:: Injective Spaces, Part { II }
:: by Artur Korni{\l}owicz and Jaros{\l}aw Gryko
::
:: Received July 3, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

theorem :: WAYBEL25:1

for p being Point of Sierpinski_Space st p = 0 holds
{p} is closed

proof end;

theorem Th2: :: WAYBEL25:2

for p being Point of Sierpinski_Space st p = 1 holds
not {p} is closed

proof end;

registration

cluster Sierpinski_Space -> non T_1 ;

coherence
not Sierpinski_Space is T_1

proof end;

end;

registration

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

coherence
for b₁ being TopLattice st b₁ is complete & b₁ is Scott holds
b₁ is T_0 by WAYBEL11:10;

end;

registration

cluster non empty strict TopSpace-like T_0 injective for TopStruct ;

existence
ex b₁ being T_0-TopSpace st
( b₁ is injective & b₁ is strict )

proof end;

end;

registration

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

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

proof end;

end;

theorem Th3: :: WAYBEL25:3

for I being non empty set
for T being Scott TopAugmentation of product (I --> (BoolePoset {0})) holds the carrier of T = the carrier of (product (I --> Sierpinski_Space))

proof end;

theorem Th4: :: WAYBEL25:4

for L1, L2 being complete LATTICE
for T1 being Scott TopAugmentation of L1
for T2 being Scott TopAugmentation of L2
for h being Function of L1,L2
for H being Function of T1,T2 st h = H & h is isomorphic holds
H is being_homeomorphism

proof end;

theorem Th5: :: WAYBEL25:5

for L1, L2 being complete LATTICE
for T1 being Scott TopAugmentation of L1
for T2 being Scott TopAugmentation of L2 st L1,L2 are_isomorphic holds
T1,T2 are_homeomorphic

proof end;

theorem Th6: :: WAYBEL25:6

for S, T being non empty TopSpace st S is injective & S,T are_homeomorphic holds
T is injective

proof end;

theorem :: WAYBEL25:7

for L1, L2 being complete LATTICE
for T1 being Scott TopAugmentation of L1
for T2 being Scott TopAugmentation of L2 st L1,L2 are_isomorphic & T1 is injective holds
T2 is injective by Th5, Th6;

definition

let X, Y be non empty TopSpace;

redefine pred X is_Retract_of Y means :: WAYBEL25:def 1
ex c being continuous Function of X,Y ex r being continuous Function of Y,X st r * c = id X;

compatibility
( X is_Retract_of Y iff ex c being continuous Function of X,Y ex r being continuous Function of Y,X st r * c = id X )

proof end;

end;

:: deftheorem defines is_Retract_of WAYBEL25:def 1 :
for X, Y being non empty TopSpace holds
( X is_Retract_of Y iff ex c being continuous Function of X,Y ex r being continuous Function of Y,X st r * c = id X );

theorem :: WAYBEL25:8

for S, T, X, Y being non empty TopSpace st TopStruct(# the carrier of S, the topology of S #) = TopStruct(# the carrier of T, the topology of T #) & TopStruct(# the carrier of X, the topology of X #) = TopStruct(# the carrier of Y, the topology of Y #) & S is_Retract_of X holds
T is_Retract_of Y

proof end;

theorem :: WAYBEL25:9

for T, S1, S2 being non empty TopStruct st S1,S2 are_homeomorphic & S1 is_Retract_of T holds
S2 is_Retract_of T

proof end;

theorem :: WAYBEL25:10

for S, T being non empty TopSpace st T is injective & S is_Retract_of T holds
S is injective

proof end;

theorem :: WAYBEL25:11

for J being non empty injective TopSpace
for Y being non empty TopSpace st J is SubSpace of Y holds
J is_Retract_of Y

proof end;

:: p.123 proposition 3.5
:: p.124 theorem 3.8 (i, part a)
:: p.126 exercise 3.14

theorem Th12: :: WAYBEL25:12

for L being complete continuous LATTICE
for T being Scott TopAugmentation of L holds T is injective

proof end;

registration

let L be complete continuous LATTICE;

cluster Scott -> injective for TopAugmentation of L;

coherence
for b₁ being TopAugmentation of L st b₁ is Scott holds
b₁ is injective by Th12;

end;

registration

let T be non empty injective TopSpace;

cluster TopStruct(# the carrier of T, the topology of T #) -> injective ;

coherence
TopStruct(# the carrier of T, the topology of T #) is injective by WAYBEL18:16;

end;

