:: WAYBEL29 semantic presentation
theorem Th1: :: WAYBEL29:1
theorem Th2: :: WAYBEL29:2
theorem Th3: :: WAYBEL29:3
theorem Th4: :: WAYBEL29:4
theorem Th5: :: WAYBEL29:5
theorem Th6: :: WAYBEL29:6
theorem Th7: :: WAYBEL29:7
theorem Th8: :: WAYBEL29:8
canceled;
theorem Th9: :: WAYBEL29:9
canceled;
theorem Th10: :: WAYBEL29:10
theorem Th11: :: WAYBEL29:11
theorem Th12: :: WAYBEL29:12
theorem Th13: :: WAYBEL29:13
for
b1,
b2 being
TopStruct st
TopStruct(# the
carrier of
b1,the
topology of
b1 #)
= TopStruct(# the
carrier of
b2,the
topology of
b2 #) holds
for
b3,
b4 being non
empty TopRelStr st
TopRelStr(# the
carrier of
b3,the
InternalRel of
b3,the
topology of
b3 #)
= TopRelStr(# the
carrier of
b4,the
InternalRel of
b4,the
topology of
b4 #) holds
ContMaps b1,
b3 = ContMaps b2,
b4
theorem Th14: :: WAYBEL29:14
theorem Th15: :: WAYBEL29:15
theorem Th16: :: WAYBEL29:16
canceled;
theorem Th17: :: WAYBEL29:17
theorem Th18: :: WAYBEL29:18
:: deftheorem Def1 defines Sigma WAYBEL29:def 1 :
theorem Th19: :: WAYBEL29:19
theorem Th20: :: WAYBEL29:20
:: deftheorem Def2 defines Sigma WAYBEL29:def 2 :
theorem Th21: :: WAYBEL29:21
theorem Th22: :: WAYBEL29:22
definition
let c1,
c2 be non
empty TopSpace;
func Theta c1,
c2 -> Function of
(InclPoset the topology of [:a1,a2:]),
(ContMaps a1,(Sigma (InclPoset the topology of a2))) means :
Def3:
:: WAYBEL29:def 3
for
b1 being
open Subset of
[:a1,a2:] holds
a3 . b1 = b1,the
carrier of
a1 *graph ;
existence
ex b1 being Function of (InclPoset the topology of [:c1,c2:]),(ContMaps c1,(Sigma (InclPoset the topology of c2))) st
for b2 being open Subset of [:c1,c2:] holds b1 . b2 = b2,the carrier of c1 *graph
correctness
uniqueness
for b1, b2 being Function of (InclPoset the topology of [:c1,c2:]),(ContMaps c1,(Sigma (InclPoset the topology of c2))) st ( for b3 being open Subset of [:c1,c2:] holds b1 . b3 = b3,the carrier of c1 *graph ) & ( for b3 being open Subset of [:c1,c2:] holds b2 . b3 = b3,the carrier of c1 *graph ) holds
b1 = b2;
end;
:: deftheorem Def3 defines Theta WAYBEL29:def 3 :
defpred S1[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being complete Scott continuous TopLattice
for b3 being Scott TopAugmentation of ContMaps a1,b2ex b4 being Function of (ContMaps b1,b3),(ContMaps [:b1,a1:],b2)ex b5 being Function of (ContMaps [:b1,a1:],b2),(ContMaps b1,b3) st
( b4 is uncurrying & b4 is one-to-one & b4 is onto & b5 is currying & b5 is one-to-one & b5 is onto );
defpred S2[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being complete Scott continuous TopLattice
for b3 being Scott TopAugmentation of ContMaps a1,b2ex b4 being Function of (ContMaps b1,b3),(ContMaps [:b1,a1:],b2)ex b5 being Function of (ContMaps [:b1,a1:],b2),(ContMaps b1,b3) st
( b4 is uncurrying & b4 is isomorphic & b5 is currying & b5 is isomorphic );
defpred S3[ T_0-TopSpace] means for b1 being non empty TopSpace holds Theta b1,a1 is isomorphic;
defpred S4[ T_0-TopSpace] means for b1 being non empty TopSpace
for b2 being Scott TopAugmentation of InclPoset the topology of a1
for b3 being continuous Function of b1,b2 holds *graph b3 is open Subset of [:b1,a1:];
defpred S5[ T_0-TopSpace] means for b1 being Scott TopAugmentation of InclPoset the topology of a1 holds { [b2,b3] where B is open Subset of a1, B is Element of a1 : b3 in b2 } is open Subset of [:b1,a1:];
defpred S6[ T_0-TopSpace] means for b1 being Scott TopAugmentation of InclPoset the topology of a1
for b2 being Element of a1
for b3 being open a_neighborhood of b2ex b4 being open Subset of b1 st
( b3 in b4 & meet b4 is a_neighborhood of b2 );
Lemma18:
for b1 being T_0-TopSpace holds
( S1[b1] iff S2[b1] )
definition
let c1 be non
empty TopSpace;
func alpha c1 -> Function of
(oContMaps a1,Sierpinski_Space ),
(InclPoset the topology of a1) means :
Def4:
:: WAYBEL29:def 4
for
b1 being
continuous Function of
