
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1108
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138
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1101
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139 _Top⊗_ : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q)
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140 _Top⊗_ {P} {Q} TP TQ = record {
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1106
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141 OS = POS TP TQ
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142 ; OS⊆PL = tp10
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1108
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143 ; o∪ = tp13
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1109
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144 ; o∩ = tp14
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1101
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145 } where
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1102
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146 -- B : (OS P ∋ x → proj⁻¹ x ) ∨ (OS Q ∋ y → proj⁻¹ y )
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147 -- U ⊂ ZFP P Q ∧ ( U ∋ ∀ x → B ∋ ∃ b → b ∋ x ∧ b ⊂ U )
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1108
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148 tp11 : {x z : Ordinal } → Subbase (base TP TQ) z → odef (* z) x → ZFProduct P Q x
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149 tp11 {x} {z} (gi (case1 record { p = p ; q = q ; op = op ; prod = z=zfp })) zx = tp12 where
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150 tp12 : ZFProduct P Q x
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151 tp12 with subst (λ k → odef k x) (trans (cong (*) z=zfp) *iso) zx
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152 ... | ab-pair pa qb = ZFP→
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153 (os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op ) (subst (λ k → odef (* p) k) (sym &iso) pa) )
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154 (subst (λ k → odef Q k ) (sym &iso) qb )
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155 tp11 {x} {z} (gi (case2 record { p = p ; q = q ; oq = oq ; prod = z=zfp })) zx = tp12 where
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156 tp12 : ZFProduct P Q x
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157 tp12 with subst (λ k → odef k x) (trans (cong (*) z=zfp) *iso) zx
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158 ... | ab-pair pa qb = ZFP→
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159 (subst (λ k → odef P k ) (sym &iso) pa )
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160 ((os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq )) (subst (λ k → odef (* q) k) (sym &iso) qb) )
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161 tp11 {x} (g∩ {a} {b} sb sb₁) zx = tp11 sb (proj1 (subst (λ k → odef k x) *iso zx ))
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1106
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162 tp10 : POS TP TQ ⊆ Power (ZFP P Q)
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1108
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163 tp10 {x} record { b = b ; pb = pb ; bx = bx } z xz = tp11 (pb _ bx) xz
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164 tp13 : {U : HOD} → U ⊆ POS TP TQ → POS TP TQ ∋ Union U
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1109
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165 tp13 {U} U⊆O = tp20 U U⊆O where
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166 ind : {x : HOD} → ({y : HOD} → x ∋ y → y ⊆ POS TP TQ → POS TP TQ ∋ Union y) → x ⊆ POS TP TQ → POS TP TQ ∋ Union x
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167 ind {x} prev x⊆O = record { b = ? ; pb = ? ; bx = ? }
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168 tp20 : (U : HOD ) → U ⊆ POS TP TQ → POS TP TQ ∋ Union U
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169 tp20 U U⊆O = ε-induction0 { λ U → U ⊆ POS TP TQ → POS TP TQ ∋ Union U } ind U U⊆O
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170 tp14 : {p q : HOD} → POS TP TQ ∋ p → POS TP TQ ∋ q → POS TP TQ ∋ (p ∩ q)
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171 tp14 {p} {q} op oq = record { b = & tp15 ; pb = ? ; bx = ? } where
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172 tp15 : HOD
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173 tp15 = ?
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431
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174
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175 -- existence of Ultra Filter
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176
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1102
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177 open Filter
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178
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431
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179 -- Ultra Filter has limit point
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180
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1102
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181 record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) : Set (suc (suc n)) where
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182 field
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183 limit : Ordinal
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184 P∋limit : odef P limit
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185 is-limit : {o : Ordinal} → odef (OS TP) o → odef (* o) limit → (* o) ⊆ filter F
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186
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431
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187 -- FIP is UFL
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188
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1102
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189 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP
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190 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf
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191 FIP→UFLP {P} TP fip {L} LP F uf = record { limit = ? ; P∋limit = ? ; is-limit = ? }
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192
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193 UFLP→FIP : {P : HOD} (TP : Topology P) →
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194 ( {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf ) → FIP TP
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195 UFLP→FIP {P} TP uflp = record { fipS⊆PL = ? ; fip≠φ = ? }
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196
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431
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197 -- Product of UFL has limit point (Tychonoff)
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198
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1102
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199 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (TP Top⊗ TQ)
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200 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (TP Top⊗ TQ) (UFLP→FIP (TP Top⊗ TQ) uflp ) where
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1104
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201 uflp : {L : HOD} (LPQ : L ⊆ Power (ZFP P Q)) (F : Filter LPQ)
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202 (uf : ultra-filter {L} {_} {LPQ} F) → UFLP (TP Top⊗ TQ) LPQ F uf
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1106
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203 uflp {L} LPQ F uf = record { limit = & < * ( UFLP.limit uflpp) , ? > ; P∋limit = ? ; is-limit = ? } where
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204 LP : (L : HOD ) (LPQ : L ⊆ Power (ZFP P Q)) → HOD
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205 LP L LPQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ1 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) )
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206 LPP : (L : HOD) (LPQ : L ⊆ Power (ZFP P Q)) → LP L LPQ ⊆ Power P
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207 LPP L LPQ record { z = z ; az = az ; x=ψz = x=ψz } w xw = tp02 (subst (λ k → odef k w)
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1105
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208 (subst₂ (λ j k → j ≡ k) refl *iso (cong (*) x=ψz) ) xw) where
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209 tp02 : Replace' (* z) (λ z₁ xz → * (zπ1 (LPQ (subst (odef L) (sym &iso) az) (& z₁) (subst (λ k → odef k (& z₁)) (sym *iso) xz)))) ⊆ P
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210 tp02 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k ) (trans (sym &iso) (sym x=ψz1) )
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211 (zp1 (LPQ (subst (λ k → odef L k) (sym &iso) az) _ (tp03 az1 ))) where
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212 tp03 : odef (* z) z1 → odef (* (& (* z))) (& (* z1))
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213 tp03 lt = subst (λ k → odef k (& (* z1))) (sym *iso) (subst (odef (* z)) (sym &iso) lt)
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1106
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214 FP : Filter (LPP L LPQ)
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215 FP = record { filter = LP (filter F) (λ x → LPQ (f⊆L F x )) ; f⊆L = tp04 ; filter1 = ? ; filter2 = ? } where
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216 tp04 : LP (filter F) (λ x → LPQ (f⊆L F x )) ⊆ LP L LPQ
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217 tp04 record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = f⊆L F az ; x=ψz = ? }
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1104
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218 uFP : ultra-filter FP
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219 uFP = record { proper = ? ; ultra = ? }
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1106
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220 uflpp : UFLP {P} TP {LP L LPQ} (LPP L LPQ) FP uFP
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221 uflpp = FIP→UFLP TP (Compact→FIP TP CP) (LPP L LPQ) FP uFP
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1104
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222 LQ : HOD
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1105
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223 LQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ2 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) )
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1104
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224 LQQ : LQ ⊆ Power Q
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225 LQQ = ?
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1102
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226
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