:: TEX_4 semantic presentation

registration
let c1 be non empty TopSpace;
let c2 be non empty Subset of c1;
cluster Cl a2 -> non empty ;
coherence
not Cl c2 is empty
by PCOMPS_1:2;
end;

registration
let c1 be non empty TopSpace;
let c2 be empty Subset of c1;
cluster Cl a2 -> empty ;
coherence
Cl c2 is empty
by PRE_TOPC:52;
end;

registration
let c1 be non empty TopSpace;
let c2 be non proper Subset of c1;
cluster Cl a2 -> non empty non proper ;
coherence
not Cl c2 is proper
proof end;
end;

registration
let c1 be non empty non trivial TopSpace;
let c2 be non empty non trivial Subset of c1;
cluster Cl a2 -> non empty non trivial ;
coherence
not Cl c2 is trivial
proof end;
end;

theorem Th1: :: TEX_4:1
for b1 being non empty TopSpace
for b2 being Subset of b1 holds Cl b2 = meet { b3 where B is Subset of b1 : ( b3 is closed & b2 c= b3 ) }
proof end;

theorem Th2: :: TEX_4:2
for b1 being non empty TopSpace
for b2 being Point of b1 holds Cl {b2} = meet { b3 where B is Subset of b1 : ( b3 is closed & b2 in b3 ) }
proof end;

registration
let c1 be non empty TopSpace;
let c2 be non proper Subset of c1;
cluster Int a2 -> non proper ;
coherence
not Int c2 is proper
proof end;
end;

registration
let c1 be non empty TopSpace;
let c2 be proper Subset of c1;
cluster Int a2 -> proper ;
coherence
Int c2 is proper
proof end;
end;

registration
let c1 be non empty TopSpace;
let c2 be empty Subset of c1;
cluster Int a2 -> empty proper ;
coherence
Int c2 is empty
;
end;

theorem Th3: :: TEX_4:3
canceled;

theorem Th4: :: TEX_4:4
for b1 being non empty TopSpace
for b2 being Subset of b1 holds Int b2 = union { b3 where B is Subset of b1 : ( b3 is open & b3 c= b2 ) }
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
attr a2 is anti-discrete means :Def1: :: TEX_4:def 1
for b1 being Point of a1
for b2 being Subset of a1 holds
( b2 is open & b1 in b2 & b1 in a2 implies a2 c= b2 );
end;

:: deftheorem Def1 defines anti-discrete TEX_4:def 1 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Point of b1
for b4 being Subset of b1 holds
( b4 is open & b3 in b4 & b3 in b2 implies b2 c= b4 ) );

definition
let c1 be non empty TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :Def2: :: TEX_4:def 2
for b1 being Point of a1
for b2 being Subset of a1 holds
( b2 is closed & b1 in b2 & b1 in a2 implies a2 c= b2 );
compatibility
( c2 is anti-discrete iff for b1 being Point of c1
for b2 being Subset of c1 holds
( b2 is closed & b1 in b2 & b1 in c2 implies c2 c= b2 ) )
proof end;
end;

:: deftheorem Def2 defines anti-discrete TEX_4:def 2 :
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Point of b1
for b4 being Subset of b1 holds
( b4 is closed & b3 in b4 & b3 in b2 implies b2 c= b4 ) );

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :Def3: :: TEX_4:def 3
for b1 being Subset of a1 holds
not ( b1 is open & not a2 misses b1 & not a2 c= b1 );
compatibility
( c2 is anti-discrete iff for b1 being Subset of c1 holds
not ( b1 is open & not c2 misses b1 & not c2 c= b1 ) )
proof end;
end;

:: deftheorem Def3 defines anti-discrete TEX_4:def 3 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Subset of b1 holds
not ( b3 is open & not b2 misses b3 & not b2 c= b3 ) );

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :Def4: :: TEX_4:def 4
for b1 being Subset of a1 holds
not ( b1 is closed & not a2 misses b1 & not a2 c= b1 );
compatibility
( c2 is anti-discrete iff for b1 being Subset of c1 holds
not ( b1 is closed & not c2 misses b1 & not c2 c= b1 ) )
proof end;
end;

