:: YELLOW13 semantic presentation

registration
let c1 be finite 1-sorted ;
cluster the carrier of a1 -> finite ;
coherence
the carrier of c1 is finite
by GROUP_1:def 14;
end;

registration
let c1 be trivial 1-sorted ;
cluster the carrier of a1 -> trivial ;
coherence
the carrier of c1 is trivial
by REALSET2:def 5;
end;

registration
cluster trivial -> finite set ;
coherence
for b1 being set holds
( b1 is trivial implies b1 is finite )
proof end;
end;

registration
cluster infinite -> non trivial set ;
coherence
for b1 being set holds
not ( not b1 is finite & b1 is trivial )
proof end;
end;

registration
cluster trivial -> finite 1-sorted ;
coherence
for b1 being 1-sorted holds
( b1 is trivial implies b1 is finite )
proof end;
end;

registration
cluster infinite -> non trivial 1-sorted ;
coherence
for b1 being 1-sorted holds
not ( not b1 is finite & b1 is trivial )
proof end;
end;

registration
cluster non trivial -> non empty 1-sorted ;
coherence
for b1 being 1-sorted holds
not ( not b1 is trivial & b1 is empty )
proof end;
end;

registration
cluster strict non empty finite trivial 1-sorted ;
existence
ex b1 being 1-sorted st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
cluster non empty finite trivial strict RelStr ;
existence
ex b1 being RelStr st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
cluster non empty finite trivial strict TopRelStr ;
existence
ex b1 being TopRelStr st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
end;

theorem Th1: :: YELLOW13:1
for b1 being non empty being_T1 TopSpace
for b2 being finite Subset of b1 holds b2 is closed
proof end;

registration
let c1 be non empty being_T1 TopSpace;
cluster finite -> closed Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds
( b1 is finite implies b1 is closed )
by Th1;
end;

registration
let c1 be compact TopStruct ;
cluster [#] a1 -> compact ;
coherence
[#] c1 is compact
by COMPTS_1:10;
end;

registration
cluster non empty strict finite trivial TopStruct ;
existence
ex b1 being TopSpace st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
end;

registration
cluster non empty finite being_T1 -> non empty discrete TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is finite & b1 is being_T1 implies b1 is discrete )
proof end;
end;

registration
cluster finite -> compact TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is finite implies b1 is compact )
proof end;
end;

theorem Th2: :: YELLOW13:2
for b1 being non empty discrete TopSpace holds b1 is_T4
proof end;

theorem Th3: :: YELLOW13:3
for b1 being non empty discrete TopSpace holds b1 is_T3
proof end;

theorem Th4: :: YELLOW13:4
for b1 being non empty discrete TopSpace holds b1 is_T2
proof end;

theorem Th5: :: YELLOW13:5
for b1 being non empty discrete TopSpace holds b1 is_T1
proof end;

registration
cluster non empty discrete -> being_T1 being_T2 being_T3 being_T4 TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is discrete & not b1 is empty implies ( b1 is being_T4 & b1 is being_T3 & b1 is being_T2 & b1 is being_T1 ) )
by Th2, Th3, Th4, Th5;
end;

registration
cluster non empty being_T1 being_T4 -> non empty being_T3 TopStruct ;
coherence
for b1 being non empty TopSpace holds
( b1 is being_T4 & b1 is being_T1 implies b1 is being_T3 )
proof end;
end;

registration
cluster being_T1 being_T3 -> being_T2 TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is being_T3 & b1 is being_T1 implies b1 is being_T2 )
proof end;
end;

registration
cluster being_T2 -> being_T1 TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is being_T2 implies b1 is being_T1 )
proof end;
end;

registration
cluster being_T1 -> T_0 TopStruct ;
coherence
for b1 being TopSpace holds
( b1 is being_T1 implies b1 is T_0 )
proof end;
end;

theorem Th6: :: YELLOW13:6
for b1 being reflexive RelStr
for b2 being reflexive transitive RelStr
for b3 being Function of b1,b2
for b4 being Subset of b1 holds downarrow (b3 .: b4) c= downarrow (b3 .: (downarrow b4))
proof end;

