The Mizar article:

Injective Spaces. Part II

by
Artur Kornilowicz, and
Jaroslaw Gryko

Received July 3, 1999

Copyright (c) 1999 Association of Mizar Users

MML identifier: WAYBEL25
[ MML identifier index ]


environ

 vocabulary PRE_TOPC, WAYBEL18, BOOLE, SUBSET_1, URYSOHN1, BHSP_3, WAYBEL11,
      METRIC_1, WAYBEL_9, FUNCTOR0, T_0TOPSP, YELLOW_9, CARD_3, FUNCOP_1,
      YELLOW_1, RELAT_1, RLVECT_2, FUNCT_1, ORDERS_1, WAYBEL_0, CAT_1, TOPS_2,
      ORDINAL2, WELLORD1, GROUP_6, FUNCT_3, WAYBEL_1, BORSUK_1, LATTICES,
      QUANTAL1, LATTICE3, YELLOW_0, WAYBEL_3, RELAT_2, PROB_1, BINOP_1, SEQM_3,
      WAYBEL24, FUNCT_2, GROUP_1, PRALG_2, PRALG_1, YELLOW_2, REALSET1,
      CONNSP_2, TOPS_1, SEQ_2, COMPTS_1, YELLOW_8, TARSKI, SETFAM_1, WAYBEL25;
 notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1,
      FUNCT_1, PARTFUN1, FUNCT_2, REALSET1, NATTRA_1, TOLER_1, QUANTAL1,
      CARD_3, PRALG_1, PRALG_2, PRE_CIRC, WAYBEL18, STRUCT_0, PRE_TOPC,
      GRCAT_1, TOPS_1, TOPS_2, COMPTS_1, BORSUK_1, TMAP_1, T_0TOPSP, URYSOHN1,
      BORSUK_3, ORDERS_1, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_2, WAYBEL_1,
      WAYBEL_2, YELLOW_6, WAYBEL_3, YELLOW_8, WAYBEL_9, WAYBEL11, YELLOW_9,
      WAYBEL17, YELLOW_1, WAYBEL24, YELLOW14;
 constructors BORSUK_3, CANTOR_1, ENUMSET1, GRCAT_1, NATTRA_1, ORDERS_3,
      PRALG_2, QUANTAL1, RELAT_2, TMAP_1, TOLER_1, TOPS_1, TOPS_2, URYSOHN1,
      WAYBEL_1, WAYBEL_8, WAYBEL17, WAYBEL24, YELLOW_8, YELLOW_9, YELLOW14,
      MONOID_0, MEMBERED;
 clusters BORSUK_3, FUNCT_1, LATTICE3, PRALG_1, PRE_TOPC, RELSET_1, STRUCT_0,
      TEX_1, TSP_1, TOPS_1, YELLOW_0, YELLOW_1, YELLOW_9, YELLOW12, WAYBEL_0,
      WAYBEL_2, WAYBEL_8, WAYBEL10, WAYBEL12, WAYBEL17, WAYBEL18, WAYBEL19,
      WAYBEL21, WAYBEL24, YELLOW14, SUBSET_1, BORSUK_1, MEMBERED, ZFMISC_1,
      FUNCT_2;
 requirements SUBSET, BOOLE;
 definitions TARSKI, XBOOLE_0, PRE_TOPC, TOPS_2, ORDERS_1, RELAT_1, RELAT_2,
      STRUCT_0, WAYBEL_1, T_0TOPSP, WAYBEL18, LATTICE3, YELLOW_6, WAYBEL_0,
      WAYBEL_9, CONNSP_2, YELLOW14;
 theorems TARSKI, PRE_TOPC, ZFMISC_1, STRUCT_0, ORDERS_1, T_0TOPSP, FUNCT_1,
      FUNCT_2, QUANTAL1, RELAT_1, FINSEQ_5, TEX_1, PBOOLE, PRALG_1, FUNCOP_1,
      CARD_3, BORSUK_1, WELLORD1, WAYBEL14, TOPS_1, TOPS_2, TOPS_3, YELLOW_0,
      YELLOW_1, YELLOW_2, YELLOW_4, YELLOW_6, YELLOW_9, YELLOW12, WAYBEL_0,
      WAYBEL_1, WAYBEL_2, WAYBEL_7, WAYBEL_9, WAYBEL11, LATTICE3, WAYBEL10,
      WAYBEL13, WAYBEL15, WAYBEL17, WAYBEL18, WAYBEL20, WAYBEL24, CONNSP_2,
      WAYBEL_3, PRALG_2, FUNCT_5, WAYBEL_8, TOPMETR, GRCAT_1, TSEP_1, EQUATION,
      TOPGRP_1, WAYBEL21, FRECHET2, ENUMSET1, YELLOW14, RELSET_1, SETFAM_1,
      XBOOLE_0, XBOOLE_1;
 schemes FUNCT_2, RELSET_1;

begin  :: Injective spaces

theorem
   for p being Point of Sierpinski_Space st p = 0 holds {p} is closed
  proof
    set S = Sierpinski_Space;
    let p be Point of S;
    assume
A1:   p = 0;
A2: the carrier of S = {0,1} by WAYBEL18:def 9;
A3: the topology of S = {{}, {1}, {0,1}} by WAYBEL18:def 9;
      [#]S \ {p} = {1}
    proof
      thus [#]S \ {p} c= {1}
      proof
        let a be set;
        assume a in [#]S \ {p};
        then a in [#]S & not a in {p} by XBOOLE_0:def 4;
        then a in the carrier of S & a <> 0 by A1,TARSKI:def 1;
        then a = 1 by A2,TARSKI:def 2;
        hence a in {1} by TARSKI:def 1;
      end;
      let a be set;
      assume a in {1};
      then a = 1 by TARSKI:def 1;
      then a in the carrier of S & a <> 0 by A2,TARSKI:def 2;
      then a in [#]S & not a in {p} by A1,PRE_TOPC:12,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 4;
    end;
    hence [#]S \ {p} in the topology of S by A3,ENUMSET1:14;
  end;

theorem Th2:
 for p being Point of Sierpinski_Space st p = 1 holds {p} is non closed
  proof
    set S = Sierpinski_Space;
    let p be Point of S;
    assume
A1:   p = 1;
A2: the carrier of S = {0,1} by WAYBEL18:def 9;
A3: the topology of S = {{}, {1}, {0,1}} by WAYBEL18:def 9;
A4: {0} <> {1} & {0} <> {0,1} by ZFMISC_1:6,9;
      [#]S \ {p} = {0}
    proof
      thus [#]S \ {p} c= {0}
      proof
        let a be set;
        assume a in [#]S \ {p};
        then a in [#]S & not a in {p} by XBOOLE_0:def 4;
        then a in the carrier of S & a <> 1 by A1,TARSKI:def 1;
        then a = 0 by A2,TARSKI:def 2;
        hence a in {0} by TARSKI:def 1;
      end;
      let a be set;
      assume a in {0};
      then a = 0 by TARSKI:def 1;
      then a in the carrier of S & a <> 1 by A2,TARSKI:def 2;
      then a in [#]S & not a in {p} by A1,PRE_TOPC:12,TARSKI:def 1;
      hence thesis by XBOOLE_0:def 4;
    end;
    hence not [#]S \ {p} in the topology of S by A3,A4,ENUMSET1:13;
  end;

definition
 cluster Sierpinski_Space -> non being_T1;
coherence
  proof
    set S = Sierpinski_Space;
      ex p being Point of S st Cl {p} <> {p}
    proof
        the carrier of S = {0,1} by WAYBEL18:def 9;
      then reconsider p = 1 as Point of S by TARSKI:def 2;
      take p;
      thus thesis by Th2;
    end;
    hence thesis by FRECHET2:47;
  end;
end;

definition
 cluster complete Scott -> discerning TopLattice;
coherence by WAYBEL11:10;
end;

definition
 cluster injective strict T_0-TopSpace;
existence
  proof
    take Sierpinski_Space;
    thus thesis;
  end;
end;

definition
 cluster complete Scott strict TopLattice;
existence
  proof
    consider T being complete Scott strict TopLattice;
    take T;
    thus thesis;
  end;
end;