:: deftheorem Def4 defines anti-discrete TEX_4:def 4 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Subset of b1 holds
not ( b3 is closed & not b2 misses b3 & not b2 c= b3 ) );

theorem Th5: :: TEX_4:5
canceled;

theorem Th6: :: TEX_4:6
for b1, b2 being TopStruct
for b3 being Subset of b1
for b4 being Subset of b2 holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 = b4 & b3 is anti-discrete implies b4 is anti-discrete )
proof end;

theorem Th7: :: TEX_4:7
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b3 c= b2 & b2 is anti-discrete implies b3 is anti-discrete )
proof end;

theorem Th8: :: TEX_4:8
for b1 being non empty TopStruct
for b2 being Point of b1 holds {b2} is anti-discrete
proof end;

theorem Th9: :: TEX_4:9
for b1 being non empty TopStruct
for b2 being empty Subset of b1 holds b2 is anti-discrete
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset-Family of c1;
attr a2 is anti-discrete-set-family means :Def5: :: TEX_4:def 5
for b1 being Subset of a1 holds
( b1 in a2 implies b1 is anti-discrete );
end;

:: deftheorem Def5 defines anti-discrete-set-family TEX_4:def 5 :
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is anti-discrete-set-family iff for b3 being Subset of b1 holds
( b3 in b2 implies b3 is anti-discrete ) );

theorem Th10: :: TEX_4:10
for b1 being non empty TopStruct
for b2 being Subset-Family of b1 holds
( b2 is anti-discrete-set-family & meet b2 <> {} implies union b2 is anti-discrete )
proof end;

theorem Th11: :: TEX_4:11
for b1 being non empty TopStruct
for b2 being Subset-Family of b1 holds
( b2 is anti-discrete-set-family implies meet b2 is anti-discrete )
proof end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
func MaxADSF c2 -> Subset-Family of a1 equals :: TEX_4:def 6
{ b1 where B is Subset of a1 : ( b1 is anti-discrete & a2 in b1 ) } ;
coherence
{ b1 where B is Subset of c1 : ( b1 is anti-discrete & c2 in b1 ) } is Subset-Family of c1
proof end;
end;

:: deftheorem Def6 defines MaxADSF TEX_4:def 6 :
for b1 being TopStruct
for b2 being Point of b1 holds MaxADSF b2 = { b3 where B is Subset of b1 : ( b3 is anti-discrete & b2 in b3 ) } ;

registration
let c1 be non empty TopStruct ;
let c2 be Point of c1;
cluster MaxADSF a2 -> non empty ;
coherence
not MaxADSF c2 is empty
proof end;
end;

theorem Th12: :: TEX_4:12
for b1 being non empty TopStruct
for b2 being Point of b1 holds MaxADSF b2 is anti-discrete-set-family
proof end;

theorem Th13: :: TEX_4:13
for b1 being non empty TopStruct
for b2 being Point of b1 holds {b2} = meet (MaxADSF b2)
proof end;

theorem Th14: :: TEX_4:14
for b1 being non empty TopStruct
for b2 being Point of b1 holds {b2} c= union (MaxADSF b2)
proof end;

theorem Th15: :: TEX_4:15
for b1 being non empty TopStruct
for b2 being Point of b1 holds union (MaxADSF b2) is anti-discrete
proof end;

definition
let c1 be TopStruct ;
let c2 be Subset of c1;
attr a2 is maximal_anti-discrete means :Def7: :: TEX_4:def 7
( a2 is anti-discrete & ( for b1 being Subset of a1 holds
( b1 is anti-discrete & a2 c= b1 implies a2 = b1 ) ) );
end;

:: deftheorem Def7 defines maximal_anti-discrete TEX_4:def 7 :
for b1 being TopStruct
for b2 being Subset of b1 holds
( b2 is maximal_anti-discrete iff ( b2 is anti-discrete & ( for b3 being Subset of b1 holds
( b3 is anti-discrete & b2 c= b3 implies b2 = b3 ) ) ) );

theorem Th16: :: TEX_4:16
for b1, b2 being TopStruct
for b3 being Subset of b1
for b4 being Subset of b2 holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 = b4 & b3 is maximal_anti-discrete implies b4 is maximal_anti-discrete )
proof end;

theorem Th17: :: TEX_4:17
for b1 being non empty TopStruct
for b2 being empty Subset of b1 holds
not b2 is maximal_anti-discrete
proof end;

theorem Th18: :: TEX_4:18
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
( b2 is anti-discrete & b2 is open implies b2 is maximal_anti-discrete )
proof end;

theorem Th19: :: TEX_4:19
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
( b2 is anti-discrete & b2 is closed implies b2 is maximal_anti-discrete )
proof end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
func MaxADSet c2 -> Subset of a1 equals :: TEX_4:def 8
union (MaxADSF a2);
coherence
union (MaxADSF c2) is Subset of c1
;
end;

