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comparison src/Topology.agda @ 1122:1c7474446754

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add OS ∋ od∅
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 04 Jan 2023 09:39:25 +0900
parents 98af35c9711f
children 256a3ba634f6
comparison
equal deleted inserted replaced
1121:98af35c9711f 1122:1c7474446754
37 field 37 field
38 OS : HOD 38 OS : HOD
39 OS⊆PL : OS ⊆ Power L 39 OS⊆PL : OS ⊆ Power L
40 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) 40 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q)
41 o∪ : { P : HOD } → P ⊂ OS → OS ∋ Union P 41 o∪ : { P : HOD } → P ⊂ OS → OS ∋ Union P
42 OS∋od∅ : OS ∋ od∅
42 -- closed Set 43 -- closed Set
43 CS : HOD 44 CS : HOD
44 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where 45 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where
45 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) 46 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L)
46 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) 47 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx ))
47 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L 48 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L
48 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy ) 49 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy )
50 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L
51 cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) (sym *iso) xy )
52 CS∋L : CS ∋ L
53 CS∋L = ⟪ ? , ? ⟫
54 --- we may add
55 -- OS∋L : OS ∋ L
56 -- OS∋od∅ : OS ∋ od∅
49 57
50 open Topology 58 open Topology
59
60 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → A ⊆ L → HOD
61 Cl {L} top A A⊆L = record { od = record { def = λ x → (c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x }
62 ; odmax = & L ; <odmax = ? }
51 63
52 -- Subbase P 64 -- Subbase P
53 -- A set of countable intersection of P will be a base (x ix an element of the base) 65 -- A set of countable intersection of P will be a base (x ix an element of the base)
54 66
55 data Subbase (P : HOD) : Ordinal → Set n where 67 data Subbase (P : HOD) : Ordinal → Set n where
86 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) 98 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L)
87 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) 99 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) ))
88 100
89 record IsSubBase (L P : HOD) : Set (suc n) where 101 record IsSubBase (L P : HOD) : Set (suc n) where
90 field 102 field
91 P⊆PL : P ⊆ Power L 103 P⊆PL : P ⊆ Power L
92 -- we may need these if OS ∋ L is necessary 104 -- we may need these if OS ∋ L is necessary
93 -- p : {x : HOD} → L ∋ x → HOD 105 -- p : {x : HOD} → L ∋ x → HOD
94 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx 106 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx
95 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x 107 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x
96 108
97 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L 109 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L
98 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00 110 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00
99 ; o∪ = tp02 ; o∩ = tp01 } where 111 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where
112 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x
113 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x ))
100 tp00 : SO L P ⊆ Power L 114 tp00 : SO L P ⊆ Power L
101 tp00 {u} ou x ux with ou ux 115 tp00 {u} ou x ux with ou ux
102 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊂L (b⊆u bx) 116 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊂L (b⊆u bx)
103 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) 117 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q)
104 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) ul 118 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) ul
151 165
152 -- Finite Intersection Property 166 -- Finite Intersection Property
153 167
154 record FIP {L : HOD} (top : Topology L) : Set n where 168 record FIP {L : HOD} (top : Topology L) : Set n where
155 field 169 field
156 finite : {X : Ordinal } → * X ⊆ CS top 170 limit : {X : Ordinal } → * X ⊆ CS top → * X ∋ L
157 → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal 171 → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal
158 limit : {X : Ordinal } → (CX : * X ⊆ CS top ) 172 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) → (XL : * X ∋ L )
159 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) 173 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x )
160 → {x : Ordinal } → odef (* X) x → odef (* x) (finite CX fip) 174 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX XL fip)
175 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) → (XL : * X ∋ L)
176 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x )
177 → odef L (limit CX XL fip)
178 L∋limit {X} CX XL fip = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX XL)) (is-limit CX XL fip XL)
161 179
162 -- Compact 180 -- Compact
163 181
164 data Finite-∪ (S : HOD) : Ordinal → Set n where 182 data Finite-∪ (S : HOD) : Ordinal → Set n where
165 fin-e : {x : Ordinal } → odef S x → Finite-∪ S x 183 fin-e : {x : Ordinal } → odef S x → Finite-∪ S x
194 fip03 : {x z : Ordinal } → odef (* x) z → (¬ odef (* x) z) → ⊥ 212 fip03 : {x z : Ordinal } → odef (* x) z → (¬ odef (* x) z) → ⊥
195 fip03 {x} {z} xz nxz = nxz xz 213 fip03 {x} {z} xz nxz = nxz xz
196 fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x 214 fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x
197 fip02 = ? 215 fip02 = ?
