:: The Characterization of Continuity of Topologies
:: by Grzegorz Bancerek and Adam Naumowicz
::
:: Received January 6, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users

theorem Th1: :: WAYBEL29:1

for X, Y being non empty set
for Z being non empty RelStr
for S being non empty SubRelStr of Z |^ [:X,Y:]
for T being non empty SubRelStr of (Z |^ Y) |^ X
for f being Function of S,T st f is currying & f is V7() & f is onto holds
f " is uncurrying

proof end;

theorem Th2: :: WAYBEL29:2

for X, Y being non empty set
for Z being non empty RelStr
for S being non empty SubRelStr of Z |^ [:X,Y:]
for T being non empty SubRelStr of (Z |^ Y) |^ X
for f being Function of T,S st f is uncurrying & f is V7() & f is onto holds
f " is currying

proof end;

theorem :: WAYBEL29:3

for X, Y being non empty set
for Z being non empty Poset
for S being non empty full SubRelStr of Z |^ [:X,Y:]
for T being non empty full SubRelStr of (Z |^ Y) |^ X
for f being Function of S,T st f is currying & f is V7() & f is onto holds
f is isomorphic

proof end;

theorem :: WAYBEL29:4

for X, Y being non empty set
for Z being non empty Poset
for T being non empty full SubRelStr of Z |^ [:X,Y:]
for S being non empty full SubRelStr of (Z |^ Y) |^ X
for f being Function of S,T st f is uncurrying & f is V7() & f is onto holds
f is isomorphic

proof end;

theorem Th5: :: WAYBEL29:5

for S1, S2, T1, T2 being RelStr st RelStr(# the carrier of S1, the InternalRel of S1 #) = RelStr(# the carrier of S2, the InternalRel of S2 #) & RelStr(# the carrier of T1, the InternalRel of T1 #) = RelStr(# the carrier of T2, the InternalRel of T2 #) holds
for f being Function of S1,T1 st f is isomorphic holds
for g being Function of S2,T2 st g = f holds
g is isomorphic

proof end;

:: Przywlaszczone

theorem Th6: :: WAYBEL29:6

for R, S, T being RelStr
for f being Function of R,S st f is isomorphic holds
for g being Function of S,T st g is isomorphic holds
for h being Function of R,T st h = g * f holds
h is isomorphic

proof end;

theorem Th7: :: WAYBEL29:7

for X, Y, X1, Y1 being TopSpace st TopStruct(# the carrier of X, the topology of X #) = TopStruct(# the carrier of X1, the topology of X1 #) & TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of Y1, the topology of Y1 #) holds
[:X,Y:] = [:X1,Y1:]

proof end;

theorem Th8: :: WAYBEL29:8

for X being non empty TopSpace
for L being non empty up-complete Scott TopPoset
for F being non empty directed Subset of (ContMaps (X,L)) holds "\/" (F,(L |^ the carrier of X)) is continuous Function of X,L

proof end;

theorem Th9: :: WAYBEL29:9

for X being non empty TopSpace
for L being non empty up-complete Scott TopPoset holds ContMaps (X,L) is directed-sups-inheriting SubRelStr of L |^ the carrier of X

proof end;

theorem Th10: :: WAYBEL29:10

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)

proof end;

registration

cluster TopSpace-like reflexive transitive antisymmetric Scott continuous with_suprema with_infima complete -> T_0 continuous injective complete for TopRelStr ;

coherence
for b₁ being continuous complete TopLattice st b₁ is Scott holds
( b₁ is injective & b₁ is T_0 )

proof end;

end;

registration

cluster non empty TopSpace-like reflexive transitive antisymmetric Scott continuous non void with_suprema with_infima complete for TopRelStr ;

existence
ex b₁ being TopLattice st
( b₁ is Scott & b₁ is continuous & b₁ is complete )

proof end;

end;

