:: LOPCLSET semantic presentation

definition

let T be non empty TopStruct ;
func OpenClosedSet T -> Subset-Family of T equals :: LOPCLSET:def 1
{ x where x is Subset of T : ( x is open & x is closed ) } ;
coherence

proof end;

end;

:: deftheorem defines OpenClosedSet LOPCLSET:def 1 :

registration

let T be non empty TopSpace;
cluster OpenClosedSet T -> non empty ;
coherence

proof end;

end;

theorem :: LOPCLSET:1

canceled;

theorem Th2: :: LOPCLSET:2

for T being non empty TopSpace
for X being Subset of T st X in OpenClosedSet T holds
X is open

proof end;

theorem Th3: :: LOPCLSET:3

for T being non empty TopSpace
for X being Subset of T st X in OpenClosedSet T holds
X is closed

proof end;

theorem Th4: :: LOPCLSET:4

for T being non empty TopSpace
for X being Subset of T st X is open & X is closed holds
X in OpenClosedSet T ;

definition

let T be non empty TopSpace;
let C, D be Element of OpenClosedSet T;
:: original: \/
redefine func C \/ D -> Element of OpenClosedSet T;
coherence

proof end;

:: original: /\
redefine func C /\ D -> Element of OpenClosedSet T;
coherence

proof end;

end;

definition

let T be non empty TopSpace;
deffunc H₁( Element of OpenClosedSet T, Element of OpenClosedSet T) -> Element of OpenClosedSet T = $1 \/ $2;
func T_join T -> BinOp of OpenClosedSet T means :Def2: :: LOPCLSET:def 2
for A, B being Element of OpenClosedSet T holds it . A,B = A \/ B;
existence

proof end;

uniqueness

proof end;

deffunc H₂( Element of OpenClosedSet T, Element of OpenClosedSet T) -> Element of OpenClosedSet T = $1 /\ $2;
func T_meet T -> BinOp of OpenClosedSet T means :Def3: :: LOPCLSET:def 3
for A, B being Element of OpenClosedSet T holds it . A,B = A /\ B;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines T_join LOPCLSET:def 2 :

:: deftheorem Def3 defines T_meet LOPCLSET:def 3 :

theorem Th5: :: LOPCLSET:5

for T being non empty TopSpace
for x, y being Element of LattStr(# (OpenClosedSet T),(T_join T),(T_meet T) #)
for x', y' being Element of OpenClosedSet T st x = x' & y = y' holds
x "\/" y = x' \/ y' by Def2;

theorem Th6: :: LOPCLSET:6

for T being non empty TopSpace
for x, y being Element of LattStr(# (OpenClosedSet T),(T_join T),(T_meet T) #)
for x', y' being Element of OpenClosedSet T st x = x' & y = y' holds
x "/\" y = x' /\ y' by Def3;

theorem Th7: :: LOPCLSET:7

for T being non empty TopSpace holds {} T is Element of OpenClosedSet T

proof end;

theorem Th8: :: LOPCLSET:8

for T being non empty TopSpace holds [#] T is Element of OpenClosedSet T

proof end;

theorem Th9: :: LOPCLSET:9

for T being non empty TopSpace
for x being Element of OpenClosedSet T holds x ` is Element of OpenClosedSet T

proof end;

theorem Th10: :: LOPCLSET:10

for T being non empty TopSpace holds LattStr(# (OpenClosedSet T),(T_join T),(T_meet T) #) is Lattice

proof end;

definition

let T be non empty TopSpace;
func OpenClosedSetLatt T -> Lattice equals :: LOPCLSET:def 4
LattStr(# (OpenClosedSet T),(T_join T),(T_meet T) #);
coherence by Th10;

end;