theorem Th7: :: YELLOW13:7
for b1 being reflexive RelStr
for b2 being reflexive transitive RelStr
for b3 being Function of b1,b2
for b4 being Subset of b1 holds
( b3 is monotone implies downarrow (b3 .: b4) = downarrow (b3 .: (downarrow b4)) )
proof end;

theorem Th8: :: YELLOW13:8
for b1 being non empty Poset holds IdsMap b1 is one-to-one
proof end;

registration
let c1 be non empty Poset;
cluster IdsMap a1 -> V13 ;
coherence
IdsMap c1 is one-to-one
by Th8;
end;

theorem Th9: :: YELLOW13:9
for b1 being finite LATTICE holds SupMap b1 is one-to-one
proof end;

registration
let c1 be finite LATTICE;
cluster SupMap a1 -> V13 ;
coherence
SupMap c1 is one-to-one
by Th9;
end;

theorem Th10: :: YELLOW13:10
for b1 being finite LATTICE holds b1, InclPoset (Ids b1) are_isomorphic
proof end;

theorem Th11: :: YELLOW13:11
for b1 being non empty complete Poset
for b2 being Element of b1
for b3 being non empty Subset of b1 holds b2 "/\" preserves_inf_of b3
proof end;

theorem Th12: :: YELLOW13:12
for b1 being non empty complete Poset
for b2 being Element of b1 holds b2 "/\" is meet-preserving
proof end;

registration
let c1 be non empty complete Poset;
let c2 be Element of c1;
cluster a2 "/\" -> meet-preserving ;
coherence
c2 "/\" is meet-preserving
by Th12;
end;

theorem Th13: :: YELLOW13:13
for b1 being non empty anti-discrete TopStruct
for b2 being Point of b1 holds
{the carrier of b1} is Basis of b2
proof end;

theorem Th14: :: YELLOW13:14
for b1 being non empty anti-discrete TopStruct
for b2 being Point of b1
for b3 being Basis of b2 holds b3 = {the carrier of b1}
proof end;

theorem Th15: :: YELLOW13:15
for b1 being non empty TopSpace
for b2 being Basis of b1
for b3 being Point of b1 holds
{ b4 where B is Subset of b1 : ( b4 in b2 & b3 in b4 ) } is Basis of b3
proof end;

Lemma11: for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b3 in Cl b2 implies for b4 being Basis of b3
for b5 being Subset of b1 holds
not ( b5 in b4 & not b2 meets b5 ) )
proof end;

Lemma12: for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
not ( ( for b4 being Basis of b3
for b5 being Subset of b1 holds
not ( b5 in b4 & not b2 meets b5 ) ) & ( for b4 being Basis of b3 holds
ex b5 being Subset of b1 st
( b5 in b4 & not b2 meets b5 ) ) )
proof end;

Lemma13: for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( ex b4 being Basis of b3 st
for b5 being Subset of b1 holds
not ( b5 in b4 & not b2 meets b5 ) implies b3 in Cl b2 )
proof end;

theorem Th16: :: YELLOW13:16
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b3 in Cl b2 iff for b4 being Basis of b3
for b5 being Subset of b1 holds
not ( b5 in b4 & not b2 meets b5 ) )
proof end;

theorem Th17: :: YELLOW13:17
for b1 being non empty TopStruct
for b2 being Subset of b1
for b3 being Point of b1 holds
( b3 in Cl b2 iff ex b4 being Basis of b3 st
for b5 being Subset of b1 holds
not ( b5 in b4 & not b2 meets b5 ) )
proof end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
mode basis of c2 -> Subset-Family of a1 means :Def1: :: YELLOW13:def 1
for b1 being Subset of a1 holds
not ( a2 in Int b1 & ( for b2 being Subset of a1 holds
not ( b2 in a3 & a2 in Int b2 & b2 c= b1 ) ) );
existence
ex b1 being Subset-Family of c1 st
for b2 being Subset of c1 holds
not ( c2 in Int b2 & ( for b3 being Subset of c1 holds
not ( b3 in b1 & c2 in Int b3 & b3 c= b2 ) ) )
proof end;
end;