theorem Th3:  :: see WAYBEL18:16
 for I being non empty set,
     T being Scott TopAugmentation of product(I --> BoolePoset 1) holds
   the carrier of T = the carrier of product(I --> Sierpinski_Space)
  proof
    let I be non empty set,
        T be Scott TopAugmentation of product(I --> BoolePoset 1);
    set S = Sierpinski_Space,
        B = BoolePoset 1;
A1: dom Carrier (I --> B) = I by PBOOLE:def 3;
A2: dom Carrier (I --> S) = I by PBOOLE:def 3;
A3: for x being set st x in dom Carrier (I --> S) holds
     (Carrier (I --> B)).x = (Carrier (I --> S)).x
    proof
      let x be set; assume
A4:     x in dom Carrier (I --> S);
      then consider R being 1-sorted such that
A5:     R = (I --> B).x &
        (Carrier (I --> B)).x = the carrier of R by A2,PRALG_1:def 13;
      consider T being 1-sorted such that
A6:     T = (I --> S).x &
        (Carrier (I --> S)).x = the carrier of T by A2,A4,PRALG_1:def 13;
      thus (Carrier (I --> B)).x
        = the carrier of B by A2,A4,A5,FUNCOP_1:13
       .= bool 1 by WAYBEL_7:4
       .= the carrier of S by WAYBEL18:def 9,YELLOW14:1
       .= (Carrier (I --> S)).x by A2,A4,A6,FUNCOP_1:13;
    end;
      the RelStr of T = the RelStr of product(I --> B)
      by YELLOW_9:def 4;
    hence the carrier of T = product Carrier (I --> B) by YELLOW_1:def 4
      .= product Carrier (I --> S) by A1,A2,A3,FUNCT_1:9
      .= the carrier of product(I --> S) by WAYBEL18:def 3;
  end;

theorem Th4:
 for L1, L2 being complete LATTICE,
     T1 being Scott TopAugmentation of L1,
     T2 being Scott TopAugmentation of L2,
     h being map of L1, L2, H being map of T1, T2 st h = H & h is isomorphic
  holds H is_homeomorphism
  proof
    let L1, L2 be complete LATTICE,
        T1 be Scott TopAugmentation of L1,
        T2 be Scott TopAugmentation of L2,
        h be map of L1, L2,
        H be map of T1, T2 such that
A1:   h = H and
A2:   h is isomorphic;
A3: the RelStr of L1 = the RelStr of T1 by YELLOW_9:def 4;
A4: the RelStr of L2 = the RelStr of T2 by YELLOW_9:def 4;
A5: h is one-to-one by A2,WAYBEL_0:66;
A6: rng h = the carrier of L2 by A2,WAYBEL_0:66
         .= [#]L2 by PRE_TOPC:12;
    consider g being map of L2,L1 such that
A7:   g = h";
      g = (h qua Function)" by A5,A6,A7,TOPS_2:def 4;
then A8: g is isomorphic by A2,WAYBEL_0:68;
    reconsider h1 = h as sups-preserving map of L1,L2 by A2,WAYBEL13:20;
    reconsider h2 = h" as sups-preserving map of L2,L1 by A7,A8,WAYBEL13:20;
A9: h1 is directed-sups-preserving;
A10: h2 is directed-sups-preserving;
    reconsider H' = h" as map of T2, T1 by A3,A4;
      H is directed-sups-preserving by A1,A3,A4,A9,WAYBEL21:6;
then A11: H is continuous by WAYBEL17:22;
      H' is directed-sups-preserving by A3,A4,A10,WAYBEL21:6;
then A12: H' is continuous by WAYBEL17:22;
A13: dom H = the carrier of T1 by FUNCT_2:def 1 .= [#]T1 by PRE_TOPC:12;
A14: rng H = the carrier of T2 by A1,A2,A4,WAYBEL_0:66
         .= [#]T2 by PRE_TOPC:12;
A15: H is one-to-one by A1,A2,WAYBEL_0:66;
A16: dom (H") = the carrier of T2 by FUNCT_2:def 1
            .= dom H' by FUNCT_2:def 1;
      for x being set st x in dom H' holds H'.x = H".x
    proof
      let x be set;
      assume x in dom H';
      thus H'.x = (h qua Function)".x by A5,A6,TOPS_2:def 4
               .= H".x by A1,A5,A14,TOPS_2:def 4;
    end;
    then H" = H' by A16,FUNCT_1:9;
    hence H is_homeomorphism by A11,A12,A13,A14,A15,TOPS_2:def 5;
  end;

theorem Th5:
 for L1, L2 being complete LATTICE,
     T1 being Scott TopAugmentation of L1,
     T2 being Scott TopAugmentation of L2 st L1, L2 are_isomorphic holds
  T1, T2 are_homeomorphic
  proof
    let L1, L2 be complete LATTICE,
        T1 be Scott TopAugmentation of L1,
        T2 be Scott TopAugmentation of L2;
    given h being map of L1, L2 such that
A1:   h is isomorphic;
      the RelStr of L1 = the RelStr of T1 & the RelStr of L2 = the RelStr of T2
      by YELLOW_9:def 4;
    then reconsider H = h as map of T1, T2 ;
    take H;
    thus thesis by A1,Th4;
  end;

theorem Th6:
 for S, T being non empty TopSpace st S is injective & S, T are_homeomorphic
  holds T is injective
  proof
    let S, T be non empty TopSpace such that
A1:   S is injective and
A2:   S, T are_homeomorphic;
    consider p being map of S,T such that
A3:   p is_homeomorphism by A2,T_0TOPSP:def 1;
A4: p is continuous by A3,TOPS_2:def 5;
    let X be non empty TopSpace,
        f be map of X, T;
    assume
A5:   f is continuous;
    let Y be non empty TopSpace;
    assume
A6:   X is SubSpace of Y;
    set F = p"*f;
      p" is continuous by A3,TOPS_2:def 5;
    then F is continuous by A5,TOPS_2:58;
    then consider G being map of Y,S such that
A7:   G is continuous and
A8:   G|(the carrier of X) = F by A1,A6,WAYBEL18:def 5;
    take g = p*G;
    thus g is continuous by A4,A7,TOPS_2:58;
      [#]X c= [#]Y by A6,PRE_TOPC:def 9;
    then the carrier of X c= [#]Y by PRE_TOPC:12;
then A9: the carrier of X c= the carrier of Y by PRE_TOPC:12;
A10: rng p = [#]T & p is one-to-one by A3,TOPS_2:72;
A11: dom f = the carrier of X by FUNCT_2:def 1
         .= (the carrier of Y) /\ (the carrier of X) by A9,XBOOLE_1:28
         .= dom g /\ (the carrier of X) by FUNCT_2:def 1;
      for x being set st x in dom f holds g.x = f.x
    proof
      let x be set;
      assume
A12:     x in dom f;
      then x in the carrier of X;
      then x in the carrier of Y by A9;
then A13:   x in dom G by FUNCT_2:def 1;
A14:   f.x in rng f by A12,FUNCT_1:def 5;
      then f.x in the carrier of T;
then A15:   f.x in dom (p") by FUNCT_2:def 1;
      thus g.x = p.(G.x) by A13,FUNCT_1:23
              .= p.((p"*f).x) by A8,A12,FUNCT_1:72
              .= p.(p".(f.x)) by A12,FUNCT_1:23
              .= (p*p").(f.x) by A15,FUNCT_1:23
              .= (id [#]T).(f.x) by A10,TOPS_2:65
              .= (id the carrier of T).(f.x) by PRE_TOPC:12
              .= f.x by A14,FUNCT_1:34;
    end;
    hence g|(the carrier of X) = f by A11,FUNCT_1:68;
  end;

theorem
   for L1, L2 being complete LATTICE,
     T1 being Scott TopAugmentation of L1,
     T2 being Scott TopAugmentation of L2 st L1, L2 are_isomorphic &
     T1 is injective holds
  T2 is injective
  proof
    let L1, L2 be complete LATTICE,
        T1 be Scott TopAugmentation of L1,
        T2 be Scott TopAugmentation of L2 such that
A1:  L1, L2 are_isomorphic and
A2:  T1 is injective;
      T1, T2 are_homeomorphic by A1,Th5;
    hence thesis by A2,Th6;
  end;