:: deftheorem Def8 defines MaxADSet TEX_4:def 8 :
for b1 being TopStruct
for b2 being Point of b1 holds MaxADSet b2 = union (MaxADSF b2);

registration
let c1 be non empty TopStruct ;
let c2 be Point of c1;
cluster MaxADSet a2 -> non empty ;
coherence
not MaxADSet c2 is empty
proof end;
end;

theorem Th20: :: TEX_4:20
for b1 being non empty TopStruct
for b2 being Point of b1 holds {b2} c= MaxADSet b2 by Th14;

theorem Th21: :: TEX_4:21
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 is anti-discrete & b3 in b2 implies b2 c= MaxADSet b3 )
proof end;

theorem Th22: :: TEX_4:22
for b1 being non empty TopStruct
for b2 being Point of b1 holds MaxADSet b2 is maximal_anti-discrete
proof end;

theorem Th23: :: TEX_4:23
for b1 being non empty TopStruct
for b2, b3 being Point of b1 holds
( b3 in MaxADSet b2 iff MaxADSet b3 = MaxADSet b2 )
proof end;

theorem Th24: :: TEX_4:24
for b1 being non empty TopStruct
for b2, b3 being Point of b1 holds
( MaxADSet b2 misses MaxADSet b3 or MaxADSet b2 = MaxADSet b3 )
proof end;

theorem Th25: :: TEX_4:25
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 is closed & b3 in b2 implies MaxADSet b3 c= b2 )
proof end;

theorem Th26: :: TEX_4:26
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b2 is open & b3 in b2 implies MaxADSet b3 c= b2 )
proof end;

theorem Th27: :: TEX_4:27
for b1 being non empty TopStruct
for b2 being Point of b1 holds
( { b3 where B is Subset of b1 : ( b3 is closed & b2 in b3 ) } <> {} implies MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is closed & b2 in b3 ) } )
proof end;

theorem Th28: :: TEX_4:28
for b1 being non empty TopStruct
for b2 being Point of b1 holds
( { b3 where B is Subset of b1 : ( b3 is open & b2 in b3 ) } <> {} implies MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is open & b2 in b3 ) } )
proof end;

definition
let c1 be non empty TopStruct ;
let c2 be Subset of c1;
redefine attr a2 is maximal_anti-discrete means :Def9: :: TEX_4:def 9
ex b1 being Point of a1 st
( b1 in a2 & a2 = MaxADSet b1 );
compatibility
( c2 is maximal_anti-discrete iff ex b1 being Point of c1 st
( b1 in c2 & c2 = MaxADSet b1 ) )
proof end;
end;

:: deftheorem Def9 defines maximal_anti-discrete TEX_4:def 9 :
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( b2 is maximal_anti-discrete iff ex b3 being Point of b1 st
( b3 in b2 & b2 = MaxADSet b3 ) );

theorem Th29: :: TEX_4:29
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b3 in b2 & b2 is maximal_anti-discrete implies b2 = MaxADSet b3 )
proof end;

definition
let c1 be non empty TopStruct ;
let c2 be non empty Subset of c1;
redefine attr a2 is maximal_anti-discrete means :: TEX_4:def 10
for b1 being Point of a1 holds
( b1 in a2 implies a2 = MaxADSet b1 );
compatibility
( c2 is maximal_anti-discrete iff for b1 being Point of c1 holds
( b1 in c2 implies c2 = MaxADSet b1 ) )
proof end;
end;

:: deftheorem Def10 defines maximal_anti-discrete TEX_4:def 10 :
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
( b2 is maximal_anti-discrete iff for b3 being Point of b1 holds
( b3 in b2 implies b2 = MaxADSet b3 ) );

definition
let c1 be non empty TopStruct ;
let c2 be Subset of c1;
func MaxADSet c2 -> Subset of a1 equals :: TEX_4:def 11
union { (MaxADSet b1) where B is Point of a1 : b1 in a2 } ;
coherence
union { (MaxADSet b1) where B is Point of c1 : b1 in c2 } is Subset of c1
proof end;
end;

:: deftheorem Def11 defines MaxADSet TEX_4:def 11 :
for b1 being non empty TopStruct
for b2 being Subset of b1 holds MaxADSet b2 = union { (MaxADSet b3) where B is Point of b1 : b3 in b2 } ;

theorem Th30: :: TEX_4:30
for b1 being non empty TopStruct
for b2 being Point of b1 holds MaxADSet b2 = MaxADSet {b2}
proof end;

theorem Th31: :: TEX_4:31
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
not ( MaxADSet b3 meets MaxADSet b2 & not MaxADSet b3 meets b2 )
proof end;

theorem Th32: :: TEX_4:32
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( MaxADSet b3 meets MaxADSet b2 implies MaxADSet b3 c= MaxADSet b2 )
proof end;

theorem Th33: :: TEX_4:33
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b2 c= b3 implies MaxADSet b2 c= MaxADSet b3 )
proof end;

theorem Th34: :: TEX_4:34
for b1 being non empty TopStruct
for b2 being Subset of b1 holds b2 c= MaxADSet b2
proof end;