::p.124 definition 3.6

definition

let T be TopStruct ;

func Omega T -> strict TopRelStr means :Def2: :: WAYBEL25:def 2
( TopStruct(# the carrier of it, the topology of it #) = TopStruct(# the carrier of T, the topology of T #) & ( for x, y being Element of it holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) );

existence
ex b₁ being strict TopRelStr st
( TopStruct(# the carrier of b₁, the topology of b₁ #) = TopStruct(# the carrier of T, the topology of T #) & ( for x, y being Element of b₁ holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) )

proof end;

uniqueness
for b₁, b₂ being strict TopRelStr st TopStruct(# the carrier of b₁, the topology of b₁ #) = TopStruct(# the carrier of T, the topology of T #) & ( for x, y being Element of b₁ holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) & TopStruct(# the carrier of b₂, the topology of b₂ #) = TopStruct(# the carrier of T, the topology of T #) & ( for x, y being Element of b₂ holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines Omega WAYBEL25:def 2 :
for T being TopStruct
for b₂ being strict TopRelStr holds
( b₂ = Omega T iff ( TopStruct(# the carrier of b₂, the topology of b₂ #) = TopStruct(# the carrier of T, the topology of T #) & ( for x, y being Element of b₂ holds
( x <= y iff ex Y being Subset of T st
( Y = {y} & x in Cl Y ) ) ) ) );

Lm1: for T being TopStruct holds the carrier of T = the carrier of (Omega T)

proof end;

registration

let T be empty TopStruct ;

cluster Omega T -> empty strict ;

coherence
Omega T is empty

proof end;

end;

registration

let T be non empty TopStruct ;

cluster Omega T -> non empty strict ;

coherence
not Omega T is empty

proof end;

end;

registration

let T be TopSpace;

cluster Omega T -> TopSpace-like strict ;

coherence
Omega T is TopSpace-like

proof end;

end;

registration

let T be TopStruct ;

cluster Omega T -> reflexive strict ;

coherence
Omega T is reflexive

proof end;

end;

Lm2: for T being TopStruct
for x, y being Element of T
for X, Y being Subset of T st X = {x} & Y = {y} holds
( x in Cl Y iff Cl X c= Cl Y )

proof end;

registration

let T be TopStruct ;

cluster Omega T -> transitive strict ;

coherence
Omega T is transitive

proof end;

end;

registration

let T be T_0-TopSpace;

cluster Omega T -> antisymmetric strict ;

coherence
Omega T is antisymmetric

proof end;

end;

theorem Th13: :: WAYBEL25:13

for S, T being TopSpace st TopStruct(# the carrier of S, the topology of S #) = TopStruct(# the carrier of T, the topology of T #) holds
Omega S = Omega T

proof end;

theorem :: WAYBEL25:14

for M being non empty set
for T being non empty TopSpace holds RelStr(# the carrier of (Omega (product (M --> T))), the InternalRel of (Omega (product (M --> T))) #) = RelStr(# the carrier of (product (M --> (Omega T))), the InternalRel of (product (M --> (Omega T))) #)

proof end;

::p.124 theorem 3.8 (i, part b)

theorem Th15: :: WAYBEL25:15

for S being complete Scott TopLattice holds Omega S = TopRelStr(# the carrier of S, the InternalRel of S, the topology of S #)

proof end;

::p.124 theorem 3.8 (i, part b)

theorem Th16: :: WAYBEL25:16

for L being complete LATTICE
for S being Scott TopAugmentation of L holds RelStr(# the carrier of (Omega S), the InternalRel of (Omega S) #) = RelStr(# the carrier of L, the InternalRel of L #)

proof end;

registration

let S be complete Scott TopLattice;

cluster Omega S -> complete strict ;

coherence
Omega S is complete

proof end;

end;

theorem Th17: :: WAYBEL25:17

for T being non empty TopSpace
for S being non empty SubSpace of T holds Omega S is full SubRelStr of Omega T

proof end;

theorem Th18: :: WAYBEL25:18

for S, T being TopSpace
for h being Function of S,T
for g being Function of (Omega S),(Omega T) st h = g & h is being_homeomorphism holds
g is isomorphic

proof end;

theorem Th19: :: WAYBEL25:19

for S, T being TopSpace st S,T are_homeomorphic holds
Omega S, Omega T are_isomorphic

proof end;