definition let X, Y be non empty TopSpace;
 redefine pred X is_Retract_of Y means :Def1:
  ex c being continuous map of X, Y,
     r being continuous map of Y, X st r * c = id X;
compatibility
  proof
    thus X is_Retract_of Y implies ex c being continuous map of X, Y,
       r being continuous map of Y, X st r * c = id X
    proof
      given f being map of Y, Y such that
A1:     f is continuous and
A2:     f*f = f and
A3:     Image f, X are_homeomorphic;
      consider h being map of Image f, X such that
A4:     h is_homeomorphism by A3,T_0TOPSP:def 1;
        h" is continuous by A4,TOPS_2:def 5;
      then reconsider c = incl Image f * h" as continuous map of X, Y by TOPS_2
:58;
        h is continuous & corestr f is continuous
        by A1,A4,TOPS_2:def 5,WAYBEL18:11;
      then reconsider r = h * corestr f as continuous map of Y, X by TOPS_2:58;
      take c, r;
A5:   rng h = [#]X & h is one-to-one by A4,TOPS_2:def 5;
      thus r * c = h * (corestr f * (incl Image f * h")) by RELAT_1:55
        .= h * (corestr f * incl Image f * h") by RELAT_1:55
        .= h * (id Image f * h") by A2,YELLOW14:36
        .= h * h" by GRCAT_1:12
        .= id rng h by A5,TOPS_2:65
        .= id the carrier of X by A5,PRE_TOPC:12
        .= id X by GRCAT_1:def 11;
    end;
    given c being continuous map of X, Y,
          r being continuous map of Y, X such that
A6:   r * c = id X;
    take f = c * r;
    thus f is continuous;
    thus f * f = c * (r * (c * r)) by RELAT_1:55
      .= c * (id X * r) by A6,RELAT_1:55
      .= f by GRCAT_1:12;
A7: r * c = id the carrier of X by A6,GRCAT_1:def 11;
A8: dom c = the carrier of X by FUNCT_2:def 1
         .= rng r by A7,FUNCT_2:24;
A9: Image f = Y|rng f by WAYBEL18:def 6
           .= Y|rng c by A8,RELAT_1:47
           .= Image c by WAYBEL18:def 6;
    then reconsider cf = corestr c as map of X, Image f;
    take h = cf";
    thus dom h = the carrier of Image f by FUNCT_2:def 1
       .= [#]Image f by PRE_TOPC:12;
A10: rng corestr c = the carrier of Image c by FUNCT_2:def 3
       .= [#]Image c by PRE_TOPC:12;
      c is one-to-one by A7,FUNCT_2:29;
then A11: corestr c is one-to-one by WAYBEL18:def 7;
    hence rng h = [#]X by A9,A10,TOPS_2:62;
      (corestr c) qua Function" is one-to-one by A11,FUNCT_1:62;
    hence h is one-to-one by A9,A10,A11,TOPS_2:def 4;
A12: corestr c is continuous by WAYBEL18:11;
      r * cf * cf" = id X * cf" by A6,WAYBEL18:def 7;
    then r * (cf * cf") = id X * cf" by RELAT_1:55;
    then r * id rng cf = id X * cf" by A9,A10,A11,TOPS_2:65;
    then r * id the carrier of Image c = id X * cf" by A10,PRE_TOPC:12;
    then r * id Image c = id X * cf" by GRCAT_1:def 11;
then A13: r * id Image c = cf" by GRCAT_1:12;
      the carrier of Image c c= the carrier of Y by BORSUK_1:35;
    then id Image c is Function of the carrier of Image c, the carrier of Y
      by FUNCT_2:9;
    then r * id Image c is Function of the carrier of Image c, the carrier of
X
      by FUNCT_2:19;
    then reconsider q = r * id Image c as map of Image f, X by A9;
      for P being Subset of X st P is open holds q"P is open
    proof
      let P be Subset of X;
      assume
A14:     P is open;
A15:   q"P = (id Image c)"(r"P) by RELAT_1:181;
        r"P is open by A14,TOPS_2:55;
then A16:   r"P in the topology of Y by PRE_TOPC:def 5;
A17:   dom id Image c = the carrier of Image c by FUNCT_2:def 1
           .= [#]Image c by PRE_TOPC:12;
        (id Image c)"(r"P) = (r"P) /\ [#]Image c
      proof
        thus (id Image c)"(r"P) c= (r"P) /\ [#]Image c
        proof
          let a be set;
          assume
A18:         a in (id Image c)"(r"P);
then A19:       a in dom id Image c by FUNCT_1:def 13;
            (id Image c).a in r"P by A18,FUNCT_1:def 13;
          then a in r"P by A18,GRCAT_1:11;
          hence a in (r"P) /\ [#]Image c by A17,A19,XBOOLE_0:def 3;
        end;
        let a be set;
        assume a in (r"P) /\ [#]Image c;
then A20:     a in r"P & a in [#]Image c by XBOOLE_0:def 3;
        then (id Image c).a in r"P by GRCAT_1:11;
        hence thesis by A17,A20,FUNCT_1:def 13;
      end;
      hence q"P in the topology of Image f by A9,A15,A16,PRE_TOPC:def 9;
    end;
    hence h is continuous by A13,TOPS_2:55;
    thus h" is continuous by A9,A10,A11,A12,TOPS_2:64;
  end;
end;

theorem Th8:
 for S, T, X, Y being non empty TopSpace st
   the TopStruct of S = the TopStruct of T &
   the TopStruct of X = the TopStruct of Y &
   S is_Retract_of X holds
  T is_Retract_of Y
  proof
    let S, T, X, Y be non empty TopSpace such that
A1:   the TopStruct of S = the TopStruct of T and
A2:   the TopStruct of X = the TopStruct of Y;
    given c being continuous map of S, X,
          r being continuous map of X, S such that
A3:   r * c = id S;
    reconsider cc = c as continuous map of T, Y
      by A1,A2,YELLOW12:36;
    reconsider rr = r as continuous map of Y, T
      by A1,A2,YELLOW12:36;
    take cc, rr;
      id S = id the carrier of S & id T = id the carrier of T by GRCAT_1:def 11
;
    hence thesis by A1,A3;
  end;

theorem
   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
    let T, S1, S2 be non empty TopStruct such that
A1:   S1, S2 are_homeomorphic and
A2:   S1 is_Retract_of T;
    consider G being map of S1,S2 such that
A3:   G is_homeomorphism by A1,T_0TOPSP:def 1;
    consider H being map of T,T such that
A4:   H is continuous and
A5:   H*H = H and
A6:   Image H, S1 are_homeomorphic by A2,WAYBEL18:def 8;
    take H;
    consider F being map of Image H,S1 such that
A7:   F is_homeomorphism by A6,T_0TOPSP:def 1;
      G*F is_homeomorphism by A3,A7,TOPS_2:71;
    hence thesis by A4,A5,T_0TOPSP:def 1;
  end;

theorem
   for S, T being non empty TopSpace st T is injective & S is_Retract_of T
  holds S is injective
  proof
    let S, T be non empty TopSpace such that
A1:   T is injective and
A2:   S is_Retract_of T;
    consider h being map of T,T such that
A3:   h is continuous and
A4:   h*h = h and
A5:   Image h, S are_homeomorphic by A2,WAYBEL18:def 8;
A6: corestr h is continuous by A3,WAYBEL18:11;
    set F = corestr h;
      for W being Point of T st W in the carrier of Image h holds F.W = W
    proof
      let W be Point of T;
      assume W in the carrier of Image h;
      then W in rng h by WAYBEL18:10;
      then consider x being set such that
A7:     x in dom h and
A8:     W = h.x by FUNCT_1:def 5;
      thus F.W = h.(h.x) by A8,WAYBEL18:def 7
              .= W by A4,A7,A8,FUNCT_1:23;
    end;
    then F is_a_retraction by BORSUK_1:def 19;
    then Image h is_a_retract_of T by A6,BORSUK_1:def 20;
    then Image h is injective by A1,WAYBEL18:9;
    hence S is injective by A5,Th6;
  end;

::p.126 exercise 3.13, 1 => 2
theorem
   for J being injective non empty TopSpace, Y being non empty TopSpace st
  J is SubSpace of Y holds J is_Retract_of Y
  proof
    let J be injective non empty TopSpace,
        Y be non empty TopSpace; assume
A1:   J is SubSpace of Y;
    then consider f being map of Y, J such that
A2:   f is continuous and
A3:   f|(the carrier of J) = id J by WAYBEL18:def 5;
A4: the carrier of J c= the carrier of Y by A1,BORSUK_1:35;
    then f is Function of the carrier of Y, the carrier of Y by FUNCT_2:9;
    then reconsider ff = f as map of Y, Y ;
      ex ff being map of Y, Y st ff is continuous & ff*ff = ff &
      Image ff, J are_homeomorphic
    proof
      take ff;
      thus ff is continuous by A1,A2,TOPMETR:7;
A5:   dom (ff*ff) = the carrier of Y by FUNCT_2:def 1;
A6:   dom f = the carrier of Y by FUNCT_2:def 1;
        for x being set st x in the carrier of Y holds (f*f).x = f.x
      proof
        let x be set;
        assume
A7:       x in the carrier of Y;
then A8:     f.x in the carrier of J by FUNCT_2:7;
        thus (f*f).x = f.(f.x) by A6,A7,FUNCT_1:23
           .= (id J).(f.x) by A3,A8,FUNCT_1:72
           .= f.x by A8,GRCAT_1:11;
      end;
      hence ff*ff = ff by A5,A6,FUNCT_1:9;
A9:   rng f = the carrier of J
      proof
        thus rng f c= the carrier of J;
        let y be set;
        assume
A10:       y in the carrier of J;
        then y in the carrier of Y by A4;
then A11:     y in dom f by FUNCT_2:def 1;
          f.y = (f|(the carrier of J)).y by A10,FUNCT_1:72
           .= y by A3,A10,GRCAT_1:11;
        hence y in rng f by A11,FUNCT_1:def 5;
      end;
      reconsider M = [#]J as non empty Subset of Y
        by A4,PRE_TOPC:12;
A12:   the carrier of Y|M = [#](Y|M) by PRE_TOPC:12
        .= M by PRE_TOPC:def 10
        .= the carrier of J by PRE_TOPC:12;
        Image ff = Y|rng ff by WAYBEL18:def 6
              .= Y|M by A9,PRE_TOPC:12
              .= the TopStruct of J by A1,A12,TSEP_1:5;
      hence Image ff, J are_homeomorphic by YELLOW14:20;
    end;
    hence thesis by WAYBEL18:def 8;
  end;