theorem Th35: :: TEX_4:35
for b1 being non empty TopStruct
for b2 being Subset of b1 holds MaxADSet b2 = MaxADSet (MaxADSet b2)
proof end;

theorem Th36: :: TEX_4:36
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b2 c= MaxADSet b3 implies MaxADSet b2 c= MaxADSet b3 )
proof end;

theorem Th37: :: TEX_4:37
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b3 c= MaxADSet b2 & b2 c= MaxADSet b3 implies MaxADSet b2 = MaxADSet b3 )
proof end;

theorem Th38: :: TEX_4:38
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds MaxADSet (b2 \/ b3) = (MaxADSet b2) \/ (MaxADSet b3)
proof end;

theorem Th39: :: TEX_4:39
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds MaxADSet (b2 /\ b3) c= (MaxADSet b2) /\ (MaxADSet b3)
proof end;

registration
let c1 be non empty TopStruct ;
let c2 be non empty Subset of c1;
cluster MaxADSet a2 -> non empty ;
coherence
not MaxADSet c2 is empty
proof end;
end;

registration
let c1 be non empty TopStruct ;
let c2 be empty Subset of c1;
cluster MaxADSet a2 -> empty ;
coherence
MaxADSet c2 is empty
proof end;
end;

registration
let c1 be non empty TopStruct ;
let c2 be non proper Subset of c1;
cluster MaxADSet a2 -> non empty non proper ;
coherence
not MaxADSet c2 is proper
proof end;
end;

registration
let c1 be non empty non trivial TopStruct ;
let c2 be non empty non trivial Subset of c1;
cluster MaxADSet a2 -> non empty non trivial ;
coherence
not MaxADSet c2 is trivial
proof end;
end;

theorem Th40: :: TEX_4:40
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b2 is open & b3 c= b2 implies MaxADSet b3 c= b2 )
proof end;

theorem Th41: :: TEX_4:41
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( { b3 where B is Subset of b1 : ( b3 is open & b2 c= b3 ) } <> {} implies MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is open & b2 c= b3 ) } )
proof end;

theorem Th42: :: TEX_4:42
for b1 being non empty TopStruct
for b2, b3 being Subset of b1 holds
( b2 is closed & b3 c= b2 implies MaxADSet b3 c= b2 )
proof end;

theorem Th43: :: TEX_4:43
for b1 being non empty TopStruct
for b2 being Subset of b1 holds
( { b3 where B is Subset of b1 : ( b3 is closed & b2 c= b3 ) } <> {} implies MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is closed & b2 c= b3 ) } )
proof end;

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :Def12: :: TEX_4:def 12
for b1 being Point of a1 holds
( b1 in a2 implies a2 c= Cl {b1} );
compatibility
( c2 is anti-discrete iff for b1 being Point of c1 holds
( b1 in c2 implies c2 c= Cl {b1} ) )
proof end;
end;

:: deftheorem Def12 defines anti-discrete TEX_4:def 12 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Point of b1 holds
( b3 in b2 implies b2 c= Cl {b3} ) );

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :: TEX_4:def 13
for b1 being Point of a1 holds
( b1 in a2 implies Cl a2 = Cl {b1} );
compatibility
( c2 is anti-discrete iff for b1 being Point of c1 holds
( b1 in c2 implies Cl c2 = Cl {b1} ) )
proof end;
end;

:: deftheorem Def13 defines anti-discrete TEX_4:def 13 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3 being Point of b1 holds
( b3 in b2 implies Cl b2 = Cl {b3} ) );

definition
let c1 be non empty TopSpace;
let c2 be Subset of c1;
redefine attr a2 is anti-discrete means :Def14: :: TEX_4:def 14
for b1, b2 being Point of a1 holds
( b1 in a2 & b2 in a2 implies Cl {b1} = Cl {b2} );
compatibility
( c2 is anti-discrete iff for b1, b2 being Point of c1 holds
( b1 in c2 & b2 in c2 implies Cl {b1} = Cl {b2} ) )
proof end;
end;