:: deftheorem Def1 defines basis YELLOW13:def 1 :
for b1 being TopStruct
for b2 being Point of b1
for b3 being Subset-Family of b1 holds
( b3 is basis of b2 iff for b4 being Subset of b1 holds
not ( b2 in Int b4 & ( for b5 being Subset of b1 holds
not ( b5 in b3 & b2 in Int b5 & b5 c= b4 ) ) ) );

definition
let c1 be non empty TopSpace;
let c2 be Point of c1;
redefine mode basis of c2 -> Subset-Family of a1 means :: YELLOW13:def 2
for b1 being a_neighborhood of a2 holds
ex b2 being a_neighborhood of a2 st
( b2 in a3 & b2 c= b1 );
compatibility
for b1 being Subset-Family of c1 holds
( b1 is basis of c2 iff for b2 being a_neighborhood of c2 holds
ex b3 being a_neighborhood of c2 st
( b3 in b1 & b3 c= b2 ) )
proof end;
end;

:: deftheorem Def2 defines basis YELLOW13:def 2 :
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being Subset-Family of b1 holds
( b3 is basis of b2 iff for b4 being a_neighborhood of b2 holds
ex b5 being a_neighborhood of b2 st
( b5 in b3 & b5 c= b4 ) );

theorem Th18: :: YELLOW13:18
for b1 being TopStruct
for b2 being Point of b1 holds
bool the carrier of b1 is basis of b2
proof end;

theorem Th19: :: YELLOW13:19
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being basis of b2 holds
not b3 is empty
proof end;

registration
let c1 be non empty TopSpace;
let c2 be Point of c1;
cluster -> non empty basis of a2;
coherence
for b1 being basis of c2 holds
not b1 is empty
by Th19;
end;

registration
let c1 be TopStruct ;
let c2 be Point of c1;
cluster non empty basis of a2;
existence
not for b1 being basis of c2 holds b1 is empty
proof end;
end;

definition
let c1 be TopStruct ;
let c2 be Point of c1;
let c3 be basis of c2;
attr a3 is correct means :Def3: :: YELLOW13:def 3
for b1 being Subset of a1 holds
( b1 in a3 iff a2 in Int b1 );
end;

:: deftheorem Def3 defines correct YELLOW13:def 3 :
for b1 being TopStruct
for b2 being Point of b1
for b3 being basis of b2 holds
( b3 is correct iff for b4 being Subset of b1 holds
( b4 in b3 iff b2 in Int b4 ) );

E18: now
let c1 be TopStruct ;
let c2 be Point of c1;
let c3 be set ;
assume E19: c3 = { b1 where B is Subset of c1 : c2 in Int b1 } ;
c3 c= bool the carrier of c1
proof
let c4 be set ; :: according to TARSKI:def 3
assume c4 in c3 ;
then consider c5 being Subset of c1 such that
E20: c4 = c5 and
c2 in Int c5 by E19;
thus c4 in bool the carrier of c1 by E20;
end;
then reconsider c4 = c3 as Subset-Family of c1 ;
reconsider c5 = c4 as Subset-Family of c1 ;
for b1 being Subset of c1 holds
not ( c2 in Int b1 & ( for b2 being Subset of c1 holds
not ( b2 in c5 & c2 in Int b2 & b2 c= b1 ) ) )
proof
let c6 be Subset of c1;
assume E20: c2 in Int c6 ;
take c7 = c6;
thus ( c7 in c5 & c2 in Int c7 & c7 c= c6 ) by E19, E20;
end;
hence c3 is basis of c2 by Def1;
end;

E19: now
let c1 be TopStruct ;
let c2 be Point of c1;
let c3 be basis of c2;
assume E20: c3 = { b1 where B is Subset of c1 : c2 in Int b1 } ;
thus c3 is correct
proof
let c4 be Subset of c1; :: according to YELLOW13:def 3
hereby
assume c4 in c3 ;
then consider c5 being Subset of c1 such that
E21: ( c4 = c5 & c2 in Int c5 ) by E20;
thus c2 in Int c4 by E21;
end;
thus ( c2 in Int c4 implies c4 in c3 ) by E20;
end;
end;

registration
let c1 be TopStruct ;
let c2 be Point of c1;
cluster correct basis of a2;
existence
ex b1 being basis of c2 st b1 is correct
proof end;
end;