:: p.123 proposition 3.5
:: p.124 theorem 3.8 (i, part a)
:: p.126 exercise 3.14
theorem Th12:
 for L being complete continuous LATTICE, T being Scott TopAugmentation of L
  holds T is injective
  proof
    let L be complete continuous LATTICE,
        T be Scott TopAugmentation of L;
    let X be non empty TopSpace,
        f be map of X, T such that
A1:   f is continuous;
    let Y be non empty TopSpace such that
A2:   X is SubSpace of Y;
deffunc F(set) =
  "\/"
({inf (f.:(V /\ the carrier of X)) where V is open Subset of Y: $1 in V},T);
    consider g being Function of the carrier of Y, the carrier of T such that
A3:   for x being Element of Y holds g.x = F(x) from LambdaD;
    reconsider g as map of Y, T ;
    take g;
      for P being Subset of T st P is open holds g"P is open
    proof
      let P be Subset of T;
      assume P is open;
      then reconsider P as open Subset of T;
        for x being set holds x in g"P iff
        ex Q being Subset of Y st Q is open & Q c= g"P & x in Q
      proof
        let x be set;
        thus x in g"P implies
          ex Q being Subset of Y st Q is open & Q c= g"P & x in Q
        proof
          assume
A4:         x in g"P;
          then reconsider y = x as Point of Y;
A5:       g.y in P by A4,FUNCT_2:46;
          set A = {inf (f.:
(V /\ the carrier of X)) where V is open Subset of Y:
                   y in V};
            A c= the carrier of T
          proof
            let a be set;
            assume a in A;
            then ex V being open Subset of Y st a = inf (f.:(V /\ the carrier
of X))
              & y in V;
            hence thesis;
          end;
          then reconsider A as Subset of T;
A6:       g.y = sup A by A3;
            the carrier of Y = [#]Y by PRE_TOPC:12;
then A7:       inf (f.:([#]Y /\ the carrier of X)) in A;
            A is directed
          proof
            let a, b be Element of T;
            assume a in A;
            then consider Va being open Subset of Y such that
A8:           a = inf (f.:(Va /\ the carrier of X)) and
A9:           y in Va;
            assume b in A;
            then consider Vb being open Subset of Y such that
A10:           b = inf (f.:(Vb /\ the carrier of X)) and
A11:           y in Vb;
            take inf (f.:(Va /\ Vb /\ the carrier of X));
A12:         Va /\ Vb is open by TOPS_1:38;
              y in Va /\ Vb by A9,A11,XBOOLE_0:def 3;
            hence inf (f.:(Va /\ Vb /\ the carrier of X)) in A by A12;
              Va /\ Vb c= Va & Va /\ Vb c= Vb by XBOOLE_1:17;
            then Va /\ Vb /\ the carrier of X c= Va /\ the carrier of X &
              Va /\ Vb /\ the carrier of X c= Vb /\ the carrier of X
                by XBOOLE_1:26;
            then f.:(Va /\ Vb /\ the carrier of X) c= f.:(Va /\ the carrier of
X) &
              f.:(Va /\ Vb /\ the carrier of X) c= f.:(Vb /\ the carrier of X)
                by RELAT_1:156;
            hence thesis by A8,A10,WAYBEL_7:3;
          end;
          then A meets P by A5,A6,A7,WAYBEL11:def 1;
          then consider b being set such that
A13:         b in A and
A14:         b in P by XBOOLE_0:3;
          consider B being open Subset of Y such that
A15:         b = inf (f.:(B /\ the carrier of X)) and
A16:         y in B by A13;
          reconsider b as Element of T by A15;
          take B;
          thus B is open;
          thus B c= g"P
          proof
            let a be set;
            assume
A17:           a in B;
            then reconsider a as Point of Y;
A18:         g.a = F(a) by A3;
              b in {inf (f.:(V /\ the carrier of X)) where V is open Subset of
Y
:
                a in V} by A15,A17;
            then b <= g.a by A18,YELLOW_2:24;
            then g.a in P by A14,WAYBEL_0:def 20;
            hence thesis by FUNCT_2:46;
          end;
          thus x in B by A16;
        end;
        thus thesis;
      end;
      hence thesis by TOPS_1:57;
    end;
    hence g is continuous by TOPS_2:55;
    set gX = g|(the carrier of X);
A19: the carrier of X c= the carrier of Y by A2,BORSUK_1:35;
A20: dom gX = dom g /\ the carrier of X by RELAT_1:90
          .= (the carrier of Y) /\ the carrier of X by FUNCT_2:def 1
          .= the carrier of X by A19,XBOOLE_1:28;
A21: dom f = the carrier of X by FUNCT_2:def 1;
      for a being set st a in the carrier of X holds gX.a = f.a
    proof
      let a be set; assume
A22:     a in the carrier of X;
      then reconsider x = a as Point of X;
      reconsider y = x as Point of Y by A19,A22;
      set A = {inf (f.:(V /\ the carrier of X)) where V is open Subset of Y:
               y in V};
        A c= the carrier of T
      proof
        let a be set;
        assume a in A;
        then ex V being open Subset of Y st a = inf (f.:(V /\ the carrier of X
))
          & y in V;
        hence thesis;
      end;
      then reconsider A as Subset of T;
A23:   f.x is_>=_than A
      proof
        let z be Element of T;
        assume z in A;
        then consider V being open Subset of Y such that
A24:       z = inf (f.:(V /\ the carrier of X)) and
A25:       y in V;
A26:     ex_inf_of f.:(V /\ the carrier of X),T by YELLOW_0:17;
          y in V /\ the carrier of X by A25,XBOOLE_0:def 3;
        then f.x in f.:(V /\ the carrier of X) by FUNCT_2:43;
        hence z <= f.x by A24,A26,YELLOW_4:2;
      end;
A27:   for b being Element of T st b is_>=_than A holds f.x <= b
      proof
        let b be Element of T such that
A28:       for k being Element of T st k in A holds k <= b;
A29:     for V being open Subset of X st x in V holds inf (f.:V) <= b
        proof
          let V be open Subset of X;
          assume
A30:         x in V;
            V in the topology of X by PRE_TOPC:def 5;
          then consider Q being Subset of Y such that
A31:         Q in the topology of Y and
A32:         V = Q /\ [#]X by A2,PRE_TOPC:def 9;
          reconsider Q as open Subset of Y by A31,PRE_TOPC:def 5
;
A33:       y in Q by A30,A32,XBOOLE_0:def 3;
A34:       V = Q /\ the carrier of X by A32,PRE_TOPC:12;
            inf (f.:(Q /\ the carrier of X)) in A by A33;
          hence thesis by A28,A34;
        end;
A35:     b is_>=_than waybelow f.x
        proof
          let w be Element of T;
          assume w in waybelow f.x;
          then w << f.x by WAYBEL_3:7;
          then f.x in wayabove w by WAYBEL_3:8;
then A36:       x in f"wayabove w by FUNCT_2:46;
            wayabove w is open by WAYBEL11:36;
          then f"wayabove w is open by A1,TOPS_2:55;
then A37:       inf (f.:f"wayabove w) <= b by A29,A36;
A38:       ex_inf_of wayabove w,T & ex_inf_of f.:f"wayabove w,T by YELLOW_0:17;
            f.:f"wayabove w c= wayabove w by FUNCT_1:145;
then A39:       inf wayabove w <= inf (f.:f"wayabove w) by A38,YELLOW_0:35;
            w <= inf wayabove w by WAYBEL14:9;
          then w <= inf (f.:f"wayabove w) by A39,ORDERS_1:26;
          hence w <= b by A37,ORDERS_1:26;
        end;
          f.x = sup waybelow f.x by WAYBEL_3:def 5;
        hence f.x <= b by A35,YELLOW_0:32;
      end;
      thus gX.a = g.y by FUNCT_1:72
        .= F(y) by A3
        .= f.a by A23,A27,YELLOW_0:30;
    end;
    hence g|(the carrier of X) = f by A20,A21,FUNCT_1:9;
  end;

definition let L be complete continuous LATTICE;
 cluster Scott -> injective TopAugmentation of L;
coherence by Th12;
end;

definition let T be injective non empty TopSpace;
 cluster the TopStruct of T -> injective;
coherence by WAYBEL18:17;
end;