:: deftheorem Def14 defines anti-discrete TEX_4:def 14 :
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is anti-discrete iff for b3, b4 being Point of b1 holds
( b3 in b2 & b4 in b2 implies Cl {b3} = Cl {b4} ) );

theorem Th44: :: TEX_4:44
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being Subset of b1 holds
( b3 is anti-discrete & Cl {b2} c= b3 implies b3 = Cl {b2} )
proof end;

theorem Th45: :: TEX_4:45
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( ( b2 is anti-discrete & b2 is closed ) iff for b3 being Point of b1 holds
( b3 in b2 implies b2 = Cl {b3} ) )
proof end;

theorem Th46: :: TEX_4:46
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is anti-discrete & not b2 is open implies b2 is boundary )
proof end;

theorem Th47: :: TEX_4:47
for b1 being non empty TopSpace
for b2 being Point of b1 holds
( Cl {b2} = {b2} implies {b2} is maximal_anti-discrete )
proof end;

theorem Th48: :: TEX_4:48
for b1 being non empty TopSpace
for b2 being Point of b1 holds MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is open & b2 in b3 ) }
proof end;

theorem Th49: :: TEX_4:49
for b1 being non empty TopSpace
for b2 being Point of b1 holds MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is closed & b2 in b3 ) }
proof end;

theorem Th50: :: TEX_4:50
for b1 being non empty TopSpace
for b2 being Point of b1 holds MaxADSet b2 c= Cl {b2}
proof end;

Lemma49: for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( MaxADSet b2 = MaxADSet b3 implies Cl {b2} = Cl {b3} )
proof end;

theorem Th51: :: TEX_4:51
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( MaxADSet b2 = MaxADSet b3 iff Cl {b2} = Cl {b3} )
proof end;

theorem Th52: :: TEX_4:52
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( MaxADSet b2 misses MaxADSet b3 iff Cl {b2} <> Cl {b3} )
proof end;

definition
let c1 be non empty TopSpace;
let c2 be Point of c1;
redefine func MaxADSet as MaxADSet c2 -> non empty Subset of a1 equals :: TEX_4:def 15
(Cl {a2}) /\ (meet { b1 where B is Subset of a1 : ( b1 is open & a2 in b1 ) } );
compatibility
for b1 being non empty Subset of c1 holds
( b1 = MaxADSet c2 iff b1 = (Cl {c2}) /\ (meet { b2 where B is Subset of c1 : ( b2 is open & c2 in b2 ) } ) )
proof end;
coherence
MaxADSet c2 is non empty Subset of c1
;
end;

:: deftheorem Def15 defines MaxADSet TEX_4:def 15 :
for b1 being non empty TopSpace
for b2 being Point of b1 holds MaxADSet b2 = (Cl {b2}) /\ (meet { b3 where B is Subset of b1 : ( b3 is open & b2 in b3 ) } );

theorem Th53: :: TEX_4:53
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( Cl {b2} c= Cl {b3} iff meet { b4 where B is Subset of b1 : ( b4 is open & b3 in b4 ) } c= meet { b4 where B is Subset of b1 : ( b4 is open & b2 in b4 ) } )
proof end;

theorem Th54: :: TEX_4:54
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( Cl {b2} c= Cl {b3} iff MaxADSet b3 c= meet { b4 where B is Subset of b1 : ( b4 is open & b2 in b4 ) } )
proof end;

theorem Th55: :: TEX_4:55
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( MaxADSet b2 misses MaxADSet b3 iff not ( ( for b4 being Subset of b1 holds
not ( b4 is open & MaxADSet b2 c= b4 & b4 misses MaxADSet b3 ) ) & ( for b4 being Subset of b1 holds
not ( b4 is open & b4 misses MaxADSet b2 & MaxADSet b3 c= b4 ) ) ) )
proof end;

theorem Th56: :: TEX_4:56
for b1 being non empty TopSpace
for b2, b3 being Point of b1 holds
( MaxADSet b2 misses MaxADSet b3 iff not ( ( for b4 being Subset of b1 holds
not ( b4 is closed & MaxADSet b2 c= b4 & b4 misses MaxADSet b3 ) ) & ( for b4 being Subset of b1 holds
not ( b4 is closed & b4 misses MaxADSet b2 & MaxADSet b3 c= b4 ) ) ) )
proof end;

theorem Th57: :: TEX_4:57
for b1 being non empty TopSpace
for b2 being Subset of b1 holds MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is open & b2 c= b3 ) }
proof end;

theorem Th58: :: TEX_4:58
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is open implies MaxADSet b2 = b2 )
proof end;

theorem Th59: :: TEX_4:59
for b1 being non empty TopSpace
for b2 being Subset of b1 holds MaxADSet (Int b2) = Int b2 by Th58;

theorem Th60: :: TEX_4:60
for b1 being non empty TopSpace
for b2 being Subset of b1 holds MaxADSet b2 c= meet { b3 where B is Subset of b1 : ( b3 is closed & b2 c= b3 ) }
proof end;

theorem Th61: :: TEX_4:61
for b1 being non empty TopSpace
for b2 being Subset of b1 holds MaxADSet b2 c= Cl b2
proof end;

theorem Th62: :: TEX_4:62
for b1 being non empty TopSpace
for b2 being Subset of b1 holds
( b2 is closed implies MaxADSet b2 = b2 )
proof end;

