:: Continuous Lattices between T$_0$ Spaces
:: by Grzegorz Bancerek
::
:: Received September 24, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

notation

let I be set ;
let J be RelStr-yielding ManySortedSet of I;
synonym I -POS_prod J for product J;

end;

notation

let I be set ;
let J be non-Empty TopStruct-yielding ManySortedSet of I;
synonym I -TOP_prod J for product J;

end;

:: 4.1. DEFINITION (a)

definition

let X, Y be non empty TopSpace;

func oContMaps (X,Y) -> non empty strict RelStr equals :: WAYBEL26:def 1
ContMaps (X,(Omega Y));

coherence
ContMaps (X,(Omega Y)) is non empty strict RelStr ;

end;

:: deftheorem defines oContMaps WAYBEL26:def 1 :
for X, Y being non empty TopSpace holds oContMaps (X,Y) = ContMaps (X,(Omega Y));

registration

let X, Y be non empty TopSpace;

cluster oContMaps (X,Y) -> non empty constituted-Functions strict reflexive transitive ;

coherence
( oContMaps (X,Y) is reflexive & oContMaps (X,Y) is transitive & oContMaps (X,Y) is constituted-Functions ) ;

end;

registration

let X be non empty TopSpace;
let Y be non empty T_0 TopSpace;

cluster oContMaps (X,Y) -> non empty strict antisymmetric ;

coherence
oContMaps (X,Y) is antisymmetric ;

end;

theorem Th1: :: WAYBEL26:1

for X, Y being non empty TopSpace
for a being set holds
( a is Element of (oContMaps (X,Y)) iff a is continuous Function of X,(Omega Y) )

proof end;

theorem Th2: :: WAYBEL26:2

for X, Y being non empty TopSpace
for a being set holds
( a is Element of (oContMaps (X,Y)) iff a is continuous Function of X,Y )

proof end;

theorem Th3: :: WAYBEL26:3

for X, Y being non empty TopSpace
for a, b being Element of (oContMaps (X,Y))
for f, g being Function of X,(Omega Y) st a = f & b = g holds
( a <= b iff f <= g )

proof end;

definition

let X, Y be non empty TopSpace;
let x be Point of X;
let A be Subset of (oContMaps (X,Y));
:: original: pi
redefine func pi (A,x) -> Subset of (Omega Y);
coherence
pi (A,x) is Subset of (Omega Y)

proof end;

end;

registration

let X, Y be non empty TopSpace;
let x be set ;
let A be non empty Subset of (oContMaps (X,Y));

cluster pi (A,x) -> non empty ;

coherence
not pi (A,x) is empty

proof end;

end;

theorem Th4: :: WAYBEL26:4

Omega Sierpinski_Space is TopAugmentation of BoolePoset {0}

proof end;

theorem Th5: :: WAYBEL26:5

for X being non empty TopSpace ex f being Function of (InclPoset the topology of X),(oContMaps (X,Sierpinski_Space)) st
( f is isomorphic & ( for V being open Subset of X holds f . V = chi (V, the carrier of X) ) )

proof end;

theorem Th6: :: WAYBEL26:6

for X being non empty TopSpace holds InclPoset the topology of X, oContMaps (X,Sierpinski_Space) are_isomorphic

proof end;

:: 4.1. DEFINITION (b)

definition

let X, Y, Z be non empty TopSpace;
let f be continuous Function of Y,Z;

func oContMaps (X,f) -> Function of (oContMaps (X,Y)),(oContMaps (X,Z)) means :Def2: :: WAYBEL26:def 2
for g being continuous Function of X,Y holds it . g = f * g;

uniqueness
for b₁, b₂ being Function of (oContMaps (X,Y)),(oContMaps (X,Z)) st ( for g being continuous Function of X,Y holds b₁ . g = f * g ) & ( for g being continuous Function of X,Y holds b₂ . g = f * g ) holds
b₁ = b₂

proof end;

existence
ex b₁ being Function of (oContMaps (X,Y)),(oContMaps (X,Z)) st
for g being continuous Function of X,Y holds b₁ . g = f * g

proof end;

func oContMaps (f,X) -> Function of (oContMaps (Z,X)),(oContMaps (Y,X)) means :Def3: :: WAYBEL26:def 3
for g being continuous Function of Z,X holds it . g = g * f;

uniqueness
for b₁, b₂ being Function of (oContMaps (Z,X)),(oContMaps (Y,X)) st ( for g being continuous Function of Z,X holds b₁ . g = g * f ) & ( for g being continuous Function of Z,X holds b₂ . g = g * f ) holds
b₁ = b₂

proof end;

existence
ex b₁ being Function of (oContMaps (Z,X)),(oContMaps (Y,X)) st
for g being continuous Function of Z,X holds b₁ . g = g * f

proof end;

end;