begin  :: Specialization order

::p.124 definition 3.6
definition let T be TopStruct;
 func Omega T -> strict TopRelStr means :Def2:
  the TopStruct of it = the TopStruct 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
  proof
    defpred P[set, set] means ex Y being Subset of T st Y = {$2} & $1 in Cl Y;
    consider R being Relation of the carrier of T such that
A1:   for x,y being set holds [x,y] in R iff
      x in the carrier of T & y in the carrier of T & P[x,y]
       from Rel_On_Set_Ex;
    take G = TopRelStr (# the carrier of T, R, the topology of T #);
    thus the TopStruct of G = the TopStruct of T;
    thus for x,y being Element of G holds x <= y iff
         ex Y being Subset of T st Y = {y} & x in Cl Y
    proof
      let x, y be Element of G;
      hereby assume
          x <= y;
        then [x,y] in R by ORDERS_1:def 9;
        hence ex Y being Subset of T st Y = {y} & x in Cl Y by A1;
      end;
      given Y being Subset of T such that
A2:      Y = {y} & x in Cl Y;
        [x,y] in R by A1,A2;
      hence x <= y by ORDERS_1:def 9;
    end;
  end;
uniqueness
 proof
  let R1,R2 be strict TopRelStr such that
A3: the TopStruct of R1 = the TopStruct of T and
A4: for x,y being Element of R1 holds x <= y iff
      ex Y being Subset of T st Y = {y} & x in Cl Y and
A5: the TopStruct of R2 = the TopStruct of T and
A6: for x,y being Element of R2 holds x <= y iff
      ex Y being Subset of T st Y = {y} & x in Cl Y;
      the InternalRel of R1 = the InternalRel of R2
     proof let x,y be set;
      hereby assume
     A7: [x,y] in the InternalRel of R1;
     then A8: x in the carrier of R1 & y in the carrier of R1 by ZFMISC_1:106;
       then reconsider x1 = x, y1 = y as Element of R1;
       reconsider x2 = x, y2 = y as Element of R2 by A3,A5,A8;
           x1 <= y1 by A7,ORDERS_1:def 9;
       then consider Y being Subset of T such that
     A9: Y = {y1} & x1 in Cl Y by A4;
           x2 <= y2 by A6,A9;
       hence [x,y] in the InternalRel of R2 by ORDERS_1:def 9;
      end; assume
    A10: [x,y] in the InternalRel of R2;
    then A11: x in the carrier of R2 & y in the carrier of R2 by ZFMISC_1:106;
      then reconsider x2 = x, y2 = y as Element of R2;
      reconsider x1 = x, y1 = y as Element of R1 by A3,A5,A11;
          x2 <= y2 by A10,ORDERS_1:def 9;
      then consider Y being Subset of T such that
    A12: Y = {y2} & x2 in Cl Y by A6;
          x1 <= y1 by A4,A12;
      hence [x,y] in the InternalRel of R1 by ORDERS_1:def 9;
     end;
  hence thesis by A3,A5;
 end;
end;

Lm1:
 for T being TopStruct holds the carrier of T = the carrier of Omega T
  proof
    let T be TopStruct;
      the TopStruct of T = the TopStruct of Omega T by Def2;
    hence the carrier of T = the carrier of Omega T;
  end;

Lm2:
    for T be TopStruct,
        a be set holds a is Subset of T iff a is Subset of Omega T by Lm1;

definition let T be empty TopStruct;
 cluster Omega T -> empty;
coherence
  proof
      the carrier of Omega T = the carrier of T by Lm1;
    hence the carrier of Omega T is empty;
  end;
end;

definition let T be non empty TopStruct;
 cluster Omega T -> non empty;
coherence
  proof
      the carrier of Omega T = the carrier of T by Lm1;
    hence the carrier of Omega T is non empty;
  end;
end;

definition let T be TopSpace;
 cluster Omega T -> TopSpace-like;
coherence
  proof
A1: the TopStruct of Omega T = the TopStruct of T by Def2;
    hence the carrier of Omega T in the topology of Omega T by PRE_TOPC:def 1;
    thus thesis by A1,PRE_TOPC:def 1;
  end;
end;

definition let T be TopStruct;
 cluster Omega T -> reflexive;
coherence
  proof
    let x be set;
    assume
A1:   x in the carrier of Omega T;
      the carrier of Omega T = the carrier of T by Lm1;
    then reconsider T' = T as non empty TopStruct by A1,STRUCT_0:def 1;
    reconsider t' = x as Element of T' by A1,Lm1;
    reconsider a = x as Element of Omega T by A1;
    consider Y being Subset of T such that
A2:   Y = {t'};
A3: Y c= Cl Y by PRE_TOPC:48;
      a in Y by A2,TARSKI:def 1;
    then a <= a by A2,A3,Def2;
    hence [x,x] in the InternalRel of Omega T by ORDERS_1:def 9;
  end;
end;

Lm3:
 for T being TopStruct, x, y being Element of T,
     X, Y being Subset of T st X = {x} & Y = {y} holds
  x in Cl Y iff Cl X c= Cl Y
  proof
    let T be TopStruct,
        x, y be Element of T,
        X, Y be Subset of T;
    assume
A1:   X = {x} & Y = {y};
    hereby assume
        x in Cl Y;
      then for V being Subset of T st V is open holds (x in V implies y in V)
        by A1,YELLOW14:22;
      hence Cl X c= Cl Y by A1,YELLOW14:23;
    end;
    assume Cl X c= Cl Y;
    hence x in Cl Y by A1,YELLOW14:21;
  end;

definition let T be TopStruct;
 cluster Omega T -> transitive;
coherence
  proof
    let x, y, z be set;
    assume that
A1:   x in the carrier of Omega T & y in the carrier of Omega T &
      z in the carrier of Omega T and
A2:   [x,y] in the InternalRel of Omega T and
A3:   [y,z] in the InternalRel of Omega T;
A4: the TopStruct of Omega T = the TopStruct of T by Def2;
    reconsider a = x, b = y, c = z as Element of Omega T by A1;
    reconsider t2=y, t3=z as Element of T by A1,A4;
      a <= b by A2,ORDERS_1:def 9;
    then consider Y2 being Subset of T such that
A5:   Y2 = {b} and
A6:   a in Cl Y2 by Def2;
      b <= c by A3,ORDERS_1:def 9;
    then consider Y3 being Subset of T such that
A7:   Y3 = {c} and
A8:   b in Cl Y3 by Def2;
      Y2 = {t2} & Y3 = {t3} by A5,A7;
    then Cl Y2 c= Cl Y3 by A8,Lm3;
    then a <= c by A6,A7,Def2;
    hence [x,z] in the InternalRel of Omega T by ORDERS_1:def 9;
  end;
end;

definition let T be T_0-TopSpace;
 cluster Omega T -> antisymmetric;
coherence
  proof
    let x, y be set;
    assume that
A1:   x in the carrier of Omega T & y in the carrier of Omega T and
A2:   [x,y] in the InternalRel of Omega T and
A3:   [y,x] in the InternalRel of Omega T;
    reconsider a = x, b = y as Element of Omega T by A1;
      the TopStruct of Omega T = the TopStruct of T by Def2;
    then reconsider t1 = x, t2 = y as Element of T by A1;
      a <= b by A2,ORDERS_1:def 9;
    then consider Y2 being Subset of T such that
A4:   Y2 = {b} and
A5:   a in Cl Y2 by Def2;
      b <= a by A3,ORDERS_1:def 9;
    then consider Y1 being Subset of T such that
A6:   Y1 = {a} and
A7:   b in Cl Y1 by Def2;
      for V being Subset of T holds not V is open or
      (t1 in V implies t2 in V) & (t2 in V implies t1 in V)
        by A4,A5,A6,A7,YELLOW14:22;
    hence x = y by T_0TOPSP:def 7;
  end;
end;

theorem Th13:
 for S, T being TopSpace st the TopStruct of S = the TopStruct of T
  holds Omega S = Omega T
  proof
    let S, T be TopSpace such that
A1:   the TopStruct of S = the TopStruct of T;
A2: the TopStruct of Omega S = the TopStruct of S by Def2
      .= the TopStruct of Omega T by A1,Def2;
      the InternalRel of Omega S = the InternalRel of Omega T
    proof
      let a,b be set;
      thus [a,b] in the InternalRel of Omega S implies
       [a,b] in the InternalRel of Omega T
      proof
        assume
A3:       [a,b] in the InternalRel of Omega S;
        then a in the carrier of Omega S & b in the carrier of Omega S
          by ZFMISC_1:106;
        then reconsider s1 = a, s2 = b as Element of Omega S;
        reconsider t1 = s1, t2 = s2 as Element of Omega T by A2;
          s1 <= s2 by A3,ORDERS_1:def 9;
        then consider Y being Subset of S such that
A4:       Y = {s2} & s1 in Cl Y by Def2;
        reconsider Z = Y as Subset of T by A1;
          Z = {t2} & t1 in Cl Z by A1,A4,TOPS_3:80;
        then t1 <= t2 by Def2;
        hence thesis by ORDERS_1:def 9;
      end;
      assume
A5:     [a,b] in the InternalRel of Omega T;
      then a in the carrier of Omega T & b in the carrier of Omega T
        by ZFMISC_1:106;
      then reconsider s1 = a, s2 = b as Element of Omega T;
      reconsider t1 = s1, t2 = s2 as Element of Omega S by A2;
        s1 <= s2 by A5,ORDERS_1:def 9;
      then consider Y being Subset of T such that
A6:     Y = {s2} & s1 in Cl Y by Def2;
      reconsider Z = Y as Subset of S by A1;
        Z = {t2} & t1 in Cl Z by A1,A6,TOPS_3:80;
      then t1 <= t2 by Def2;
      hence thesis by ORDERS_1:def 9;
    end;
    hence Omega S = Omega T by A2;
  end;