registration

let X be non empty TopSpace;
let L be non empty up-complete Scott TopPoset;

cluster ContMaps (X,L) -> up-complete ;

coherence
ContMaps (X,L) is up-complete

proof end;

end;

theorem Th11: :: WAYBEL29:11

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I st ( for i being Element of I holds J . i is up-complete ) holds
I -POS_prod J is up-complete

proof end;

theorem :: WAYBEL29:12

for I being non empty set
for J being non-Empty Poset-yielding ManySortedSet of I st ( for i being Element of I holds
( J . i is up-complete & J . i is lower-bounded ) ) holds
for x, y being Element of (product J) holds
( x << y iff ( ( for i being Element of I holds x . i << y . i ) & ex K being finite Subset of I st
for i being Element of I st not i in K holds
x . i = Bottom (J . i) ) )

proof end;

registration

let X be set ;
let L be non empty reflexive antisymmetric lower-bounded RelStr ;

cluster L |^ X -> lower-bounded ;

coherence
L |^ X is lower-bounded

proof end;

end;

registration

let X be non empty TopSpace;
let L be non empty lower-bounded TopPoset;

cluster ContMaps (X,L) -> lower-bounded ;

coherence
ContMaps (X,L) is lower-bounded

proof end;

end;

registration

let L be non empty up-complete Poset;

cluster -> up-complete for TopAugmentation of L;

coherence
for b₁ being TopAugmentation of L holds b₁ is up-complete

proof end;

cluster Scott -> correct for TopAugmentation of L;

coherence
for b₁ being TopAugmentation of L st b₁ is Scott holds
b₁ is correct

proof end;

end;

registration

let L be non empty up-complete Poset;

cluster non empty reflexive transitive antisymmetric up-complete strict Scott non void for TopAugmentation of L;

existence
ex b₁ being TopAugmentation of L st
( b₁ is strict & b₁ is Scott )

proof end;

end;

theorem Th13: :: WAYBEL29:13

for L being non empty up-complete Poset
for S1, S2 being Scott TopAugmentation of L holds the topology of S1 = the topology of S2

proof end;

theorem Th14: :: WAYBEL29:14

for S1, S2 being non empty reflexive antisymmetric up-complete TopRelStr st TopRelStr(# the carrier of S1, the InternalRel of S1, the topology of S1 #) = TopRelStr(# the carrier of S2, the InternalRel of S2, the topology of S2 #) & S1 is Scott holds
S2 is Scott

proof end;

definition

let L be non empty up-complete Poset;

func Sigma L -> strict Scott TopAugmentation of L means :Def1: :: WAYBEL29:def 1
verum;

uniqueness
for b₁, b₂ being strict Scott TopAugmentation of L holds b₁ = b₂

proof end;

existence
ex b₁ being strict Scott TopAugmentation of L st verum ;

end;

:: deftheorem Def1 defines Sigma WAYBEL29:def 1 :
for L being non empty up-complete Poset
for b₂ being strict Scott TopAugmentation of L holds
( b₂ = Sigma L iff verum );

theorem Th15: :: WAYBEL29:15

for S being non empty up-complete Scott TopPoset holds Sigma S = TopRelStr(# the carrier of S, the InternalRel of S, the topology of S #)

proof end;

theorem Th16: :: WAYBEL29:16

for L1, L2 being non empty up-complete Poset st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) holds
Sigma L1 = Sigma L2

proof end;

definition

let S, T be non empty up-complete Poset;
let f be Function of S,T;

func Sigma f -> Function of (Sigma S),(Sigma T) equals :: WAYBEL29:def 2
f;

coherence
f is Function of (Sigma S),(Sigma T)

proof end;

end;