Lm3: for S, T being non empty RelStr
for f being Function of S,S
for g being Function of T,T st RelStr(# the carrier of S, the InternalRel of S #) = RelStr(# the carrier of T, the InternalRel of T #) & f = g & f is projection holds
g is projection
by WAYBEL_9:1;

::p.124 proposition 3.7
::p.124 theorem 3.8 (ii, part a)

theorem Th20: :: WAYBEL25:20

for T being injective T_0-TopSpace holds Omega T is complete continuous LATTICE

proof end;

registration

let T be injective T_0-TopSpace;

cluster Omega T -> complete strict continuous ;

coherence
( Omega T is complete & Omega T is continuous ) by Th20;

end;

theorem Th21: :: WAYBEL25:21

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

proof end;

registration

let X, Y be non empty TopSpace;

cluster Function-like quasi_total continuous -> monotone for Element of bool [: the carrier of (Omega X), the carrier of (Omega Y):];

coherence
for b₁ being Function of (Omega X),(Omega Y) st b₁ is continuous holds
b₁ is monotone by Th21;

end;

theorem Th22: :: WAYBEL25:22

for T being non empty TopSpace
for x being Element of (Omega T) holds Cl {x} = downarrow x

proof end;

registration

let T be non empty TopSpace;
let x be Element of (Omega T);

cluster Cl {x} -> non empty directed lower ;

coherence
( not Cl {x} is empty & Cl {x} is lower & Cl {x} is directed )

proof end;

cluster downarrow x -> closed ;

coherence
downarrow x is closed

proof end;

end;

theorem Th23: :: WAYBEL25:23

for X being TopSpace
for A being open Subset of (Omega X) holds A is upper

proof end;

registration

let T be TopSpace;

cluster open -> upper for Element of bool the carrier of (Omega T);

coherence
for b₁ being Subset of (Omega T) st b₁ is open holds
b₁ is upper by Th23;

end;

Lm4: now :: thesis: for X, Y being non empty TopSpace
for N being net of ContMaps (X,(Omega Y)) holds the mapping of N in Funcs ( the carrier of N,(Funcs ( the carrier of X, the carrier of Y)))

let X, Y be non empty TopSpace; :: thesis: for N being net of ContMaps (X,(Omega Y)) holds the mapping of N in Funcs ( the carrier of N,(Funcs ( the carrier of X, the carrier of Y)))
let N be net of ContMaps (X,(Omega Y)); :: thesis: the mapping of N in Funcs ( the carrier of N,(Funcs ( the carrier of X, the carrier of Y)))
A1: the carrier of Y = the carrier of (Omega Y) by Lm1;
the carrier of (ContMaps (X,(Omega Y))) c= Funcs ( the carrier of X, the carrier of Y)

proof

let f be object ; :: according to TARSKI:def 3 :: thesis: ( not f in the carrier of (ContMaps (X,(Omega Y))) or f in Funcs ( the carrier of X, the carrier of Y) )
assume f in the carrier of (ContMaps (X,(Omega Y))) ; :: thesis: f in Funcs ( the carrier of X, the carrier of Y)
then ex x being Function of X,(Omega Y) st
( x = f & x is continuous ) by WAYBEL24:def 3;
hence f in Funcs ( the carrier of X, the carrier of Y) by A1, FUNCT_2:8; :: thesis: verum

end;

then A2: Funcs ( the carrier of N, the carrier of (ContMaps (X,(Omega Y)))) c= Funcs ( the carrier of N,(Funcs ( the carrier of X, the carrier of Y))) by FUNCT_5:56;
the mapping of N in Funcs ( the carrier of N, the carrier of (ContMaps (X,(Omega Y)))) by FUNCT_2:8;
hence the mapping of N in Funcs ( the carrier of N,(Funcs ( the carrier of X, the carrier of Y))) by A2; :: thesis: verum

end;