theorem
   for M being non empty set, T being non empty TopSpace holds
  the RelStr of Omega product(M --> T) = the RelStr of product(M --> Omega T)
  proof
    let M be non empty set,
        T be non empty TopSpace;
A1: dom Carrier (M --> T) = M by PBOOLE:def 3
      .= dom Carrier (M --> Omega T) by PBOOLE:def 3;
A2: for i being set st i in dom Carrier (M --> T) holds
       (Carrier (M --> T)).i = (Carrier (M --> Omega T)).i
    proof
      let i be set;
      assume
          i in dom Carrier (M --> T);
then A3:   i in M by PBOOLE:def 3;
      then consider R1 being 1-sorted such that
A4:   R1 = (M --> T).i and
A5:   (Carrier (M --> T)).i = the carrier of R1 by PRALG_1:def 13;
      consider R2 being 1-sorted such that
A6:   R2 = (M --> Omega T).i and
A7:   (Carrier (M --> Omega T)).i = the carrier of R2 by A3,PRALG_1:def 13;
        the carrier of R1 = the carrier of T by A3,A4,FUNCOP_1:13
         .= the carrier of Omega T by Lm1
         .= the carrier of R2 by A3,A6,FUNCOP_1:13;
      hence (Carrier (M --> T)).i = (Carrier (M --> Omega T)).i by A5,A7;
    end;
A8: the carrier of Omega product (M --> T)
     = the carrier of product (M --> T) by Lm1
    .= product Carrier (M --> T) by WAYBEL18:def 3
    .= product Carrier (M --> Omega T) by A1,A2,FUNCT_1:9;
then A9:  the carrier of Omega product (M --> T)
     = the carrier of product (M --> Omega T) by YELLOW_1:def 4;
A10: the carrier of Omega product (M --> T)
     = the carrier of product (M --> T) by Lm1;
      the InternalRel of Omega product(M --> T) =
        the InternalRel of product (M --> Omega T)
    proof
      let x, y be set;
      hereby
        assume
A11:      [x,y] in the InternalRel of Omega product(M --> T);
then A12:     x in the carrier of Omega product(M --> T) &
         y in the carrier of Omega product(M --> T) by ZFMISC_1:106;
        then reconsider xo = x, yo = y as Element of Omega product(M --> T)
         ;
        reconsider p1 = x, p2 = y as Element of product(M --> T)
          by A10,A12;
        reconsider xp = x, yp = y as Element of product(M --> Omega T)
          by A9,A12;
          xo <= yo by A11,ORDERS_1:def 9;
        then consider Yp being Subset of product(M --> T) such that
A13:       Yp = {p2} and
A14:       p1 in Cl Yp by Def2;
A15:     xp in product Carrier (M --> Omega T) by A8,A11,ZFMISC_1:106;
        then consider f being Function such that
A16:       xp = f and
            dom f = dom Carrier (M --> Omega T) and
            for i being set st i in dom Carrier (M --> Omega T) holds
           f.i in Carrier (M --> Omega T).i by CARD_3:def 5;
          yp in product Carrier (M --> Omega T) by A8,A11,ZFMISC_1:106;
        then consider g being Function such that
A17:       yp = g and
            dom g = dom Carrier (M --> Omega T) and
            for i being set st i in dom Carrier (M --> Omega T) holds
           g.i in Carrier (M --> Omega T).i by CARD_3:def 5;
          for i being set st i in M
         ex R being RelStr, xi, yi being Element of R
           st R = (M --> Omega T).i & xi = f.i & yi = g.i & xi <= yi
        proof
          let i be set;
          assume
A18:         i in M;
          then reconsider j = i as Element of M;
A19:       p1.j in Cl {p2.j} by A13,A14,YELLOW14:31;
          set t1 = p1.j, t2 = p2.j;
            t1 in the carrier of T & t2 in the carrier of T;
          then t1 in the carrier of Omega T &
          t2 in the carrier of Omega T by Lm1;
          then reconsider xi = t1, yi = t2 as Element of Omega T;
          take Omega T, xi, yi;
          thus Omega T = (M --> Omega T).i by A18,FUNCOP_1:13;
          thus xi = f.i by A16;
          thus yi = g.i by A17;
          thus xi <= yi by A19,Def2;
        end;
        then xp <= yp by A15,A16,A17,YELLOW_1:def 4;
        hence [x,y] in the InternalRel of product (M --> Omega T)
          by ORDERS_1:def 9;
      end;
      assume
A20:     [x,y] in the InternalRel of product (M --> Omega T);
then A21:   x in the carrier of product (M --> Omega T) &
      y in the carrier of product (M --> Omega T) by ZFMISC_1:106;
      then reconsider xp = x, yp = y as Element of product(M --> Omega T)
       ;
      reconsider xo = x, yo = y as Element of Omega product(M --> T)
        by A9,A21;
      reconsider p1 = x, p2 = y as Element of product(M --> T)
        by A9,A10,A21;
A22:   xp in product Carrier (M --> Omega T) by A21,YELLOW_1:def 4;
        xp <= yp by A20,ORDERS_1:def 9;
      then consider f, g being Function such that
A23:     f = xp and
A24:     g = yp and
A25:     for i being set st i in M ex R being RelStr, xi, yi being Element of R
          st R = (M --> Omega T).i & xi = f.i & yi = g.i & xi <= yi
             by A22,YELLOW_1:def 4;
        for i being Element of M holds p1.i in Cl {p2.i}
      proof
        let i be Element of M;
        consider R1 being RelStr,
                xi, yi being Element of R1 such that
A26:       R1 = (M --> Omega T).i and
A27:       xi = f.i & yi = g.i and
A28:       xi <= yi by A25;
        reconsider xi, yi as Element of Omega T by A26,FUNCOP_1:13;
          xi <= yi by A26,A28,FUNCOP_1:13;
        then ex Y being Subset of T st Y = {yi} & xi in Cl Y by Def2;
        hence p1.i in Cl {p2.i} by A23,A24,A27;
      end;
      then p1 in Cl {p2} by YELLOW14:31;
      then xo <= yo by Def2;
      hence [x,y] in the InternalRel of Omega product(M --> T)
        by ORDERS_1:def 9;
    end;
    hence thesis by A8,YELLOW_1:def 4;
  end;

::p.124 theorem 3.8 (i, part b)
theorem Th15:
 for S being Scott complete TopLattice holds Omega S = the TopRelStr of S
  proof
    let S be Scott complete TopLattice;
    consider L being Scott TopAugmentation of S;
A1: the RelStr of S = the RelStr of L by YELLOW_9:def 4;
A2: the TopStruct of Omega S = the TopStruct of S by Def2;
      the InternalRel of Omega S = the InternalRel of S
    proof
      let x, y be set;
      hereby
        assume
A3:       [x,y] in the InternalRel of Omega S;
then A4:     x in the carrier of Omega S &
        y in the carrier of Omega S by ZFMISC_1:106;
        then reconsider o1 = x, o2 = y as Element of Omega S;
          o1 <= o2 by A3,ORDERS_1:def 9;
        then consider Y2 being Subset of S such that
A5:       Y2 = {o2} & o1 in Cl Y2 by Def2;
          x in the carrier of S & y in the carrier of S by A4,Lm1;
        then reconsider t1 = x, t2 = y as Element of S;
          t1 in downarrow t2 by A5,WAYBEL11:9;
        then t1 in downarrow {t2} by WAYBEL_0:def 17;
        then consider t3 being Element of S such that
A6:       t3 >= t1 & t3 in {t2} by WAYBEL_0:def 15;
          t1 <= t2 by A6,TARSKI:def 1;
        hence [x,y] in the InternalRel of S by ORDERS_1:def 9;
      end;
      assume
A7:     [x,y] in the InternalRel of S;
then A8:   x in the carrier of S & y in the carrier of S by ZFMISC_1:106;
A9:   x in the carrier of L & y in the carrier of L by A1,A7,ZFMISC_1:106;
      reconsider t1 = x, t2 = y as Element of S by A8;
A10:   t1 <= t2 by A7,ORDERS_1:def 9;
        t2 in {t2} by TARSKI:def 1;
      then t1 in downarrow {t2} by A10,WAYBEL_0:def 15;
      then t1 in downarrow t2 by WAYBEL_0:def 17;
then A11:   t1 in Cl {t2} by WAYBEL11:9;
      reconsider o1 = x, o2 = y as Element of Omega S
       by A1,A2,A9;
        o1 <= o2 by A11,Def2;
      hence [x,y] in the InternalRel of Omega S by ORDERS_1:def 9;
    end;
    hence thesis by A2;
  end;

