:: Injective Spaces, Part { II }
:: by Artur Korni{\l}owicz and Jaros{\l}aw Gryko
::
:: 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
coherence
not Sierpinski_Space is T_1
proof end;
end;

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

registration
existence
ex b1 being T_0-TopSpace st
( b1 is injective & b1 is strict )
proof end;
end;

registration
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 --> ) holds the carrier of T = the carrier of ()
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 ;

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 ()
proof end;

registration
let T be empty TopStruct ;
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;
coherence
proof end;
end;

registration
let T be TopStruct ;
coherence
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 ;
coherence
proof end;
end;

registration
let T be T_0-TopSpace;
coherence
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 --> ())), the InternalRel of (product (M --> ())) #)
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 (), the InternalRel of () #) = RelStr(# the carrier of L, the InternalRel of L #)
proof end;

registration
let S be complete Scott TopLattice;
coherence
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 (),() 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;
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 (),() 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 (), the carrier of ():];
coherence
for b1 being Function of (),() 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 () holds Cl {x} = downarrow x
proof end;

registration
let T be non empty TopSpace;
let x be Element of ();
coherence
( not Cl {x} is empty & Cl {x} is lower & Cl {x} is directed )
proof end;
coherence
proof end;
end;

theorem Th23: :: WAYBEL25:23
for X being TopSpace
for A being open Subset of () holds A is upper
proof end;

registration
let T be TopSpace;
cluster open -> upper for Element of bool the carrier of ();
coherence
for b1 being Subset of () 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,()) 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,()) 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,()); :: 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 () by Lm1;
the carrier of (ContMaps (X,())) 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,())) or f in Funcs ( the carrier of X, the carrier of Y) )
assume f in the carrier of (ContMaps (X,())) ; :: thesis: f in Funcs ( the carrier of X, the carrier of Y)
then ex x being Function of X,() st
( x = f & x is continuous ) by WAYBEL24:def 3;
hence f in Funcs ( the carrier of X, the carrier of Y) by ; :: thesis: verum
end;
then A2: Funcs ( the carrier of N, the carrier of (ContMaps (X,()))) 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,()))) 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,())
for i being Element of N
for x being Point of X holds the mapping of (commute (N,x,())) . 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,())
for x being Point of X holds commute (N,x,()) is eventually-directed
proof end;

registration
let X, Y be non empty TopSpace;
let N be eventually-directed net of ContMaps (X,());
let x be Point of X;
cluster commute (N,x,()) -> strict eventually-directed ;
coherence
commute (N,x,()) 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,());
coherence
for b1 being net of ContMaps (X,()) holds b1 is Function-yielding
;
end;

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

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

let N be net of ContMaps (X,()); :: thesis: for x being Point of X holds dom the mapping of N = dom the mapping of (commute (N,x,()))
let x be Point of X; :: thesis: dom the mapping of N = dom the mapping of (commute (N,x,()))
ContMaps (X,()) is SubRelStr of () |^ the carrier of X by WAYBEL24:def 3;
then the carrier of (ContMaps (X,())) c= the carrier of (() |^ 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,())), the InternalRel of (commute (N,x,())) #) by Def3;
thus dom the mapping of N = the carrier of N by FUNCT_2:def 1
.= dom the mapping of (commute (N,x,())) by ; :: 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,()) st ( for x being Point of X holds ex_sup_of commute (N,x,()) ) holds
ex_sup_of rng the mapping of N,() |^ 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 () 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 () 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 ; :: thesis: verum
end;
end;

registration
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;
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;
coherence 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;
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 (),() 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 (), the carrier of ():];
coherence
for b1 being Function of (),() 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
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,())
for f, g being Function of X,() st f = "\/" ((rng the mapping of N),(() |^ the carrier of X)) & ex_sup_of rng the mapping of N,() |^ 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,())
for x being Point of X
for f being Function of X,() st ( for a being Point of X holds commute (N,a,()) is eventually-directed ) & f = "\/" ((rng the mapping of N),(() |^ the carrier of X)) holds
f . x = sup (commute (N,x,()))
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,()) st ( for x being Point of X holds commute (N,x,()) is eventually-directed ) holds
"\/" ((rng the mapping of N),(() |^ 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,()) is directed-sups-inheriting SubRelStr of () |^ the carrier of X
proof end;