:: 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 b
1, b
2 being
TopStruct holds
(
TopStruct(# the
carrier of b
1,the
topology of b
1 #)
= TopStruct(# the
carrier of b
2,the
topology of b
2 #) implies for b
3, b
4 being non
empty TopRelStr holds
(
TopRelStr(# the
carrier of b
3,the
InternalRel of b
3,the
topology of b
3 #)
= TopRelStr(# the
carrier of b
4,the
InternalRel of b
4,the
topology of b
4 #) implies
ContMaps b
1,b
3 = ContMaps b
2,b
4 ) )
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 c
1, c
2 be non
empty TopSpace;
func Theta c
1,c
2 -> Function of
(InclPoset the topology of [:a1,a2:]),
(ContMaps a1,(Sigma (InclPoset the topology of a2))) means :
Def3:
:: WAYBEL29:def 3
for b
1 being
open Subset of
[:a1,a2:] holds a
3 . b
1 = b
1,the
carrier of a
1 *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))) holds
( ( 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 ) implies 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,b2 holds
ex 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,b2 holds
ex 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 b2 holds
ex 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 c
1 be non
empty TopSpace;
func alpha c
1 -> Function of
(oContMaps a1,Sierpinski_Space ),
(InclPoset the topology of a1) means :
Def4:
:: WAYBEL29:def 4
for b
1 being
continuous Function of a
1,
Sierpinski_Space holds a
2 . b
1 = b
1 " {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) holds
( ( 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} ) implies 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 c
1 be non
empty set ;
let c
2, c
3 be non
empty TopSpace;
func commute c
2,c
1,c
3 -> Function of
(oContMaps a2,(a1 -TOP_prod (a1 => a3))),
((oContMaps a2,a3) |^ a1) means :
Def5:
:: WAYBEL29:def 5
for b
1 being
continuous Function of a
2,
(a1 -TOP_prod (a1 => a3)) holds a
4 . b
1 = commute b
1;
uniqueness
for b1, b2 being Function of (oContMaps c2,(c1 -TOP_prod (c1 => c3))),((oContMaps c2,c3) |^ c1) holds
( ( 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 ) implies 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 holds
( S3[b1] implies S4[b1] )
theorem Th27: :: WAYBEL29:27
Lemma23:
for b1 being T_0-TopSpace holds
( S4[b1] implies S3[b1] )
Lemma24:
for b1 being T_0-TopSpace holds
( S4[b1] implies S5[b1] )
Lemma25:
for b1 being T_0-TopSpace holds
( S5[b1] implies S6[b1] )
Lemma26:
for b1 being T_0-TopSpace holds
( S6[b1] implies S4[b1] )
Lemma27:
for b1 being T_0-TopSpace holds
( S6[b1] implies InclPoset the topology of b1 is continuous )
Lemma28:
for b1 being T_0-TopSpace holds
( InclPoset the topology of b1 is continuous implies S6[b1] )
theorem Th28: :: WAYBEL29:28
for b
1 being
T_0-TopSpace holds
( ( for b
2 being non
empty TopSpacefor b
3 being
complete Scott continuous TopLatticefor b
4 being
Scott TopAugmentation of
ContMaps b
1,b
3 holds
ex b
5 being
Function of
(ContMaps b2,b4),
(ContMaps [:b2,b1:],b3)ex b
6 being
Function of
(ContMaps [:b2,b1:],b3),
(ContMaps b2,b4) st
( b
5 is
uncurrying & b
5 is
one-to-one & b
5 is
onto & b
6 is
currying & b
6 is
one-to-one & b
6 is
onto ) ) iff for b
2 being non
empty TopSpacefor b
3 being
complete Scott continuous TopLatticefor b
4 being
Scott TopAugmentation of
ContMaps b
1,b
3 holds
ex b
5 being
Function of
(ContMaps b2,b4),
(ContMaps [:b2,b1:],b3)ex b
6 being
Function of
(ContMaps [:b2,b1:],b3),
(ContMaps b2,b4) st
( b
5 is
uncurrying & b
5 is
isomorphic & b
6 is
currying & b
6 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 holds
( S7[b1] implies S8[b1] )
Lemma31:
for b1 being complete LATTICE holds
( S8[b1] implies S7[b1] )
theorem Th34: :: WAYBEL29:34
theorem Th35: :: WAYBEL29:35
theorem Th36: :: WAYBEL29:36
theorem Th37: :: WAYBEL29:37