::p.124 theorem 3.8 (i, part b)
theorem Th16:
 for L being complete LATTICE,
     S being Scott TopAugmentation of L holds
  the RelStr of Omega S = the RelStr of L
  proof
    let L be complete LATTICE,
        S be Scott TopAugmentation of L;
      the TopRelStr of S = Omega S by Th15;
    hence thesis by YELLOW_9:def 4;
  end;

definition let S be Scott complete TopLattice;
 cluster Omega S -> complete;
coherence
  proof
    consider T being Scott TopAugmentation of S;
A1: the RelStr of Omega T = the RelStr of S by Th16;
      the topology of S = sigma S by WAYBEL14:23
      .= the topology of T by YELLOW_9:51;
    then the TopStruct of S = the TopStruct of T by A1,Lm1;
    then Omega S = Omega T by Th13;
    hence thesis by A1,YELLOW_0:3;
  end;
end;

theorem Th17:
 for T being non empty TopSpace, S being non empty SubSpace of T holds
  Omega S is full SubRelStr of Omega T
  proof
    let T be non empty TopSpace,
        S be non empty SubSpace of T;
A1: the carrier of S c= the carrier of T by BORSUK_1:35;
A2: the carrier of Omega S = the carrier of S by Lm1;
A3: the carrier of Omega T = the carrier of T by Lm1;
A4: the carrier of Omega S c= the carrier of Omega T by A1,A2,Lm1;
A5: the InternalRel of Omega S c= the InternalRel of Omega T
    proof
      let x, y be set;
      assume
A6:     [x,y] in the InternalRel of Omega S;
then A7:   x in the carrier of Omega S &
       y in the carrier of Omega S by ZFMISC_1:106;
      then reconsider o1 = x, o2 = y as Element of Omega S;
        o1 <= o2 by A6,ORDERS_1:def 9;
      then consider Y2 being Subset of S such that
A8:     Y2 = {o2} and
A9:     o1 in Cl Y2 by Def2;
      reconsider s2 = y as Element of S by A2,A7;
      reconsider t2 = y as Element of T by A1,A2,A7;
        Cl {s2} = (Cl {t2}) /\ ([#]S) by PRE_TOPC:47;
then A10:   Cl {s2} c= Cl {t2} by XBOOLE_1:17;
      reconsider o3 = x, o4 = y as Element of Omega T
        by A1,A2,A3,A7;
        o3 <= o4 by A8,A9,A10,Def2;
      hence [x,y] in the InternalRel of Omega T by ORDERS_1:def 9;
    end;
      the InternalRel of Omega S
        = (the InternalRel of Omega T)|_2 the carrier of Omega S
    proof
      let x, y be set;
      thus [x,y] in the InternalRel of Omega S
        implies [x,y] in (the InternalRel of Omega T)|_2 the carrier of Omega S
          by A5,WELLORD1:16;
      assume
A11:     [x,y] in (the InternalRel of Omega T)|_2 the carrier of Omega S;
then A12:   [x,y] in the InternalRel of Omega T by WELLORD1:16;
A13:   x in the carrier of Omega S &
      y in the carrier of Omega S by A11,ZFMISC_1:106;
      then reconsider o3 = x, o4 = y as Element of Omega S;
      reconsider s1 = x, s2 = y as Element of S by A2,A13;
      reconsider t1 = x, t2 = y as Element of T by A1,A2,A13;
      reconsider o1 = x, o2 = y as Element of Omega T by A1,A2,A3,A13;
        o1 <= o2 by A12,ORDERS_1:def 9;
      then consider Y being Subset of T such that
A14:     Y = {t2} and
A15:     t1 in Cl Y by Def2;
A16:   Cl {s2} = (Cl {t2}) /\ [#]S by PRE_TOPC:47;
        t1 in [#]S by A2,A13,PRE_TOPC:12;
      then s1 in Cl {s2} by A14,A15,A16,XBOOLE_0:def 3;
      then o3 <= o4 by Def2;
      hence [x,y] in the InternalRel of Omega S by ORDERS_1:def 9;
    end;
    hence Omega S is full SubRelStr of Omega T by A4,A5,YELLOW_0:def 13,def 14
;
  end;

theorem Th18:
 for S, T being TopSpace, h being map of S, T, g being map of Omega S, Omega T
  st h = g & h is_homeomorphism holds g is isomorphic
  proof
    let S, T be TopSpace,
        h be map of S, T,
        g be map of Omega S, Omega T;
    assume that
A1:   h = g and
A2:   h is_homeomorphism;
A3: the carrier of S = the carrier of Omega S &
      the carrier of T = the carrier of Omega T by Lm1;
A4: dom h = [#]S by A2,TOPS_2:def 5;
A5: h is one-to-one & rng h = [#]T by A2,TOPS_2:def 5;
    per cases;
    suppose
A6:   S is non empty & T is non empty;
    then reconsider s = S, t = T as non empty TopSpace;
    reconsider f = g as map of Omega s, Omega t;
A7: rng f = the carrier of Omega t by A1,A3,A5,PRE_TOPC:12;
      for x, y being Element of Omega s holds x <= y iff f.x <= f.y
    proof
      let x, y be Element of Omega s;
        dom h = the carrier of S by A4,PRE_TOPC:12;
then A8:   dom h = the carrier of Omega s by Lm1;
      hereby
        assume x <= y;
        then consider Y being Subset of s such that
A9:       Y = {y} and
A10:       x in Cl Y by Def2;
        reconsider Z = {f.y} as Subset of t by A3;
          f.x in f.:Cl Y by A10,FUNCT_2:43;
then A11:     h.x in Cl (h.:Y) by A1,A2,TOPS_2:74;
          h.:Y = Z by A1,A8,A9,FUNCT_1:117;
        hence f.x <= f.y by A1,A11,Def2;
      end;
      assume f.x <= f.y;
      then consider Y being Subset of t such that
A12:     Y = {f.y} and
A13:     f.x in Cl Y by Def2;
      reconsider Z = {f".(f.y)} as Subset of s by A3;
A14:   h" is_homeomorphism by A2,A6,TOPS_2:70;
A15:   rng f = [#]Omega t by A7,PRE_TOPC:12;
then A16:   f" = f qua Function" by A1,A5,TOPS_2:def 4
        .= h" by A1,A5,TOPS_2:def 4;
        f".(f.x) in f".:Cl Y by A13,FUNCT_2:43;
then A17:   h".(h.x) in Cl (h".:Y) by A1,A14,A16,TOPS_2:74;
A18:   x = (f qua Function)".(f.x) by A1,A5,A8,FUNCT_1:56
       .= f".(f.x) by A1,A5,A15,TOPS_2:def 4;
A19:   y = (h qua Function)".(h.y) by A5,A8,FUNCT_1:56
       .= f".(f.y) by A1,A5,A15,TOPS_2:def 4;
A20:   dom f = [#]S by A1,A2,TOPS_2:def 5
           .= the carrier of S by PRE_TOPC:12;
        f".:Y = f qua Function".:Y by A1,A5,A15,TOPS_2:def 4
          .= f"Y by A1,A5,FUNCT_1:155
          .= Z by A1,A3,A5,A12,A19,A20,FINSEQ_5:4;
      hence x <= y by A1,A16,A17,A18,A19,Def2;
    end;
    hence thesis by A1,A5,A7,WAYBEL_0:66;
    suppose S is empty or T is empty;
    then reconsider s = S, t = T as empty TopSpace by A4,A5,YELLOW14:19;
      Omega s is empty & Omega t is empty;
    hence thesis by WAYBEL_0:def 38;
  end;

theorem Th19:
 for S, T being TopSpace st S, T are_homeomorphic holds
  Omega S, Omega T are_isomorphic
  proof
    let S, T be TopSpace;
    assume S,T are_homeomorphic;
    then consider h being map of S, T such that
A1:   h is_homeomorphism by T_0TOPSP:def 1;
      the carrier of S = the carrier of Omega S &
      the carrier of T = the carrier of Omega T by Lm1;
    then reconsider f = h as map of Omega S, Omega T ;
    take f;
    thus thesis by A1,Th18;
  end;

Lm4:
 for S, T being non empty RelStr, f being map of S, S, g being map of T, T
  st the RelStr of S = the RelStr of T & f = g & f is projection holds
 g is projection
  proof
    let S, T be non empty RelStr,
        f be map of S,S,
        g be map of T,T;
    assume the RelStr of S = the RelStr of T & f = g &
      f is idempotent monotone;
    hence g is idempotent monotone by WAYBEL_9:1;
  end;