theorem Th63: :: TEX_4:63
for b1 being non empty TopSpace
for b2 being Subset of b1 holds MaxADSet (Cl b2) = Cl b2 by Th62;

theorem Th64: :: TEX_4:64
for b1 being non empty TopSpace
for b2 being Subset of b1 holds Cl (MaxADSet b2) = Cl b2
proof end;

theorem Th65: :: TEX_4:65
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( MaxADSet b2 = MaxADSet b3 implies Cl b2 = Cl b3 )
proof end;

theorem Th66: :: TEX_4:66
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( ( b2 is closed or b3 is closed ) implies MaxADSet (b2 /\ b3) = (MaxADSet b2) /\ (MaxADSet b3) )
proof end;

theorem Th67: :: TEX_4:67
for b1 being non empty TopSpace
for b2, b3 being Subset of b1 holds
( ( b2 is open or b3 is open ) implies MaxADSet (b2 /\ b3) = (MaxADSet b2) /\ (MaxADSet b3) )
proof end;

theorem Th68: :: TEX_4:68
for b1 being non empty TopStruct
for b2 being SubSpace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 & b2 is anti-discrete implies b3 is anti-discrete )
proof end;

theorem Th69: :: TEX_4:69
for b1 being non empty TopStruct
for b2 being SubSpace of b1 holds
( b2 is TopSpace-like implies for b3 being Subset of b1 holds
( b3 = the carrier of b2 & b3 is anti-discrete implies b2 is anti-discrete ) )
proof end;

theorem Th70: :: TEX_4:70
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1 holds
( ( for b3 being open SubSpace of b1 holds
( b2 misses b3 or b2 is SubSpace of b3 ) ) implies b2 is anti-discrete )
proof end;

theorem Th71: :: TEX_4:71
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1 holds
( ( for b3 being closed SubSpace of b1 holds
( b2 misses b3 or b2 is SubSpace of b3 ) ) implies b2 is anti-discrete )
proof end;

theorem Th72: :: TEX_4:72
for b1 being non empty TopSpace
for b2 being anti-discrete SubSpace of b1
for b3 being open SubSpace of b1 holds
( b2 misses b3 or b2 is SubSpace of b3 )
proof end;

theorem Th73: :: TEX_4:73
for b1 being non empty TopSpace
for b2 being anti-discrete SubSpace of b1
for b3 being closed SubSpace of b1 holds
( b2 misses b3 or b2 is SubSpace of b3 )
proof end;

definition
let c1 be non empty TopStruct ;
let c2 be SubSpace of c1;
attr a2 is maximal_anti-discrete means :Def16: :: TEX_4:def 16
( a2 is anti-discrete & ( for b1 being SubSpace of a1 holds
( b1 is anti-discrete & the carrier of a2 c= the carrier of b1 implies the carrier of a2 = the carrier of b1 ) ) );
end;

:: deftheorem Def16 defines maximal_anti-discrete TEX_4:def 16 :
for b1 being non empty TopStruct
for b2 being SubSpace of b1 holds
( b2 is maximal_anti-discrete iff ( b2 is anti-discrete & ( for b3 being SubSpace of b1 holds
( b3 is anti-discrete & the carrier of b2 c= the carrier of b3 implies the carrier of b2 = the carrier of b3 ) ) ) );

registration
let c1 be non empty TopStruct ;
cluster maximal_anti-discrete -> anti-discrete SubSpace of a1;
coherence
for b1 being SubSpace of c1 holds
( b1 is maximal_anti-discrete implies b1 is anti-discrete )
by Def16;
cluster non anti-discrete -> non maximal_anti-discrete SubSpace of a1;
coherence
for b1 being SubSpace of c1 holds
not ( not b1 is anti-discrete & b1 is maximal_anti-discrete )
by Def16;
end;

theorem Th74: :: TEX_4:74
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1
for b3 being Subset of b1 holds
( b3 = the carrier of b2 implies ( b2 is maximal_anti-discrete iff b3 is maximal_anti-discrete ) )
proof end;