:: deftheorem Def2 defines oContMaps WAYBEL26:def 2 :
for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z
for b₅ being Function of (oContMaps (X,Y)),(oContMaps (X,Z)) holds
( b₅ = oContMaps (X,f) iff for g being continuous Function of X,Y holds b₅ . g = f * g );

:: deftheorem Def3 defines oContMaps WAYBEL26:def 3 :
for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z
for b₅ being Function of (oContMaps (Z,X)),(oContMaps (Y,X)) holds
( b₅ = oContMaps (f,X) iff for g being continuous Function of Z,X holds b₅ . g = g * f );

theorem Th7: :: WAYBEL26:7

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace holds oContMaps (X,Y) is up-complete

proof end;

theorem Th8: :: WAYBEL26:8

for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z holds oContMaps (X,f) is monotone

proof end;

theorem :: WAYBEL26:9

for X, Y being non empty TopSpace
for f being continuous Function of Y,Y st f is idempotent holds
oContMaps (X,f) is idempotent

proof end;

theorem Th10: :: WAYBEL26:10

for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z holds oContMaps (f,X) is monotone

proof end;

theorem Th11: :: WAYBEL26:11

for X, Y being non empty TopSpace
for f being continuous Function of Y,Y st f is idempotent holds
oContMaps (f,X) is idempotent

proof end;

theorem Th12: :: WAYBEL26:12

for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z
for x being Element of X
for A being Subset of (oContMaps (X,Y)) holds pi (((oContMaps (X,f)) .: A),x) = f .: (pi (A,x))

proof end;

:: 4.2. LEMMA (ii)

theorem Th13: :: WAYBEL26:13

for X being non empty TopSpace
for Y, Z being monotone-convergence T_0-TopSpace
for f being continuous Function of Y,Z holds oContMaps (X,f) is directed-sups-preserving

proof end;

theorem Th14: :: WAYBEL26:14

for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z
for x being Element of Y
for A being Subset of (oContMaps (Z,X)) holds pi (((oContMaps (f,X)) .: A),x) = pi (A,(f . x))

proof end;

theorem Th15: :: WAYBEL26:15

for Y, Z being non empty TopSpace
for X being monotone-convergence T_0-TopSpace
for f being continuous Function of Y,Z holds oContMaps (f,X) is directed-sups-preserving

proof end;

:: 4.3. LEMMA (i) genralized

theorem Th16: :: WAYBEL26:16

for X, Z being non empty TopSpace
for Y being non empty SubSpace of Z holds oContMaps (X,Y) is full SubRelStr of oContMaps (X,Z)

proof end;

theorem Th17: :: WAYBEL26:17

for Z being monotone-convergence T_0-TopSpace
for Y being non empty SubSpace of Z
for f being continuous Function of Z,Y st f is being_a_retraction holds
Omega Y is directed-sups-inheriting SubRelStr of Omega Z

proof end;

Lm1: for Z being monotone-convergence T_0-TopSpace
for Y being non empty SubSpace of Z
for f being continuous Function of Z,Y st f is being_a_retraction holds
Y is monotone-convergence

proof end;

theorem Th18: :: WAYBEL26:18

for X being non empty TopSpace
for Z being monotone-convergence T_0-TopSpace
for Y being non empty SubSpace of Z
for f being continuous Function of Z,Y st f is being_a_retraction holds
oContMaps (X,f) is_a_retraction_of oContMaps (X,Z), oContMaps (X,Y)

proof end;

theorem Th19: :: WAYBEL26:19

for X being non empty TopSpace
for Z being monotone-convergence T_0-TopSpace
for Y being non empty SubSpace of Z st Y is_a_retract_of Z holds
oContMaps (X,Y) is_a_retract_of oContMaps (X,Z)

proof end;

theorem Th20: :: WAYBEL26:20

for X, Y, Z being non empty TopSpace
for f being continuous Function of Y,Z st f is being_homeomorphism holds
oContMaps (X,f) is isomorphic

proof end;

theorem Th21: :: WAYBEL26:21

for X, Y, Z being non empty TopSpace st Y,Z are_homeomorphic holds
oContMaps (X,Y), oContMaps (X,Z) are_isomorphic

proof end;