:: deftheorem defines Sigma WAYBEL29:def 2 :
for S, T being non empty up-complete Poset
for f being Function of S,T holds Sigma f = f;

registration

let S, T be non empty up-complete Poset;
let f be directed-sups-preserving Function of S,T;

cluster Sigma f -> continuous ;

coherence
Sigma f is continuous

proof end;

end;

theorem :: WAYBEL29:17

for S, T being non empty up-complete Poset
for f being Function of S,T holds
( f is isomorphic iff Sigma f is isomorphic )

proof end;

theorem Th18: :: WAYBEL29:18

for X being non empty TopSpace
for S being Scott complete TopLattice holds oContMaps (X,S) = ContMaps (X,S)

proof end;

definition

let X, Y be non empty TopSpace;

func Theta (X,Y) -> Function of (InclPoset the topology of [:X,Y:]),(ContMaps (X,(Sigma (InclPoset the topology of Y)))) 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

proof end;

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;

proof end;

end;

:: deftheorem Def3 defines Theta WAYBEL29:def 3 :
for X, Y being non empty TopSpace
for b₃ being Function of (InclPoset the topology of [:X,Y:]),(ContMaps (X,(Sigma (InclPoset the topology of Y)))) holds
( b₃ = Theta (X,Y) iff for W being open Subset of [:X,Y:] holds b₃ . W = (W, the carrier of X) *graph );

defpred S₁[ T_0-TopSpace] means for X being non empty TopSpace
for L being Scott continuous complete TopLattice
for T being Scott TopAugmentation of ContMaps ($1,L) ex f being Function of (ContMaps (X,T)),(ContMaps ([:X,$1:],L)) ex g being Function of (ContMaps ([:X,$1:],L)),(ContMaps (X,T)) st
( f is uncurrying & f is V7() & f is onto & g is currying & g is V7() & g is onto );

defpred S₂[ T_0-TopSpace] means for X being non empty TopSpace
for L being Scott continuous complete TopLattice
for T being Scott TopAugmentation of ContMaps ($1,L) ex f being Function of (ContMaps (X,T)),(ContMaps ([:X,$1:],L)) ex g being Function of (ContMaps ([:X,$1:],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,$1) is isomorphic ;

defpred S₄[ T_0-TopSpace] means for X being non empty TopSpace
for T being Scott TopAugmentation of InclPoset the topology of $1
for f being continuous Function of X,T holds *graph f is open Subset of [:X,$1:];

defpred S₅[ T_0-TopSpace] means for T being Scott TopAugmentation of InclPoset the topology of $1 holds { [W,y] where W is open Subset of $1, y is Element of $1 : y in W } is open Subset of [:T,$1:];

defpred S₆[ T_0-TopSpace] means for S being Scott TopAugmentation of InclPoset the topology of $1
for y being Element of $1
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 );

Lm1: for T being T_0-TopSpace holds
( S₁[T] iff S₂[T] )

proof end;

definition

let X be non empty TopSpace;

func alpha X -> Function of (oContMaps (X,Sierpinski_Space)),(InclPoset the topology of X) 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}

proof end;

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

proof end;

end;

:: deftheorem Def4 defines alpha WAYBEL29:def 4 :
for X being non empty TopSpace
for b₂ being Function of (oContMaps (X,Sierpinski_Space)),(InclPoset the topology of X) holds
( b₂ = alpha X iff for g being continuous Function of X,Sierpinski_Space holds b₂ . g = g " {1} );

theorem :: WAYBEL29:19

for X being non empty TopSpace
for V being open Subset of X holds ((alpha X) ") . V = chi (V, the carrier of X)

proof end;

registration

let X be non empty TopSpace;

cluster alpha X -> isomorphic ;

coherence
alpha X is isomorphic

proof end;

end;

registration

let X be non empty TopSpace;

cluster (alpha X) " -> isomorphic ;

coherence
(alpha X) " is isomorphic by YELLOW14:10;

end;

registration

let S be injective T_0-TopSpace;

cluster Omega S -> Scott ;

coherence
Omega S is Scott

proof end;

end;

registration

let X be non empty TopSpace;