:: deftheorem defines OpenClosedSetLatt LOPCLSET:def 4 :

theorem Th11: :: LOPCLSET:11

for T being non empty TopSpace
for x, y being Element of (OpenClosedSetLatt T) holds x "\/" y = x \/ y by Th5;

theorem Th12: :: LOPCLSET:12

for T being non empty TopSpace
for x, y being Element of (OpenClosedSetLatt T) holds x "/\" y = x /\ y by Th6;

theorem :: LOPCLSET:13

for T being non empty TopSpace holds the carrier of (OpenClosedSetLatt T) = OpenClosedSet T ;

theorem Th14: :: LOPCLSET:14

for T being non empty TopSpace holds OpenClosedSetLatt T is Boolean

proof end;

theorem Th15: :: LOPCLSET:15

for T being non empty TopSpace holds [#] T is Element of (OpenClosedSetLatt T) by Th8;

theorem Th16: :: LOPCLSET:16

for T being non empty TopSpace holds {} T is Element of (OpenClosedSetLatt T) by Th7;

registration

cluster non trivial LattStr ;
existence

proof end;

end;

definition

let BL be non trivial B_Lattice;
func ultraset BL -> Subset-Family of BL equals :: LOPCLSET:def 5
{ F where F is Filter of BL : F is_ultrafilter } ;
coherence

proof end;

end;

:: deftheorem defines ultraset LOPCLSET:def 5 :

registration

let BL be non trivial B_Lattice;
cluster ultraset BL -> non empty ;
coherence

proof end;

end;

theorem :: LOPCLSET:17

canceled;

theorem :: LOPCLSET:18

for x being set
for BL being non trivial B_Lattice holds
( x in ultraset BL iff ex UF being Filter of BL st
( UF = x & UF is_ultrafilter ) ) ;

theorem Th19: :: LOPCLSET:19

for BL being non trivial B_Lattice
for a being Element of BL holds { F where F is Filter of BL : ( F is_ultrafilter & a in F ) } c= ultraset BL

proof end;

definition

let BL be non trivial B_Lattice;
func UFilter BL -> Function means :Def6: :: LOPCLSET:def 6
( dom it = the carrier of BL & ( for a being Element of BL holds it . a = { UF where UF is Filter of BL : ( UF is_ultrafilter & a in UF ) } ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines UFilter LOPCLSET:def 6 :

theorem Th20: :: LOPCLSET:20

for x being set
for BL being non trivial B_Lattice
for a being Element of BL holds
( x in (UFilter BL) . a iff ex F being Filter of BL st
( F = x & F is_ultrafilter & a in F ) )

proof end;

theorem Th21: :: LOPCLSET:21

for BL being non trivial B_Lattice
for a being Element of BL
for F being Filter of BL holds
( F in (UFilter BL) . a iff ( F is_ultrafilter & a in F ) )

proof end;

theorem Th22: :: LOPCLSET:22

for BL being non trivial B_Lattice
for a, b being Element of BL
for F being Filter of BL st F is_ultrafilter holds
( a "\/" b in F iff ( a in F or b in F ) )

proof end;

theorem Th23: :: LOPCLSET:23

for BL being non trivial B_Lattice
for a, b being Element of BL holds (UFilter BL) . (a "/\" b) = ((UFilter BL) . a) /\ ((UFilter BL) . b)

proof end;

theorem Th24: :: LOPCLSET:24

for BL being non trivial B_Lattice
for a, b being Element of BL holds (UFilter BL) . (a "\/" b) = ((UFilter BL) . a) \/ ((UFilter BL) . b)

proof end;

definition

let BL be non trivial B_Lattice;
:: original: UFilter
redefine func UFilter BL -> Function of the carrier of BL, bool (ultraset BL);
coherence

proof end;

end;

definition

let BL be non trivial B_Lattice;
func StoneR BL -> set equals :: LOPCLSET:def 7
rng (UFilter BL);
coherence ;

end;

