:: LOPCLSET semantic presentation
:: deftheorem Def1 defines OpenClosedSet LOPCLSET:def 1 :
theorem Th1: :: LOPCLSET:1
canceled;
theorem Th2: :: LOPCLSET:2
theorem Th3: :: LOPCLSET:3
theorem Th4: :: LOPCLSET:4
definition
let c
1 be non
empty TopSpace;
deffunc H
1(
Element of
OpenClosedSet c
1,
Element of
OpenClosedSet c
1)
-> Element of
OpenClosedSet c
1 = a
1 \/ a
2;
func T_join c
1 -> BinOp of
OpenClosedSet a
1 means :
Def2:
:: LOPCLSET:def 2
for b
1, b
2 being
Element of
OpenClosedSet a
1 holds a
2 . b
1,b
2 = b
1 \/ b
2;
existence
ex b1 being BinOp of OpenClosedSet c1 st
for b2, b3 being Element of OpenClosedSet c1 holds b1 . b2,b3 = b2 \/ b3
uniqueness
for b1, b2 being BinOp of OpenClosedSet c1 holds
( ( for b3, b4 being Element of OpenClosedSet c1 holds b1 . b3,b4 = b3 \/ b4 ) & ( for b3, b4 being Element of OpenClosedSet c1 holds b2 . b3,b4 = b3 \/ b4 ) implies b1 = b2 )
deffunc H
2(
Element of
OpenClosedSet c
1,
Element of
OpenClosedSet c
1)
-> Element of
OpenClosedSet c
1 = a
1 /\ a
2;
func T_meet c
1 -> BinOp of
OpenClosedSet a
1 means :
Def3:
:: LOPCLSET:def 3
for b
1, b
2 being
Element of
OpenClosedSet a
1 holds a
2 . b
1,b
2 = b
1 /\ b
2;
existence
ex b1 being BinOp of OpenClosedSet c1 st
for b2, b3 being Element of OpenClosedSet c1 holds b1 . b2,b3 = b2 /\ b3
uniqueness
for b1, b2 being BinOp of OpenClosedSet c1 holds
( ( for b3, b4 being Element of OpenClosedSet c1 holds b1 . b3,b4 = b3 /\ b4 ) & ( for b3, b4 being Element of OpenClosedSet c1 holds b2 . b3,b4 = b3 /\ b4 ) implies b1 = b2 )
end;
:: deftheorem Def2 defines T_join LOPCLSET:def 2 :
:: deftheorem Def3 defines T_meet LOPCLSET:def 3 :
theorem Th5: :: LOPCLSET:5
theorem Th6: :: LOPCLSET:6
theorem Th7: :: LOPCLSET:7
theorem Th8: :: LOPCLSET:8
theorem Th9: :: LOPCLSET:9
theorem Th10: :: LOPCLSET:10
:: deftheorem Def4 defines OpenClosedSetLatt LOPCLSET:def 4 :
theorem Th11: :: LOPCLSET:11
theorem Th12: :: LOPCLSET:12
theorem Th13: :: LOPCLSET:13
theorem Th14: :: LOPCLSET:14
theorem Th15: :: LOPCLSET:15
theorem Th16: :: LOPCLSET:16
:: deftheorem Def5 defines ultraset LOPCLSET:def 5 :
theorem Th17: :: LOPCLSET:17
canceled;
theorem Th18: :: LOPCLSET:18
theorem Th19: :: LOPCLSET:19
:: deftheorem Def6 defines UFilter LOPCLSET:def 6 :
theorem Th20: :: LOPCLSET:20
theorem Th21: :: LOPCLSET:21
theorem Th22: :: LOPCLSET:22
theorem Th23: :: LOPCLSET:23
theorem Th24: :: LOPCLSET:24
:: deftheorem Def7 defines StoneR LOPCLSET:def 7 :
theorem Th25: :: LOPCLSET:25
theorem Th26: :: LOPCLSET:26
:: deftheorem Def8 defines StoneSpace LOPCLSET:def 8 :
theorem Th27: :: LOPCLSET:27
theorem Th28: :: LOPCLSET:28
:: deftheorem Def9 defines StoneBLattice LOPCLSET:def 9 :
Lemma27:
for b1 being non trivial B_Lattice
for b2 being Subset of (StoneSpace b1) holds
( b2 in StoneR b1 implies b2 is open )
theorem Th29: :: LOPCLSET:29
theorem Th30: :: LOPCLSET:30
theorem Th31: :: LOPCLSET:31
canceled;
theorem Th32: :: LOPCLSET:32
theorem Th33: :: LOPCLSET:33
canceled;
theorem Th34: :: LOPCLSET:34
theorem Th35: :: LOPCLSET:35
theorem Th36: :: LOPCLSET:36
theorem Th37: :: LOPCLSET:37
theorem Th38: :: LOPCLSET:38
Lemma34:
for b1 being non trivial B_Lattice holds StoneR b1 c= OpenClosedSet (StoneSpace b1)
theorem Th39: :: LOPCLSET:39
canceled;
theorem Th40: :: LOPCLSET:40
theorem Th41: :: LOPCLSET:41
theorem Th42: :: LOPCLSET:42
theorem Th43: :: LOPCLSET:43
theorem Th44: :: LOPCLSET:44