registration
let c1 be non empty TopSpace;
cluster non empty open anti-discrete -> non empty anti-discrete maximal_anti-discrete SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
( b1 is open & b1 is anti-discrete implies b1 is maximal_anti-discrete )
proof end;
cluster non empty open non maximal_anti-discrete -> non empty non anti-discrete non maximal_anti-discrete SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
not ( b1 is open & not b1 is maximal_anti-discrete & b1 is anti-discrete )
proof end;
cluster non empty anti-discrete non maximal_anti-discrete -> non empty non open SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
not ( b1 is anti-discrete & not b1 is maximal_anti-discrete & b1 is open )
proof end;
cluster non empty closed anti-discrete -> non empty anti-discrete maximal_anti-discrete SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
( b1 is closed & b1 is anti-discrete implies b1 is maximal_anti-discrete )
proof end;
cluster non empty closed non maximal_anti-discrete -> non empty non anti-discrete non maximal_anti-discrete SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
not ( b1 is closed & not b1 is maximal_anti-discrete & b1 is anti-discrete )
proof end;
cluster non empty anti-discrete non maximal_anti-discrete -> non empty non closed SubSpace of a1;
coherence
for b1 being non empty SubSpace of c1 holds
not ( b1 is anti-discrete & not b1 is maximal_anti-discrete & b1 is closed )
proof end;
end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
func MaxADSspace c2 -> strict SubSpace of a1 means :Def17: :: TEX_4:def 17
the carrier of a3 = MaxADSet a2;
existence
ex b1 being strict SubSpace of c1 st the carrier of b1 = MaxADSet c2
proof end;
uniqueness
for b1, b2 being strict SubSpace of c1 holds
( the carrier of b1 = MaxADSet c2 & the carrier of b2 = MaxADSet c2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def17 defines MaxADSspace TEX_4:def 17 :
for b1 being TopStruct
for b2 being Point of b1
for b3 being strict SubSpace of b1 holds
( b3 = MaxADSspace b2 iff the carrier of b3 = MaxADSet b2 );

registration
let c1 be non empty TopStruct ;
let c2 be Point of c1;
cluster MaxADSspace a2 -> non empty strict ;
coherence
not MaxADSspace c2 is empty
proof end;
end;

Lemma63: for b1 being non empty TopStruct
for b2, b3 being SubSpace of b1 holds
( the carrier of b2 c= the carrier of b3 implies b2 is SubSpace of b3 )
proof end;

theorem Th75: :: TEX_4:75
for b1 being non empty TopStruct
for b2 being Point of b1 holds
Sspace b2 is SubSpace of MaxADSspace b2
proof end;

theorem Th76: :: TEX_4:76
for b1 being non empty TopStruct
for b2, b3 being Point of b1 holds
( b3 is Point of (MaxADSspace b2) iff TopStruct(# the carrier of (MaxADSspace b3),the topology of (MaxADSspace b3) #) = TopStruct(# the carrier of (MaxADSspace b2),the topology of (MaxADSspace b2) #) )
proof end;

theorem Th77: :: TEX_4:77
for b1 being non empty TopStruct
for b2, b3 being Point of b1 holds
( the carrier of (MaxADSspace b2) misses the carrier of (MaxADSspace b3) or TopStruct(# the carrier of (MaxADSspace b2),the topology of (MaxADSspace b2) #) = TopStruct(# the carrier of (MaxADSspace b3),the topology of (MaxADSspace b3) #) )
proof end;

registration
let c1 be non empty TopSpace;
cluster strict anti-discrete maximal_anti-discrete SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( b1 is maximal_anti-discrete & b1 is strict )
proof end;
end;

registration
let c1 be non empty TopSpace;
let c2 be Point of c1;
cluster MaxADSspace a2 -> non empty strict anti-discrete maximal_anti-discrete ;
coherence
MaxADSspace c2 is maximal_anti-discrete
proof end;
end;

theorem Th78: :: TEX_4:78
for b1 being non empty TopSpace
for b2 being non empty closed SubSpace of b1
for b3 being Point of b1 holds
( b3 is Point of b2 implies MaxADSspace b3 is SubSpace of b2 )
proof end;

theorem Th79: :: TEX_4:79
for b1 being non empty TopSpace
for b2 being non empty open SubSpace of b1
for b3 being Point of b1 holds
( b3 is Point of b2 implies MaxADSspace b3 is SubSpace of b2 )
proof end;

theorem Th80: :: TEX_4:80
for b1 being non empty TopSpace
for b2 being Point of b1 holds
( Cl {b2} = {b2} implies Sspace b2 is maximal_anti-discrete )
proof end;

notation
let c1 be TopStruct ;
let c2 be Subset of c1;
synonym Sspace c2 for c1 | c2;
end;