:: deftheorem defines StoneR LOPCLSET:def 7 :

registration

let BL be non trivial B_Lattice;
cluster StoneR BL -> non empty ;
coherence

proof end;

end;

theorem Th25: :: LOPCLSET:25

for BL being non trivial B_Lattice holds StoneR BL c= bool (ultraset BL) by RELSET_1:12;

theorem Th26: :: LOPCLSET:26

for x being set
for BL being non trivial B_Lattice holds
( x in StoneR BL iff ex a being Element of BL st (UFilter BL) . a = x )

proof end;

definition

let BL be non trivial B_Lattice;
func StoneSpace BL -> strict TopSpace means :Def8: :: LOPCLSET:def 8
( the carrier of it = ultraset BL & the topology of it = { (union A) where A is Subset-Family of (ultraset BL) : A c= StoneR BL } );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def8 defines StoneSpace LOPCLSET:def 8 :

registration

let BL be non trivial B_Lattice;
cluster StoneSpace BL -> non empty strict ;
coherence

proof end;

end;

theorem :: LOPCLSET:27

for BL being non trivial B_Lattice
for a being Element of BL
for F being Filter of BL st F is_ultrafilter & not F in (UFilter BL) . a holds
not a in F by Th21;

theorem Th28: :: LOPCLSET:28

for BL being non trivial B_Lattice
for a being Element of BL holds (ultraset BL) \ ((UFilter BL) . a) = (UFilter BL) . (a ` )

proof end;

definition

let BL be non trivial B_Lattice;
func StoneBLattice BL -> Lattice equals :: LOPCLSET:def 9
OpenClosedSetLatt (StoneSpace BL);
coherence ;

end;

:: deftheorem defines StoneBLattice LOPCLSET:def 9 :

Lm1: for BL being non trivial B_Lattice
for p being Subset of (StoneSpace BL) st p in StoneR BL holds
p is open

proof end;

theorem Th29: :: LOPCLSET:29

for BL being non trivial B_Lattice holds UFilter BL is one-to-one

proof end;

theorem :: LOPCLSET:30

for BL being non trivial B_Lattice holds union (StoneR BL) = ultraset BL

proof end;

theorem :: LOPCLSET:31

canceled;

theorem Th32: :: LOPCLSET:32

for X being non empty set holds
not for Y being Finite_Subset of X holds Y is empty

proof end;

registration

let D be non empty set ;
cluster non empty Element of Fin D;
existence by Th32;

end;

theorem :: LOPCLSET:33

canceled;

theorem Th34: :: LOPCLSET:34

for L being non trivial B_Lattice
for D being non empty Subset of L st Bottom L in <.D.) holds
ex B being non empty Finite_Subset of the carrier of L st
( B c= D & FinMeet B = Bottom L )

proof end;

theorem Th35: :: LOPCLSET:35

for L being 0_Lattice
for F being Filter of L holds
( not F is_ultrafilter or not Bottom L in F )

proof end;

theorem Th36: :: LOPCLSET:36

for BL being non trivial B_Lattice holds (UFilter BL) . (Bottom BL) = {}

proof end;

theorem :: LOPCLSET:37

for BL being non trivial B_Lattice holds (UFilter BL) . (Top BL) = ultraset BL

proof end;

theorem Th38: :: LOPCLSET:38

for X being set
for BL being non trivial B_Lattice st ultraset BL = union X & X is Subset of (StoneR BL) holds
ex Y being Finite_Subset of X st ultraset BL = union Y

proof end;

Lm2: for BL being non trivial B_Lattice holds StoneR BL c= OpenClosedSet (StoneSpace BL)

proof end;

theorem :: LOPCLSET:39

canceled;

theorem Th40: :: LOPCLSET:40

for BL being non trivial B_Lattice holds StoneR BL = OpenClosedSet (StoneSpace BL)

proof end;

definition

let BL be non trivial B_Lattice;
:: original: UFilter
redefine func UFilter BL -> Homomorphism of BL, StoneBLattice BL;
coherence

proof end;

end;

theorem Th41: :: LOPCLSET:41

for BL being non trivial B_Lattice holds rng (UFilter BL) = the carrier of (StoneBLattice BL)

proof end;

theorem Th42: :: LOPCLSET:42

for BL being non trivial B_Lattice holds UFilter BL is isomorphism

proof end;

theorem Th43: :: LOPCLSET:43

for BL being non trivial B_Lattice holds BL, StoneBLattice BL are_isomorphic

proof end;

theorem :: LOPCLSET:44

for BL being non trivial B_Lattice ex T being non empty TopSpace st BL, OpenClosedSetLatt T are_isomorphic

proof end;