a1,
Sierpinski_Space holds
a2 . b1 = b1 " {1};
existence
ex b1 being Function of (oContMaps c1,Sierpinski_Space ),(InclPoset the topology of c1) st
for b2 being continuous Function of c1,Sierpinski_Space holds b1 . b2 = b2 " {1}
uniqueness
for b1, b2 being Function of (oContMaps c1,Sierpinski_Space ),(InclPoset the topology of c1) st ( for b3 being continuous Function of c1,Sierpinski_Space holds b1 . b3 = b3 " {1} ) & ( for b3 being continuous Function of c1,Sierpinski_Space holds b2 . b3 = b3 " {1} ) holds
b1 = b2
end;
:: deftheorem Def4 defines alpha WAYBEL29:def 4 :
theorem Th23: :: WAYBEL29:23
theorem Th24: :: WAYBEL29:24
theorem Th25: :: WAYBEL29:25
theorem Th26: :: WAYBEL29:26
definition
let c1 be non
empty set ;
let c2,
c3 be non
empty TopSpace;
func commute c2,
c1,
c3 -> Function of
(oContMaps a2,(a1 -TOP_prod (a1 => a3))),
((oContMaps a2,a3) |^ a1) means :
Def5:
:: WAYBEL29:def 5
for
b1 being
continuous Function of
a2,
(a1 -TOP_prod (a1 => a3)) holds
a4 . b1 = commute b1;
uniqueness
for b1, b2 being Function of (oContMaps c2,(c1 -TOP_prod (c1 => c3))),((oContMaps c2,c3) |^ c1) st ( for b3 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b1 . b3 = commute b3 ) & ( for b3 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b2 . b3 = commute b3 ) holds
b1 = b2
existence
ex b1 being Function of (oContMaps c2,(c1 -TOP_prod (c1 => c3))),((oContMaps c2,c3) |^ c1) st
for b2 being continuous Function of c2,(c1 -TOP_prod (c1 => c3)) holds b1 . b2 = commute b2
end;
:: deftheorem Def5 defines commute WAYBEL29:def 5 :
Lemma21:
for b1 being T_0-TopSpace st S3[b1] holds
S4[b1]
theorem Th27: :: WAYBEL29:27
Lemma23:
for b1 being T_0-TopSpace st S4[b1] holds
S3[b1]
Lemma24:
for b1 being T_0-TopSpace st S4[b1] holds
S5[b1]
Lemma25:
for b1 being T_0-TopSpace st S5[b1] holds
S6[b1]
Lemma26:
for b1 being T_0-TopSpace st S6[b1] holds
S4[b1]
Lemma27:
for b1 being T_0-TopSpace st S6[b1] holds
InclPoset the topology of b1 is continuous
Lemma28:
for b1 being T_0-TopSpace st InclPoset the topology of b1 is continuous holds
S6[b1]
theorem Th28: :: WAYBEL29:28
for
b1 being
T_0-TopSpace holds
( ( for
b2 being non
empty TopSpacefor
b3 being
complete Scott continuous TopLatticefor
b4 being
Scott TopAugmentation of
ContMaps b1,
b3ex
b5 being
Function of
(ContMaps b2,b4),
(ContMaps [:b2,b1:],b3)ex
b6 being
Function of
(ContMaps [:b2,b1:],b3),
(ContMaps b2,b4) st
(
b5 is
uncurrying &
b5 is
one-to-one &
b5 is
onto &
b6 is
currying &
b6 is
one-to-one &
b6 is
onto ) ) iff for
b2 being non
empty TopSpacefor
b3 being
complete Scott continuous TopLatticefor
b4 being
Scott TopAugmentation of
ContMaps b1,
b3ex
b5 being
Function of
(ContMaps b2,b4),
(ContMaps [:b2,b1:],b3)ex
b6 being
Function of
(ContMaps [:b2,b1:],b3),
(ContMaps b2,b4) st
(
b5 is
uncurrying &
b5 is
isomorphic &
b6 is
currying &
b6 is
isomorphic ) )
by Lemma18;
theorem Th29: :: WAYBEL29:29
theorem Th30: :: WAYBEL29:30
theorem Th31: :: WAYBEL29:31
theorem Th32: :: WAYBEL29:32
defpred S7[ complete LATTICE] means InclPoset (sigma a1) is continuous;
defpred S8[ complete LATTICE] means for b1 being Scott TopAugmentation of a1
for b2 being complete LATTICE
for b3 being Scott TopAugmentation of b2 holds sigma [:b2,a1:] = the topology of [:b3,b1:];
defpred S9[ complete LATTICE] means for b1 being Scott TopAugmentation of a1
for b2 being complete LATTICE
for b3 being Scott TopAugmentation of b2
for b4 being Scott TopAugmentation of [:b2,a1:] holds TopStruct(# the carrier of b4,the topology of b4 #) = [:b3,b1:];
Lemma29:
for b1 being complete LATTICE holds
( S8[b1] iff S9[b1] )
theorem Th33: :: WAYBEL29:33
Lemma30:
for b1 being complete LATTICE st S7[b1] holds
S8[b1]
Lemma31:
for b1 being complete LATTICE st S8[b1] holds
S7[b1]
theorem Th34: :: WAYBEL29:34
theorem Th35: :: WAYBEL29:35
theorem Th36: :: WAYBEL29:36
theorem Th37: :: WAYBEL29:37