198 fip01 : Ordinal 216 fip01 : Ordinal
199 fip01 = FIP.finite fip (CCX ox) fip02 217 fip01 = FIP.limit fip (CCX ox) ? fip02
200 ¬CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → * (cex ox oc) ⊆ * (CX ox) → Subbase (* (cex ox oc)) o∅ 218 ¬CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → * (cex ox oc) ⊆ * (CX ox) → Subbase (* (cex ox oc)) o∅
201 ¬CXfip {X} ox oc = ? where 219 ¬CXfip {X} ox oc = ? where
202 fip04 : odef (Cex ox) (cex ox oc) 220 fip04 : odef (Cex ox) (cex ox oc)
203 fip04 = ? 221 fip04 = ?
204 -- this defines finite cover 222 -- this defines finite cover
260 278
261 open Filter 279 open Filter
262 280
263 -- Ultra Filter has limit point 281 -- Ultra Filter has limit point
264 282
265 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 283 record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP )
284 (FL : filter F ∋ P) (uf : ultra-filter {L} {P} {LP} F) : Set (suc (suc n)) where
266 field 285 field
267 limit : Ordinal 286 limit : Ordinal
268 P∋limit : odef P limit 287 P∋limit : odef P limit
269 is-limit : {o : Ordinal} → odef (OS TP) o → odef (* o) limit → (* o) ⊆ filter F 288 is-limit : {o : Ordinal} → odef (OS TP) o → odef (* o) limit → (* o) ⊆ filter F
270 289
271 -- FIP is UFL 290 -- FIP is UFL
272 291
273 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP 292 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP
274 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf 293 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (FL : filter F ∋ P) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F FL uf
275 FIP→UFLP {P} TP fip {L} LP F uf = record { limit = ? ; P∋limit = ? ; is-limit = ? } 294 FIP→UFLP {P} TP fip {L} LP F FL uf = record { limit = FIP.limit fip fip00 CFP fip01 ; P∋limit = FIP.L∋limit fip fip00 CFP fip01 ; is-limit = fip02 }
295 where
296 CF : Ordinal
297 CF = & ( Replace' (filter F) (λ z fz → Cl TP z (fip03 fz)) ) where
298 fip03 : {z : HOD} → filter F ∋ z → z ⊆ P
299 fip03 {z} fz {x} zx with LP ( f⊆L F fz )
300 ... | pw = pw x (subst (λ k → odef k x) (sym *iso) zx )
301 CFP : * CF ∋ P -- filter F ∋ P
302 CFP = ?
303 fip00 : * CF ⊆ CS TP -- replaced
304 fip00 = ?
305 fip01 : {C x : Ordinal} → * C ⊆ * CF → Subbase (* C) x → o∅ o< x
306 fip01 {C} {x} CCF (gi Cx) = ? -- filter is proper .i.e it contains no od∅
307 fip01 {C} {.(& (* _ ∩ * _))} CCF (g∩ sb sb₁) = ?
308 fip02 : {o : Ordinal} → odef (OS TP) o → odef (* o) (FIP.limit fip fip00 ? fip01) → * o ⊆ filter F
309 fip02 {p} oo ol = ? where
310 fip04 : odef ? (FIP.limit fip fip00 ? fip01)
311 fip04 = FIP.is-limit fip fip00 CFP fip01 ?