cluster oContMaps (X,Sierpinski_Space) -> complete ;

coherence
oContMaps (X,Sierpinski_Space) is complete

proof end;

end;

theorem :: WAYBEL29:20

Omega Sierpinski_Space = Sigma (BoolePoset {0})

proof end;

registration

let M be non empty set ;
let S be injective T_0-TopSpace;

cluster product (M => S) -> injective ;

coherence
M -TOP_prod (M => S) is injective

proof end;

end;

theorem :: WAYBEL29:21

for M being non empty set
for L being continuous complete LATTICE holds Omega (M -TOP_prod (M => (Sigma L))) = Sigma (M -POS_prod (M => L))

proof end;

theorem :: WAYBEL29:22

for M being non empty set
for T being injective T_0-TopSpace holds Omega (M -TOP_prod (M => T)) = Sigma (M -POS_prod (M => (Omega T)))

proof end;

definition

let M be non empty set ;
let X, Y be non empty TopSpace;

func commute (X,M,Y) -> Function of (oContMaps (X,(M -TOP_prod (M => Y)))),((oContMaps (X,Y)) |^ M) 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

proof end;

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

proof end;

end;

:: deftheorem Def5 defines commute WAYBEL29:def 5 :
for M being non empty set
for X, Y being non empty TopSpace
for b₄ being Function of (oContMaps (X,(M -TOP_prod (M => Y)))),((oContMaps (X,Y)) |^ M) holds
( b₄ = commute (X,M,Y) iff for f being continuous Function of X,(M -TOP_prod (M => Y)) holds b₄ . f = commute f );

registration

let M be non empty set ;
let X, Y be non empty TopSpace;

cluster commute (X,M,Y) -> V7() onto ;

correctness
coherence
( commute (X,M,Y) is one-to-one & commute (X,M,Y) is onto );

proof end;

end;

registration

let M be non empty set ;
let X be non empty TopSpace;

cluster commute (X,M,Sierpinski_Space) -> isomorphic ;

correctness
coherence
commute (X,M,Sierpinski_Space) is isomorphic ;

proof end;

end;

Lm2: for T being T_0-TopSpace st S₃[T] holds
S₄[T]

proof end;

theorem Th23: :: WAYBEL29:23

for X, Y being non empty TopSpace
for S being Scott TopAugmentation of InclPoset the topology of Y
for f1, f2 being Element of (ContMaps (X,S)) st f1 <= f2 holds
*graph f1 c= *graph f2

proof end;

Lm3: for T being T_0-TopSpace st S₄[T] holds
S₃[T]

proof end;

Lm4: for T being T_0-TopSpace st S₄[T] holds
S₅[T]

proof end;

Lm5: for T being T_0-TopSpace st S₅[T] holds
S₆[T]

proof end;

Lm6: for T being T_0-TopSpace st S₆[T] holds
S₄[T]

proof end;

Lm7: for T being T_0-TopSpace st S₆[T] holds
InclPoset the topology of T is continuous

proof end;

Lm8: for T being T_0-TopSpace st InclPoset the topology of T is continuous holds
S₆[T]

proof end;

:: 4.10. THEOREM, (1) <=> (1'), pp. 131-133

theorem :: WAYBEL29:24

for Y being T_0-TopSpace holds
( ( for X being non empty TopSpace
for L being Scott continuous complete 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 V7() & f is onto & g is currying & g is V7() & g is onto ) ) iff for X being non empty TopSpace
for L being Scott continuous complete 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 Lm1;

:: 4.10. THEOREM, (6) <=> (2), pp. 131-133

theorem :: WAYBEL29:25

for Y being T_0-TopSpace holds
( InclPoset the topology of Y is continuous iff for X being non empty TopSpace holds Theta (X,Y) is isomorphic )

proof end;