:: 4.3. LEMMA (ii)

theorem Th22: :: WAYBEL26:22

for X being non empty TopSpace
for Z being monotone-convergence T_0-TopSpace
for Y being non empty SubSpace of Z st Y is_a_retract_of Z & oContMaps (X,Z) is complete & oContMaps (X,Z) is continuous holds
( oContMaps (X,Y) is complete & oContMaps (X,Y) is continuous ) by Th19, YELLOW16:21, YELLOW16:22;

theorem Th23: :: WAYBEL26:23

for X being non empty TopSpace
for Y, Z being monotone-convergence T_0-TopSpace st Y is_Retract_of Z & oContMaps (X,Z) is complete & oContMaps (X,Z) is continuous holds
( oContMaps (X,Y) is complete & oContMaps (X,Y) is continuous )

proof end;

theorem Th24: :: WAYBEL26:24

for Y being non trivial T_0-TopSpace st not Y is T_1 holds
Sierpinski_Space is_Retract_of Y

proof end;

theorem Th25: :: WAYBEL26:25

for X being non empty TopSpace
for Y being non trivial T_0-TopSpace st oContMaps (X,Y) is with_suprema holds
not Y is T_1

proof end;

registration

cluster Sierpinski_Space -> non trivial monotone-convergence ;

coherence
( not Sierpinski_Space is trivial & Sierpinski_Space is monotone-convergence )

proof end;

end;

registration

cluster non empty non trivial TopSpace-like T_0 injective monotone-convergence for TopStruct ;

existence
ex b₁ being T_0-TopSpace st
( b₁ is injective & b₁ is monotone-convergence & not b₁ is trivial )

proof end;

end;

:: 4.4. PROPOSITION

theorem Th26: :: WAYBEL26:26

for X being non empty TopSpace
for Y being non trivial monotone-convergence T_0-TopSpace st oContMaps (X,Y) is complete & oContMaps (X,Y) is continuous holds
InclPoset the topology of X is continuous

proof end;

theorem Th27: :: WAYBEL26:27

for X being non empty TopSpace
for x being Point of X
for Y being monotone-convergence T_0-TopSpace ex F being directed-sups-preserving projection Function of (oContMaps (X,Y)),(oContMaps (X,Y)) st
( ( for f being continuous Function of X,Y holds F . f = X --> (f . x) ) & ex h being continuous Function of X,X st
( h = X --> x & F = oContMaps (h,Y) ) )

proof end;

:: 4.5. PROPOSITION

theorem Th28: :: WAYBEL26:28

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace st oContMaps (X,Y) is complete & oContMaps (X,Y) is continuous holds
( Omega Y is complete & Omega Y is continuous )

proof end;

theorem Th29: :: WAYBEL26:29

for X being non empty 1-sorted
for I being non empty set
for J being non-Empty TopStruct-yielding ManySortedSet of I
for f being Function of X,(I -TOP_prod J)
for i being Element of I holds (commute f) . i = (proj (J,i)) * f

proof end;

theorem Th30: :: WAYBEL26:30

for S being 1-sorted
for M being set holds Carrier (M --> S) = M --> the carrier of S

proof end;

theorem Th31: :: WAYBEL26:31

for X, Y being non empty TopSpace
for M being non empty set
for f being continuous Function of X,(M -TOP_prod (M --> Y)) holds commute f is Function of M, the carrier of (oContMaps (X,Y))

proof end;

theorem Th32: :: WAYBEL26:32

for X, Y being non empty TopSpace holds the carrier of (oContMaps (X,Y)) c= Funcs ( the carrier of X, the carrier of Y)

proof end;

theorem Th33: :: WAYBEL26:33

for X, Y being non empty TopSpace
for M being non empty set
for f being Function of M, the carrier of (oContMaps (X,Y)) holds commute f is continuous Function of X,(M -TOP_prod (M --> Y))

proof end;

theorem Th34: :: WAYBEL26:34

for X being non empty TopSpace
for M being non empty set ex F being Function of (oContMaps (X,(M -TOP_prod (M --> Sierpinski_Space)))),(M -POS_prod (M --> (oContMaps (X,Sierpinski_Space)))) st
( F is isomorphic & ( for f being continuous Function of X,(M -TOP_prod (M --> Sierpinski_Space)) holds F . f = commute f ) )

proof end;

theorem Th35: :: WAYBEL26:35

for X being non empty TopSpace
for M being non empty set holds oContMaps (X,(M -TOP_prod (M --> Sierpinski_Space))),M -POS_prod (M --> (oContMaps (X,Sierpinski_Space))) are_isomorphic

proof end;