312
276 313
277 UFLP→FIP : {P : HOD} (TP : Topology P) → 314 UFLP→FIP : {P : HOD} (TP : Topology P) →
278 ( {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F uf ) → FIP TP 315 ( {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (FL : filter F ∋ P) (uf : ultra-filter {L} {P} {LP} F) → UFLP TP LP F ? uf ) → FIP TP
279 UFLP→FIP {P} TP uflp = record { fip≠φ = ? } 316 UFLP→FIP {P} TP uflp = record { limit = ? ; is-limit = ? }
280 317
281 -- Product of UFL has limit point (Tychonoff) 318 -- Product of UFL has limit point (Tychonoff)
282 319
283 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (TP Top⊗ TQ) 320 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (TP Top⊗ TQ)
284 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (TP Top⊗ TQ) (UFLP→FIP (TP Top⊗ TQ) uflp ) where 321 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (TP Top⊗ TQ) (UFLP→FIP (TP Top⊗ TQ) uflp ) where
285 uflp : {L : HOD} (LPQ : L ⊆ Power (ZFP P Q)) (F : Filter LPQ) 322 uflp : {L : HOD} (LPQ : L ⊆ Power (ZFP P Q)) (F : Filter LPQ) (LF : filter F ∋ ZFP P Q)
286 (uf : ultra-filter {L} {_} {LPQ} F) → UFLP (TP Top⊗ TQ) LPQ F uf 323 (uf : ultra-filter {L} {_} {LPQ} F) → UFLP (TP Top⊗ TQ) LPQ F ? uf
287 uflp {L} LPQ F uf = record { limit = & < * ( UFLP.limit uflpp) , ? > ; P∋limit = ? ; is-limit = ? } where 324 uflp {L} LPQ F LF uf = record { limit = & < * ( UFLP.limit uflpp) , ? > ; P∋limit = ? ; is-limit = ? } where
288 LP : (L : HOD ) (LPQ : L ⊆ Power (ZFP P Q)) → HOD 325 LP : (L : HOD ) (LPQ : L ⊆ Power (ZFP P Q)) → HOD
289 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 )))) ) 326 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 )))) )
290 LPP : (L : HOD) (LPQ : L ⊆ Power (ZFP P Q)) → LP L LPQ ⊆ Power P 327 LPP : (L : HOD) (LPQ : L ⊆ Power (ZFP P Q)) → LP L LPQ ⊆ Power P
291 LPP L LPQ record { z = z ; az = az ; x=ψz = x=ψz } w xw = tp02 (subst (λ k → odef k w) 328 LPP L LPQ record { z = z ; az = az ; x=ψz = x=ψz } w xw = tp02 (subst (λ k → odef k w)
292 (subst₂ (λ j k → j ≡ k) refl *iso (cong (*) x=ψz) ) xw) where 329 (subst₂ (λ j k → j ≡ k) refl *iso (cong (*) x=ψz) ) xw) where
299 FP = record { filter = LP (filter F) (λ x → LPQ (f⊆L F x )) ; f⊆L = tp04 ; filter1 = ? ; filter2 = ? } where 336 FP = record { filter = LP (filter F) (λ x → LPQ (f⊆L F x )) ; f⊆L = tp04 ; filter1 = ? ; filter2 = ? } where
300 tp04 : LP (filter F) (λ x → LPQ (f⊆L F x )) ⊆ LP L LPQ 337 tp04 : LP (filter F) (λ x → LPQ (f⊆L F x )) ⊆ LP L LPQ
301 tp04 record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = f⊆L F az ; x=ψz = ? } 338 tp04 record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = f⊆L F az ; x=ψz = ? }
302 uFP : ultra-filter FP 339 uFP : ultra-filter FP
303 uFP = record { proper = ? ; ultra = ? } 340 uFP = record { proper = ? ; ultra = ? }
304 uflpp : UFLP {P} TP {LP L LPQ} (LPP L LPQ) FP uFP 341 uflpp : UFLP {P} TP {LP L LPQ} (LPP L LPQ) FP ? uFP
305 uflpp = FIP→UFLP TP (Compact→FIP TP CP) (LPP L LPQ) FP uFP 342 uflpp = FIP→UFLP TP (Compact→FIP TP CP) (LPP L LPQ) FP ? uFP
306 LQ : HOD 343 LQ : HOD
307 LQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ2 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) ) 344 LQ = Replace' L ( λ x lx → Replace' x ( λ z xz → * ( zπ2 (LPQ lx (& z) (subst (λ k → odef k (& z)) (sym *iso) xz )))) )
308 LQQ : LQ ⊆ Power Q 345 LQQ : LQ ⊆ Power Q
309 LQQ = ? 346 LQQ = ?
310 347