:: 4.10. THEOREM, (6) <=> (3), pp. 131-133

theorem :: WAYBEL29:26

for Y being T_0-TopSpace holds
( InclPoset the topology of Y is continuous iff for X being non empty TopSpace
for f being continuous Function of X,(Sigma (InclPoset the topology of Y)) holds *graph f is open Subset of [:X,Y:] )

proof end;

:: 4.10. THEOREM, (6) <=> (4), pp. 131-133

theorem :: WAYBEL29:27

for Y being T_0-TopSpace holds
( InclPoset the topology of Y is continuous iff { [W,y] where W is open Subset of Y, y is Element of Y : y in W } is open Subset of [:(Sigma (InclPoset the topology of Y)),Y:] )

proof end;

:: 4.10. THEOREM, (6) <=> (5), pp. 131-133

theorem :: WAYBEL29:28

for Y being T_0-TopSpace holds
( InclPoset the topology of Y is continuous iff for y being Element of Y
for V being open a_neighborhood of y ex H being open Subset of (Sigma (InclPoset the topology of Y)) st
( V in H & meet H is a_neighborhood of y ) )

proof end;

defpred S₇[ complete LATTICE] means InclPoset (sigma $1) is continuous ;

defpred S₈[ complete LATTICE] means for SL being Scott TopAugmentation of $1
for S being complete LATTICE
for SS being Scott TopAugmentation of S holds sigma [:S,$1:] = the topology of [:SS,SL:];

defpred S₉[ complete LATTICE] means for SL being Scott TopAugmentation of $1
for S being complete LATTICE
for SS being Scott TopAugmentation of S
for SSL being Scott TopAugmentation of [:S,$1:] holds TopStruct(# the carrier of SSL, the topology of SSL #) = [:SS,SL:];

Lm9: for L being complete LATTICE holds
( S₈[L] iff S₉[L] )

proof end;

theorem :: WAYBEL29:29

for R1, R2, R3 being non empty RelStr
for f1 being Function of R1,R3 st f1 is isomorphic holds
for f2 being Function of R2,R3 st f2 = f1 & f2 is isomorphic holds
RelStr(# the carrier of R1, the InternalRel of R1 #) = RelStr(# the carrier of R2, the InternalRel of R2 #)

proof end;

Lm10: for L being complete LATTICE st S₇[L] holds
S₈[L]

proof end;

Lm11: for L being complete LATTICE st S₈[L] holds
S₇[L]

proof end;

:: 4.11. THEOREM, (1) <=> (2), p. 133.

theorem Th30: :: WAYBEL29:30

for L being complete LATTICE holds
( InclPoset (sigma L) is continuous iff for S being complete LATTICE holds sigma [:S,L:] = the topology of [:(Sigma S),(Sigma L):] )

proof end;

:: 4.11. THEOREM, (2) <=> (3), p. 133.

theorem Th31: :: WAYBEL29:31

for L being complete LATTICE holds
( ( for S being complete LATTICE holds sigma [:S,L:] = the topology of [:(Sigma S),(Sigma L):] ) iff for S being complete LATTICE holds TopStruct(# the carrier of (Sigma [:S,L:]), the topology of (Sigma [:S,L:]) #) = [:(Sigma S),(Sigma L):] )

proof end;

:: 4.11. THEOREM, (2) <=> (3+), p. 133.

theorem Th32: :: WAYBEL29:32

for L being complete LATTICE holds
( ( for S being complete LATTICE holds sigma [:S,L:] = the topology of [:(Sigma S),(Sigma L):] ) iff for S being complete LATTICE holds Sigma [:S,L:] = Omega [:(Sigma S),(Sigma L):] )

proof end;

:: 4.11. THEOREM, (1) <=> (3+), p. 133.

theorem :: WAYBEL29:33

for L being complete LATTICE holds
( InclPoset (sigma L) is continuous iff for S being complete LATTICE holds Sigma [:S,L:] = Omega [:(Sigma S),(Sigma L):] )

proof end;