definition

let I be non empty set ;
let S, T be non empty RelStr ;
let N be net of T;
let i be Element of I;
assume A1: the carrier of T c= the carrier of (S |^ I) ;

func commute (N,i,S) -> strict net of S means :Def3: :: WAYBEL25:def 3
( RelStr(# the carrier of it, the InternalRel of it #) = RelStr(# the carrier of N, the InternalRel of N #) & the mapping of it = (commute the mapping of N) . i );

existence
ex b₁ being strict net of S st
( RelStr(# the carrier of b₁, the InternalRel of b₁ #) = RelStr(# the carrier of N, the InternalRel of N #) & the mapping of b₁ = (commute the mapping of N) . i )

proof end;

uniqueness
for b₁, b₂ being strict net of S st RelStr(# the carrier of b₁, the InternalRel of b₁ #) = RelStr(# the carrier of N, the InternalRel of N #) & the mapping of b₁ = (commute the mapping of N) . i & RelStr(# the carrier of b₂, the InternalRel of b₂ #) = RelStr(# the carrier of N, the InternalRel of N #) & the mapping of b₂ = (commute the mapping of N) . i holds
b₁ = b₂ ;

end;

:: deftheorem Def3 defines commute WAYBEL25:def 3 :
for I being non empty set
for S, T being non empty RelStr
for N being net of T
for i being Element of I st the carrier of T c= the carrier of (S |^ I) holds
for b₆ being strict net of S holds
( b₆ = commute (N,i,S) iff ( RelStr(# the carrier of b₆, the InternalRel of b₆ #) = RelStr(# the carrier of N, the InternalRel of N #) & the mapping of b₆ = (commute the mapping of N) . i ) );

theorem Th24: :: WAYBEL25:24

for X, Y being non empty TopSpace
for N being net of ContMaps (X,(Omega Y))
for i being Element of N
for x being Point of X holds the mapping of (commute (N,x,(Omega Y))) . i = ( the mapping of N . i) . x

proof end;

theorem Th25: :: WAYBEL25:25

for X, Y being non empty TopSpace
for N being eventually-directed net of ContMaps (X,(Omega Y))
for x being Point of X holds commute (N,x,(Omega Y)) is eventually-directed

proof end;

registration

let X, Y be non empty TopSpace;
let N be eventually-directed net of ContMaps (X,(Omega Y));
let x be Point of X;

cluster commute (N,x,(Omega Y)) -> strict eventually-directed ;

coherence
commute (N,x,(Omega Y)) is eventually-directed by Th25;

end;

registration

let X, Y be non empty TopSpace;

cluster non empty transitive directed -> Function-yielding for NetStr over ContMaps (X,(Omega Y));

coherence
for b₁ being net of ContMaps (X,(Omega Y)) holds b₁ is Function-yielding ;

end;

Lm5: now :: thesis: for X, Y being non empty TopSpace
for N being net of ContMaps (X,(Omega Y))
for i being Element of N holds
( the mapping of N . i is Function of X,(Omega Y) & the mapping of N . i is Function of X,Y )

let X, Y be non empty TopSpace; :: thesis: for N being net of ContMaps (X,(Omega Y))
for i being Element of N holds
( the mapping of N . i is Function of X,(Omega Y) & the mapping of N . i is Function of X,Y )
let N be net of ContMaps (X,(Omega Y)); :: thesis: for i being Element of N holds
( the mapping of N . i is Function of X,(Omega Y) & the mapping of N . i is Function of X,Y )
let i be Element of N; :: thesis: ( the mapping of N . i is Function of X,(Omega Y) & the mapping of N . i is Function of X,Y )
thus the mapping of N . i is Function of X,(Omega Y) by WAYBEL24:21; :: thesis: the mapping of N . i is Function of X,Y
the carrier of Y = the carrier of (Omega Y) by Lm1;
hence the mapping of N . i is Function of X,Y by WAYBEL24:21; :: thesis: verum

end;