:: 4.6. PROPOSITION

theorem Th36: :: WAYBEL26:36

for X being non empty TopSpace st InclPoset the topology of X is continuous holds
for Y being injective T_0-TopSpace holds
( oContMaps (X,Y) is complete & oContMaps (X,Y) is continuous )

proof end;

registration

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

existence
ex b₁ being TopLattice st
( not b₁ is trivial & b₁ is complete & b₁ is Scott )

proof end;

end;

:: 4.7. THEOREM p.129.

theorem :: WAYBEL26:37

for X being non empty TopSpace
for L being non trivial complete Scott TopLattice holds
( oContMaps (X,L) is complete & oContMaps (X,L) is continuous iff ( InclPoset the topology of X is continuous & L is continuous ) )

proof end;

registration

let f be Function;

cluster Union (disjoin f) -> Relation-like ;

coherence
Union (disjoin f) is Relation-like by CARD_3:21;

end;

definition

let f be Function;

func *graph f -> Relation equals :: WAYBEL26:def 4
(Union (disjoin f)) ~ ;

correctness
coherence
(Union (disjoin f)) ~ is Relation;
;

end;

:: deftheorem defines *graph WAYBEL26:def 4 :
for f being Function holds *graph f = (Union (disjoin f)) ~ ;

theorem Th38: :: WAYBEL26:38

for x, y being object
for f being Function holds
( [x,y] in *graph f iff ( x in dom f & y in f . x ) )

proof end;

theorem Th39: :: WAYBEL26:39

for X being finite set holds
( proj1 X is finite & proj2 X is finite )

proof end;

:: 4.8. LEMMA p.130.

theorem Th40: :: WAYBEL26:40

for X, Y being non empty TopSpace
for S being Scott TopAugmentation of InclPoset the topology of Y
for f being Function of X,S st *graph f is open Subset of [:X,Y:] holds
f is continuous

proof end;

definition

let W be Relation;
let X be set ;

func (W,X) *graph -> Function means :Def5: :: WAYBEL26:def 5
( dom it = X & ( for x being object st x in X holds
it . x = Im (W,x) ) );

existence
ex b₁ being Function st
( dom b₁ = X & ( for x being object st x in X holds
b₁ . x = Im (W,x) ) )

proof end;

correctness
uniqueness
for b₁, b₂ being Function st dom b₁ = X & ( for x being object st x in X holds
b₁ . x = Im (W,x) ) & dom b₂ = X & ( for x being object st x in X holds
b₂ . x = Im (W,x) ) holds
b₁ = b₂;

proof end;

end;

:: deftheorem Def5 defines *graph WAYBEL26:def 5 :
for W being Relation
for X being set
for b₃ being Function holds
( b₃ = (W,X) *graph iff ( dom b₃ = X & ( for x being object st x in X holds
b₃ . x = Im (W,x) ) ) );

theorem Th41: :: WAYBEL26:41

for W being Relation
for X being set st dom W c= X holds
*graph ((W,X) *graph) = W

proof end;

registration

let X, Y be TopSpace;

cluster -> Relation-like for Element of bool the carrier of [:X,Y:];

coherence
for b₁ being Subset of [:X,Y:] holds b₁ is Relation-like

proof end;

cluster -> Relation-like for Element of the topology of [:X,Y:];

coherence
for b₁ being Element of the topology of [:X,Y:] holds b₁ is Relation-like ;

end;

theorem Th42: :: WAYBEL26:42

for X, Y being non empty TopSpace
for W being open Subset of [:X,Y:]
for x being Point of X holds Im (W,x) is open Subset of Y

proof end;

:: 4.9. PROPOSITION a) p.130.

theorem Th43: :: WAYBEL26:43

for X, Y being non empty TopSpace
for S being Scott TopAugmentation of InclPoset the topology of Y
for W being open Subset of [:X,Y:] holds (W, the carrier of X) *graph is continuous Function of X,S

proof end;

:: 4.9. PROPOSITION b) p.130.

theorem Th44: :: WAYBEL26:44

for X, Y being non empty TopSpace
for S being Scott TopAugmentation of InclPoset the topology of Y
for W1, W2 being open Subset of [:X,Y:] st W1 c= W2 holds
for f1, f2 being Element of (oContMaps (X,S)) st f1 = (W1, the carrier of X) *graph & f2 = (W2, the carrier of X) *graph holds
f1 <= f2

proof end;

:: 4.9. PROPOSITION p.130.

theorem :: WAYBEL26:45

for X, Y being non empty TopSpace
for S being Scott TopAugmentation of InclPoset the topology of Y ex F being Function of (InclPoset the topology of [:X,Y:]),(oContMaps (X,S)) st
( F is monotone & ( for W being open Subset of [:X,Y:] holds F . W = (W, the carrier of X) *graph ) )

proof end;