:: 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
S1,
S2 being
TopStruct st
TopStruct(# the
carrier of
S1,the
topology of
S1 #)
= TopStruct(# the
carrier of
S2,the
topology of
S2 #) holds
for
T1,
T2 being non
empty TopRelStr st
TopRelStr(# the
carrier of
T1,the
InternalRel of
T1,the
topology of
T1 #)
= TopRelStr(# the
carrier of
T2,the
InternalRel of
T2,the
topology of
T2 #) holds
ContMaps S1,
T1 = ContMaps S2,
T2
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 X be non
empty TopSpace,
Y 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
W being
open Subset of
[:X,Y:] holds
it . W = W,the
carrier of
X *graph ;
existence
ex b1 being Function of (InclPoset the topology of [:X,Y:]),(ContMaps X,(Sigma (InclPoset the topology of Y))) st
for W being open Subset of [:X,Y:] holds b1 . W = W,the carrier of X *graph
correctness
uniqueness
for b1, b2 being Function of (InclPoset the topology of [:X,Y:]),(ContMaps X,(Sigma (InclPoset the topology of Y))) st ( for W being open Subset of [:X,Y:] holds b1 . W = W,the carrier of X *graph ) & ( for W being open Subset of [:X,Y:] holds b2 . W = W,the carrier of X *graph ) holds
b1 = b2;
end;
:: deftheorem Def3 defines Theta WAYBEL29:def 3 :
defpred S1[ T_0-TopSpace] means for X being non empty TopSpace
for L being complete Scott continuous TopLattice
for T being Scott TopAugmentation of ContMaps a1,L ex f being Function of (ContMaps X,T),(ContMaps [:X,a1:],L) ex g being Function of (ContMaps [:X,a1:],L),(ContMaps X,T) st
( f is uncurrying & f is one-to-one & f is onto & g is currying & g is one-to-one & g is onto );
defpred S2[ T_0-TopSpace] means for X being non empty TopSpace
for L being complete Scott continuous TopLattice
for T being Scott TopAugmentation of ContMaps a1,L ex f being Function of (ContMaps X,T),(ContMaps [:X,a1:],L) ex g being Function of (ContMaps [:X,a1:],L),(ContMaps X,T) st
( f is uncurrying & f is isomorphic & g is currying & g is isomorphic );
defpred S3[ T_0-TopSpace] means for X being non empty TopSpace holds Theta X,a1 is isomorphic;
defpred S4[ T_0-TopSpace] means for X being non empty TopSpace
for T being Scott TopAugmentation of InclPoset the topology of a1
for f being continuous Function of X,T holds *graph f is open Subset of [:X,a1:];
defpred S5[ T_0-TopSpace] means for T being Scott TopAugmentation of InclPoset the topology of a1 holds { [W,y] where W is open Subset of a1, y is Element of a1 : y in W } is open Subset of [:T,a1:];
defpred S6[ T_0-TopSpace] means for S being Scott TopAugmentation of InclPoset the topology of a1
for y being Element of a1
for V being open a_neighborhood of y ex H being open Subset of S st
( V in H & meet H is a_neighborhood of y );
Lemma183:
for T being T_0-TopSpace holds
( S1[T] iff S2[T] )
definition
let X be non
empty TopSpace;
func alpha c1 -> Function of
(oContMaps a1,Sierpinski_Space ),
(InclPoset the topology of a1) means :
Def4:
:: WAYBEL29:def 4
for
g being
continuous Function of
X,
Sierpinski_Space holds
it . g = g " {1};
existence
ex b1 being Function of (oContMaps X,Sierpinski_Space ),(InclPoset the topology of X) st
for g being continuous Function of X,Sierpinski_Space holds b1 . g = g " {1}
uniqueness
for b1, b2 being Function of (oContMaps X,Sierpinski_Space ),(InclPoset the topology of X) st ( for g being continuous Function of X,Sierpinski_Space holds b1 . g = g " {1} ) & ( for g being continuous Function of X,Sierpinski_Space holds b2 . g = g " {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 M be non
empty set ;
let X be non
empty TopSpace,
Y 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
f being
continuous Function of
X,
(M -TOP_prod (M => Y)) holds
it . f = commute f;
uniqueness
for b1, b2 being Function of (oContMaps X,(M -TOP_prod (M => Y))),((oContMaps X,Y) |^ M) st ( for f being continuous Function of X,(M -TOP_prod (M => Y)) holds b1 . f = commute f ) & ( for f being continuous Function of X,(M -TOP_prod (M => Y)) holds b2 . f = commute f ) holds
b1 = b2
existence
ex b1 being Function of (oContMaps X,(M -TOP_prod (M => Y))),((oContMaps X,Y) |^ M) st
for f being continuous Function of X,(M -TOP_prod (M => Y)) holds b1 . f = commute f
end;
:: deftheorem Def5 defines commute WAYBEL29:def 5 :
Lemma192:
for T being T_0-TopSpace st S3[T] holds
S4[T]
theorem Th27: :: WAYBEL29:27
Lemma194:
for T being T_0-TopSpace st S4[T] holds
S3[T]
Lemma197:
for T being T_0-TopSpace st S4[T] holds
S5[T]
Lemma198:
for T being T_0-TopSpace st S5[T] holds
S6[T]
Lemma204:
for T being T_0-TopSpace st S6[T] holds
S4[T]
Lemma210:
for T being T_0-TopSpace st S6[T] holds
InclPoset the topology of T is continuous
Lemma214:
for T being T_0-TopSpace st InclPoset the topology of T is continuous holds
S6[T]
theorem Th28: :: WAYBEL29:28
for
Y being
T_0-TopSpace holds
( ( for
X being non
empty TopSpace for
L being
complete Scott continuous TopLattice for
T being
Scott TopAugmentation of
ContMaps Y,
L ex
f being
Function of
(ContMaps X,T),
(ContMaps [:X,Y:],L) ex
g being
Function of
(ContMaps [:X,Y:],L),
(ContMaps X,T) st
(
f is
uncurrying &
f is
one-to-one &
f is
onto &
g is
currying &
g is
one-to-one &
g is
onto ) ) iff for
X being non
empty TopSpace for
L being
complete Scott continuous TopLattice for
T being
Scott TopAugmentation of
ContMaps Y,
L ex
f being
Function of
(ContMaps X,T),
(ContMaps [:X,Y:],L) ex
g being
Function of
(ContMaps [:X,Y:],L),
(ContMaps X,T) st
(
f is
uncurrying &
f is
isomorphic &
g is
currying &
g is
isomorphic ) )
by ;
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 SL being Scott TopAugmentation of a1
for S being complete LATTICE
for SS being Scott TopAugmentation of S holds sigma [:S,a1:] = the topology of [:SS,SL:];
defpred S9[ complete LATTICE] means for SL being Scott TopAugmentation of a1
for S being complete LATTICE
for SS being Scott TopAugmentation of S
for SSL being Scott TopAugmentation of [:S,a1:] holds TopStruct(# the carrier of SSL,the topology of SSL #) = [:SS,SL:];
Lemma222:
for L being complete LATTICE holds
( S8[L] iff S9[L] )
theorem Th33: :: WAYBEL29:33
Lemma228:
for L being complete LATTICE st S7[L] holds
S8[L]
Lemma247:
for L being complete LATTICE st S8[L] holds
S7[L]
theorem Th34: :: WAYBEL29:34
theorem Th35: :: WAYBEL29:35
theorem Th36: :: WAYBEL29:36
theorem Th37: :: WAYBEL29:37