Lm6: now :: thesis: for X, Y being non empty TopSpace
for N being net of ContMaps (X,(Omega Y))
for x being Point of X holds dom the mapping of N = dom the mapping of (commute (N,x,(Omega Y)))

let X, Y be non empty TopSpace; :: thesis: for N being net of ContMaps (X,(Omega Y))
for x being Point of X holds dom the mapping of N = dom the mapping of (commute (N,x,(Omega Y)))
let N be net of ContMaps (X,(Omega Y)); :: thesis: for x being Point of X holds dom the mapping of N = dom the mapping of (commute (N,x,(Omega Y)))
let x be Point of X; :: thesis: dom the mapping of N = dom the mapping of (commute (N,x,(Omega Y)))
ContMaps (X,(Omega Y)) is SubRelStr of (Omega Y) |^ the carrier of X by WAYBEL24:def 3;
then the carrier of (ContMaps (X,(Omega Y))) c= the carrier of ((Omega Y) |^ the carrier of X) by YELLOW_0:def 13;
then A1: RelStr(# the carrier of N, the InternalRel of N #) = RelStr(# the carrier of (commute (N,x,(Omega Y))), the InternalRel of (commute (N,x,(Omega Y))) #) by Def3;
thus dom the mapping of N = the carrier of N by FUNCT_2:def 1
.= dom the mapping of (commute (N,x,(Omega Y))) by A1, FUNCT_2:def 1 ; :: thesis: verum

end;

theorem Th26: :: WAYBEL25:26

for X being non empty TopSpace
for Y being T_0-TopSpace
for N being net of ContMaps (X,(Omega Y)) st ( for x being Point of X holds ex_sup_of commute (N,x,(Omega Y)) ) holds
ex_sup_of rng the mapping of N,(Omega Y) |^ the carrier of X

proof end;

::p.125 definition 3.9

definition

let T be non empty TopSpace;

attr T is monotone-convergence means :Def4: :: WAYBEL25:def 4
for D being non empty directed Subset of (Omega T) holds
( ex_sup_of D, Omega T & ( for V being open Subset of T st sup D in V holds
D meets V ) );

end;

:: deftheorem Def4 defines monotone-convergence WAYBEL25:def 4 :
for T being non empty TopSpace holds
( T is monotone-convergence iff for D being non empty directed Subset of (Omega T) holds
( ex_sup_of D, Omega T & ( for V being open Subset of T st sup D in V holds
D meets V ) ) );

theorem Th27: :: WAYBEL25:27

for S, T being non empty TopSpace st TopStruct(# the carrier of S, the topology of S #) = TopStruct(# the carrier of T, the topology of T #) & S is monotone-convergence holds
T is monotone-convergence

proof end;

Lm7: now :: thesis: for T being non empty 1-sorted
for D being non empty Subset of T
for d being Element of T st the carrier of T = {d} holds
D = {d}

let T be non empty 1-sorted ; :: thesis: for D being non empty Subset of T
for d being Element of T st the carrier of T = {d} holds
D = {d}
let D be non empty Subset of T; :: thesis: for d being Element of T st the carrier of T = {d} holds
D = {d}
let d be Element of T; :: thesis: ( the carrier of T = {d} implies D = {d} )
assume A1: the carrier of T = {d} ; :: thesis: D = {d}
thus D = {d} :: thesis: verum

proof

thus D c= {d} by A1; :: according to XBOOLE_0:def 10 :: thesis: {d} c= D
set x = the Element of D;
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in {d} or a in D )
assume a in {d} ; :: thesis: a in D
then A2: a = d by TARSKI:def 1;
the Element of D in D ;
hence a in D by A1, A2, TARSKI:def 1; :: thesis: verum

end;

end;

registration

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

coherence
for b₁ being T_0-TopSpace st b₁ is trivial holds
b₁ is monotone-convergence

proof end;

end;

theorem :: WAYBEL25:28

for S being monotone-convergence T_0-TopSpace
for T being T_0-TopSpace st S,T are_homeomorphic holds
T is monotone-convergence

proof end;

theorem Th29: :: WAYBEL25:29

for S being complete Scott TopLattice holds S is monotone-convergence

proof end;

registration

let L be complete LATTICE;

cluster Scott -> Scott monotone-convergence for TopAugmentation of L;

coherence
for b₁ being Scott TopAugmentation of L holds b₁ is monotone-convergence by Th29;

end;

registration

let L be complete LATTICE;
let S be Scott TopAugmentation of L;

cluster TopStruct(# the carrier of S, the topology of S #) -> monotone-convergence ;

coherence
TopStruct(# the carrier of S, the topology of S #) is monotone-convergence by Th27;