theorem Th20: :: YELLOW13:20
for b1 being TopStruct
for b2 being Point of b1 holds
{ b3 where B is Subset of b1 : b2 in Int b3 } is correct basis of b2 by Lemma18, Lemma19;

registration
let c1 be non empty TopSpace;
let c2 be Point of c1;
cluster non empty correct basis of a2;
existence
ex b1 being basis of c2 st
( not b1 is empty & b1 is correct )
proof end;
end;

theorem Th21: :: YELLOW13:21
for b1 being non empty anti-discrete TopStruct
for b2 being Point of b1 holds
{the carrier of b1} is correct basis of b2
proof end;

theorem Th22: :: YELLOW13:22
for b1 being non empty anti-discrete TopStruct
for b2 being Point of b1
for b3 being correct basis of b2 holds b3 = {the carrier of b1}
proof end;

theorem Th23: :: YELLOW13:23
for b1 being non empty TopSpace
for b2 being Point of b1
for b3 being Basis of b2 holds
b3 is basis of b2
proof end;

definition
let c1 be TopStruct ;
mode basis of c1 -> Subset-Family of a1 means :Def4: :: YELLOW13:def 4
for b1 being Point of a1 holds
a2 is basis of b1;
existence
ex b1 being Subset-Family of c1 st
for b2 being Point of c1 holds
b1 is basis of b2
proof end;
end;

:: deftheorem Def4 defines basis YELLOW13:def 4 :
for b1 being TopStruct
for b2 being Subset-Family of b1 holds
( b2 is basis of b1 iff for b3 being Point of b1 holds
b2 is basis of b3 );

theorem Th24: :: YELLOW13:24
for b1 being TopStruct holds
bool the carrier of b1 is basis of b1
proof end;

theorem Th25: :: YELLOW13:25
for b1 being non empty TopSpace
for b2 being basis of b1 holds
not b2 is empty
proof end;

registration
let c1 be non empty TopSpace;
cluster -> non empty basis of a1;
coherence
for b1 being basis of c1 holds
not b1 is empty
by Th25;
end;

registration
let c1 be TopStruct ;
cluster non empty basis of a1;
existence
not for b1 being basis of c1 holds b1 is empty
proof end;
end;

theorem Th26: :: YELLOW13:26
for b1 being non empty TopSpace
for b2 being basis of b1 holds the topology of b1 c= UniCl (Int b2)
proof end;

theorem Th27: :: YELLOW13:27
for b1 being TopSpace
for b2 being Basis of b1 holds
b2 is basis of b1
proof end;

definition
let c1 be non empty TopSpace-like TopRelStr ;
attr a1 is topological_semilattice means :Def5: :: YELLOW13:def 5
for b1 being Function of [:a1,a1:],a1 holds
( b1 = inf_op a1 implies b1 is continuous );
end;

:: deftheorem Def5 defines topological_semilattice YELLOW13:def 5 :
for b1 being non empty TopSpace-like TopRelStr holds
( b1 is topological_semilattice iff for b2 being Function of [:b1,b1:],b1 holds
( b2 = inf_op b1 implies b2 is continuous ) );

registration
cluster non empty TopSpace-like trivial reflexive -> non empty TopSpace-like topological_semilattice TopRelStr ;
coherence
for b1 being non empty TopSpace-like TopRelStr holds
( b1 is reflexive & b1 is trivial implies b1 is topological_semilattice )
proof end;
end;

registration
cluster non empty TopSpace-like finite trivial being_T1 reflexive T_0 compact being_T2 being_T3 being_T4 topological_semilattice TopRelStr ;
existence
ex b1 being TopRelStr st
( b1 is reflexive & b1 is trivial & not b1 is empty & b1 is TopSpace-like )
proof end;
end;

theorem Th28: :: YELLOW13:28
for b1 being non empty TopSpace-like topological_semilattice TopRelStr
for b2 being Element of b1 holds b2 "/\" is continuous
proof end;

registration
let c1 be non empty TopSpace-like topological_semilattice TopRelStr ;
let c2 be Element of c1;
cluster a2 "/\" -> continuous ;
coherence
c2 "/\" is continuous
by Th28;
end;