Lemma64: for b1 being TopStruct
for b2 being Subset of b1 holds the carrier of (b1 | b2) = b2
proof end;

registration
let c1 be non empty TopStruct ;
let c2 be non empty Subset of c1;
canceled;
cluster Sspace a2 -> non empty ;
coherence
not Sspace c2 is empty
;
end;

theorem Th81: :: TEX_4:81
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
b2 is Subset of (Sspace b2)
proof end;

theorem Th82: :: TEX_4:82
for b1 being non empty TopStruct
for b2 being SubSpace of b1
for b3 being non empty Subset of b1 holds
( b3 is Subset of b2 implies Sspace b3 is SubSpace of b2 )
proof end;

registration
let c1 be non empty non trivial TopStruct ;
cluster strict non proper SubSpace of a1;
existence
ex b1 being SubSpace of c1 st
( not b1 is proper & b1 is strict )
proof end;
end;

registration
let c1 be non empty non trivial TopStruct ;
let c2 be non empty non trivial Subset of c1;
cluster Sspace a2 -> non trivial ;
coherence
not Sspace c2 is trivial
proof end;
end;

registration
let c1 be non empty TopStruct ;
let c2 be non empty non proper Subset of c1;
cluster Sspace a2 -> non proper ;
coherence
not Sspace c2 is proper
proof end;
end;

definition
let c1 be non empty TopStruct ;
let c2 be Subset of c1;
func MaxADSspace c2 -> strict SubSpace of a1 means :Def18: :: TEX_4:def 18
the carrier of a3 = MaxADSet a2;
existence
ex b1 being strict SubSpace of c1 st the carrier of b1 = MaxADSet c2
proof end;
uniqueness
for b1, b2 being strict SubSpace of c1 holds
( the carrier of b1 = MaxADSet c2 & the carrier of b2 = MaxADSet c2 implies b1 = b2 )
proof end;
end;

:: deftheorem Def18 defines MaxADSspace TEX_4:def 18 :
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being strict SubSpace of b1 holds
( b3 = MaxADSspace b2 iff the carrier of b3 = MaxADSet b2 );

registration
let c1 be non empty TopStruct ;
let c2 be non empty Subset of c1;
cluster MaxADSspace a2 -> non empty strict ;
coherence
not MaxADSspace c2 is empty
proof end;
end;

theorem Th83: :: TEX_4:83
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
b2 is Subset of (MaxADSspace b2)
proof end;

theorem Th84: :: TEX_4:84
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds
Sspace b2 is SubSpace of MaxADSspace b2
proof end;

theorem Th85: :: TEX_4:85
for b1 being non empty TopStruct
for b2 being Point of b1 holds TopStruct(# the carrier of (MaxADSspace b2),the topology of (MaxADSspace b2) #) = TopStruct(# the carrier of (MaxADSspace {b2}),the topology of (MaxADSspace {b2}) #)
proof end;

theorem Th86: :: TEX_4:86
for b1 being non empty TopStruct
for b2, b3 being non empty Subset of b1 holds
( b2 c= b3 implies MaxADSspace b2 is SubSpace of MaxADSspace b3 )
proof end;

theorem Th87: :: TEX_4:87
for b1 being non empty TopStruct
for b2 being non empty Subset of b1 holds TopStruct(# the carrier of (MaxADSspace b2),the topology of (MaxADSspace b2) #) = TopStruct(# the carrier of (MaxADSspace (MaxADSet b2)),the topology of (MaxADSspace (MaxADSet b2)) #)
proof end;

theorem Th88: :: TEX_4:88
for b1 being non empty TopStruct
for b2, b3 being non empty Subset of b1 holds
( b2 is Subset of (MaxADSspace b3) implies MaxADSspace b2 is SubSpace of MaxADSspace b3 )
proof end;

theorem Th89: :: TEX_4:89
for b1 being non empty TopStruct
for b2, b3 being non empty Subset of b1 holds
( ( b3 is Subset of (MaxADSspace b2) & b2 is Subset of (MaxADSspace b3) ) iff TopStruct(# the carrier of (MaxADSspace b2),the topology of (MaxADSspace b2) #) = TopStruct(# the carrier of (MaxADSspace b3),the topology of (MaxADSspace b3) #) )
proof end;

registration
let c1 be non empty non trivial TopStruct ;
let c2 be non empty non trivial Subset of c1;
cluster MaxADSspace a2 -> non empty strict non trivial ;
coherence
not MaxADSspace c2 is trivial
proof end;
end;

registration
let c1 be non empty TopStruct ;
let c2 be non empty non proper Subset of c1;
cluster MaxADSspace a2 -> non empty strict non proper ;
coherence
not MaxADSspace c2 is proper
proof end;
end;

theorem Th90: :: TEX_4:90
for b1 being non empty TopSpace
for b2 being open SubSpace of b1
for b3 being non empty Subset of b1 holds
( b3 is Subset of b2 implies MaxADSspace b3 is SubSpace of b2 )
proof end;

theorem Th91: :: TEX_4:91
for b1 being non empty TopSpace
for b2 being closed SubSpace of b1
for b3 being non empty Subset of b1 holds
( b3 is Subset of b2 implies MaxADSspace b3 is SubSpace of b2 )
proof end;