end;

theorem Th30: :: WAYBEL25:30

for X being monotone-convergence T_0-TopSpace holds Omega X is up-complete

proof end;

registration

let X be monotone-convergence T_0-TopSpace;

cluster Omega X -> up-complete strict ;

coherence
Omega X is up-complete by Th30;

end;

theorem Th31: :: WAYBEL25:31

for X being non empty monotone-convergence TopSpace
for N being eventually-directed prenet of Omega X holds ex_sup_of N

proof end;

theorem Th32: :: WAYBEL25:32

for X being non empty monotone-convergence TopSpace
for N being eventually-directed net of Omega X holds sup N in Lim N

proof end;

theorem Th33: :: WAYBEL25:33

for X being non empty monotone-convergence TopSpace
for N being eventually-directed net of Omega X holds N is convergent

proof end;

registration

let X be non empty monotone-convergence TopSpace;

cluster non empty transitive directed eventually-directed -> eventually-directed convergent for NetStr over Omega X;

coherence
for b₁ being eventually-directed net of Omega X holds b₁ is convergent by Th33;

end;

theorem :: WAYBEL25:34

for X being non empty TopSpace st ( for N being eventually-directed net of Omega X holds
( ex_sup_of N & sup N in Lim N ) ) holds
X is monotone-convergence

proof end;

::p.125 lemma 3.10

theorem Th35: :: WAYBEL25:35

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

proof end;

registration

let X be non empty monotone-convergence TopSpace;
let Y be T_0-TopSpace;

cluster Function-like quasi_total continuous -> directed-sups-preserving for Element of bool [: the carrier of (Omega X), the carrier of (Omega Y):];

coherence
for b₁ being Function of (Omega X),(Omega Y) st b₁ is continuous holds
b₁ is directed-sups-preserving by Th35;

end;

theorem Th36: :: WAYBEL25:36

for T being monotone-convergence T_0-TopSpace
for R being T_0-TopSpace st R is_Retract_of T holds
R is monotone-convergence

proof end;

::p.124 theorem 3.8 (ii, part b)

theorem Th37: :: WAYBEL25:37

for T being injective T_0-TopSpace
for S being Scott TopAugmentation of Omega T holds TopStruct(# the carrier of S, the topology of S #) = TopStruct(# the carrier of T, the topology of T #)

proof end;

theorem :: WAYBEL25:38

for T being injective T_0-TopSpace holds
( T is compact & T is locally-compact & T is sober )

proof end;

theorem Th39: :: WAYBEL25:39

for T being injective T_0-TopSpace holds T is monotone-convergence

proof end;

registration

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

coherence
for b₁ being T_0-TopSpace st b₁ is injective holds
b₁ is monotone-convergence by Th39;

end;

theorem Th40: :: WAYBEL25:40

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace
for N being net of ContMaps (X,(Omega Y))
for f, g being Function of X,(Omega Y) st f = "\/" ((rng the mapping of N),((Omega Y) |^ the carrier of X)) & ex_sup_of rng the mapping of N,(Omega Y) |^ the carrier of X & g in rng the mapping of N holds
g <= f

proof end;

theorem Th41: :: WAYBEL25:41

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace
for N being net of ContMaps (X,(Omega Y))
for x being Point of X
for f being Function of X,(Omega Y) st ( for a being Point of X holds commute (N,a,(Omega Y)) is eventually-directed ) & f = "\/" ((rng the mapping of N),((Omega Y) |^ the carrier of X)) holds
f . x = sup (commute (N,x,(Omega Y)))

proof end;

::p.125 lemma 3.11

theorem Th42: :: WAYBEL25:42

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace
for N being net of ContMaps (X,(Omega Y)) st ( for x being Point of X holds commute (N,x,(Omega Y)) is eventually-directed ) holds
"\/" ((rng the mapping of N),((Omega Y) |^ the carrier of X)) is continuous Function of X,Y

proof end;

::p.126 lemma 3.12 (i)

theorem :: WAYBEL25:43

for X being non empty TopSpace
for Y being monotone-convergence T_0-TopSpace holds ContMaps (X,(Omega Y)) is directed-sups-inheriting SubRelStr of (Omega Y) |^ the carrier of X

proof end;