::p.124 proposition 3.7
::p.124 theorem 3.8 (ii, part a)
theorem Th20:
 for T being injective T_0-TopSpace holds
  Omega T is complete continuous LATTICE
  proof
    set S = Sierpinski_Space,
        B = BoolePoset 1;
    let T be injective T_0-TopSpace;
    consider M being non empty set such that
A1:   T is_Retract_of product (M --> S) by WAYBEL18:20;
    consider f being map of product(M --> S), product(M --> S) such that
A2:   f is continuous and
A3:   f*f = f and
A4:   Image f, T are_homeomorphic by A1,WAYBEL18:def 8;
    consider TL being Scott TopAugmentation of product(M --> B);
A5: the topology of TL = the topology of product (M --> S) by WAYBEL18:16;
A6: the RelStr of TL = the RelStr of product(M --> B) by YELLOW_9:def 4;
A7: the carrier of TL = the carrier of product(M --> S) by Th3;
    then reconsider ff = f as map of TL, TL ;
  the TopStruct of the TopStruct of TL = the TopStruct of TL implies
     Omega the TopStruct of TL = Omega TL by Th13;
then A8: the RelStr of Omega the TopStruct of product(M --> S)
      = the RelStr of product (M --> B) by A5,A7,Th16;
      ff is continuous by A2,A5,A7,YELLOW12:36;
then A9: ff is directed-sups-preserving by WAYBEL17:22;
      ff is projection
    proof
      thus ff is idempotent monotone by A3,A9,QUANTAL1:def 9,WAYBEL17:3;
    end;
    then reconsider g = ff as projection map of product (M --> B),
     product (M --> B) by A6,Lm4;
      g is directed-sups-preserving by A6,A9,WAYBEL21:6;
then A10: Image g is complete continuous LATTICE by WAYBEL15:17,YELLOW_2:37;
A11: the RelStr of Omega Image f, Omega Image f are_isomorphic by WAYBEL13:26;
      Omega Image f, Omega T are_isomorphic by A4,Th19;
then A12: the RelStr of Omega Image f, Omega T are_isomorphic by A11,WAYBEL_1:8
;
A13: the carrier of Image g = rng g by YELLOW_2:11
      .= the carrier of Image f by WAYBEL18:10
      .= the carrier of Omega Image f by Lm1;
A14: the InternalRel of Image g = (the InternalRel of
      (product (M --> B)))|_2 the carrier of Image g
        by YELLOW_0:def 14;
      Omega Image f is full SubRelStr of Omega product (M --> S) by Th17;
    then the InternalRel of Omega Image f = (the InternalRel of
      (Omega product (M --> S)))|_2 the carrier of Omega Image f
        by YELLOW_0:def 14;
    hence thesis by A8,A10,A12,A13,A14,WAYBEL15:11,WAYBEL20:18,YELLOW14:12,13;
  end;

definition let T be injective T_0-TopSpace;
 cluster Omega T -> complete continuous;
coherence by Th20;
end;

theorem Th21:
 for X, Y being non empty TopSpace,
     f being continuous map of Omega X, Omega Y holds
  f is monotone
  proof
    let X, Y be non empty TopSpace,
        f be continuous map of Omega X, Omega Y;
    let x, y be Element of Omega X;
    assume x <= y;
    then consider A being Subset of X such that
A1:   A = {y} and
A2:   x in Cl A by Def2;
A3: the carrier of Y = the carrier of Omega Y by Lm1;
    then reconsider Z = {f.y} as Subset of Y;
      for G being Subset of Y st G is open holds f.x in G implies Z meets G
    proof
      let G be Subset of Y such that
A4:     G is open and
A5:     f.x in G;
     the carrier of X = the carrier of Omega X by Lm1;
      then reconsider g = f as map of X, Y by A3;
        the TopStruct of X = the TopStruct of Omega X &
        the TopStruct of Y = the TopStruct of Omega Y by Def2;
      then g is continuous by YELLOW12:36;
then A6:   g"G is open by A4,TOPS_2:55;
        x in g"G by A5,FUNCT_2:46;
      then A meets g"G by A2,A6,PRE_TOPC:def 13;
      then consider m being set such that
A7:     m in A /\ g"G by XBOOLE_0:4;
A8:   m in A & m in g"G by A7,XBOOLE_0:def 3;
      then m = y by A1,TARSKI:def 1;
      then f.y in Z & f.y in G by A8,FUNCT_2:46,TARSKI:def 1;
      then Z /\ G <> {}Y by XBOOLE_0:def 3;
      hence Z meets G by XBOOLE_0:def 7;
    end;
    then f.x in Cl Z by A3,PRE_TOPC:def 13;
    hence f.x <= f.y by Def2;
  end;

definition let X, Y be non empty TopSpace;
 cluster continuous -> monotone map of Omega X, Omega Y;
coherence by Th21;
end;

theorem Th22:
 for T being non empty TopSpace, x being Element of Omega T
  holds Cl {x} = downarrow x
  proof
    let T be non empty TopSpace,
        x be Element of Omega T;
A1: the TopStruct of T = the TopStruct of Omega T by Def2;
    hereby
      let a be set;
      assume
A2:     a in Cl {x};
      then reconsider b = a as Element of Omega T;
      reconsider Z = {x} as Subset of T by A1;
        a in Cl Z by A1,A2,TOPS_3:80;
      then b <= x by Def2;
      hence a in downarrow x by WAYBEL_0:17;
    end;
    let a be set;
    assume
A3:   a in downarrow x;
    then reconsider b = a as Element of Omega T;
      b <= x by A3,WAYBEL_0:17;
    then ex Z being Subset of T st Z = {x} & b in Cl Z by Def2;
    hence a in Cl {x} by A1,TOPS_3:80;
  end;

definition let T be non empty TopSpace,
               x be Element of Omega T;
 cluster Cl {x} -> non empty lower directed;
coherence
  proof
    reconsider y = x as Element of Omega T;
      Cl {y} = downarrow y by Th22;
    hence thesis;
  end;
 cluster downarrow x -> closed;
coherence
  proof
      Cl {x} = downarrow x by Th22;
    hence thesis;
  end;
end;

theorem Th23:
 for X being TopSpace, A being open Subset of Omega X holds A is upper
  proof
    let X be TopSpace,
        A be open Subset of Omega X;
    let x, y be Element of Omega X such that
A1:   x in A;
    assume x <= y;
    then consider Z being Subset of X such that
A2:   Z = {y} & x in Cl Z by Def2;
      X is non empty
    proof
      thus the carrier of X is non empty by A2;
    end;
    then reconsider X as non empty TopSpace;
    reconsider y as Element of Omega X;
      the TopStruct of X = the TopStruct of Omega X by Def2;
    then x in Cl {y} by A2,TOPS_3:80;
    then A meets {y} by A1,PRE_TOPC:def 13;
    hence thesis by ZFMISC_1:56;
  end;

definition let T be TopSpace;
 cluster open -> upper Subset of Omega T;
coherence by Th23;
end;

Lm5:
now
  let X, Y be non empty TopSpace,
      N be net of ContMaps(X,Omega Y);
A1:the mapping of N in Funcs(the carrier of N,
     the carrier of ContMaps(X,Omega Y)) by FUNCT_2:11;
A2: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 set;
    assume f in the carrier of ContMaps(X,Omega Y);
    then ex x being map of X, Omega Y st x = f & x is continuous by WAYBEL24:
def 3;
    hence thesis by A2,FUNCT_2:11;
  end;
  then 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_5:63;
  hence the mapping of N in Funcs(the carrier of N,
   Funcs(the carrier of X, the carrier of Y)) by A1;
end;

definition let I be non empty set,
               S, T be non empty RelStr,
               N be net of T,
               i be Element of I such that
A1: the carrier of T c= the carrier of S |^ I;
 func commute(N,i,S) -> strict net of S means :Def3:
  the RelStr of it = the RelStr of N &
  the mapping of it = (commute the mapping of N).i;
existence
  proof
A2: the mapping of N in
 Funcs(the carrier of N, the carrier of T) by FUNCT_2:11;
      the carrier of T c= Funcs(I, the carrier of S) by A1,YELLOW_1:28;
    then Funcs(the carrier of N, the carrier of T) c=
      Funcs(the carrier of N, Funcs(I, the carrier of S)) by FUNCT_5:63;
    then dom ((commute the mapping of N).i) = the carrier of N &
      rng ((commute the mapping of N).i) c= the carrier of S by A2,EQUATION:3;
    then reconsider f = (commute(the mapping of N)).i
      as Function of the carrier of N, the carrier of S
      by FUNCT_2:def 1,RELSET_1:11;
    set A = NetStr (#the carrier of N, the InternalRel of N, f#);
A3: the RelStr of A = the RelStr of N;
      A is directed
    proof
A4:   [#]N is directed by WAYBEL_0:def 6;
        [#]N = the carrier of N by PRE_TOPC:12
        .= [#]A by PRE_TOPC:12;
      hence [#]A is directed by A3,A4,WAYBEL_0:3;
    end;
    then reconsider A as strict net of S by A3,WAYBEL_8:13;
    take A;
    thus thesis;
  end;
uniqueness;
end;

theorem Th24:
 for X, Y being non empty TopSpace,
     N being net of ContMaps(X,Omega Y),
     i being Element of N,
     x being Point of X holds
  (the mapping of commute(N,x,Omega Y)).i = (the mapping of N).i.x
  proof
    let X, Y be non empty TopSpace,
        N be net of ContMaps(X,Omega Y),
        i be Element of N,
        x be Point of X;
A1:  the mapping of N in Funcs(the carrier of N,
                Funcs(the carrier of X, the carrier of Y)) by Lm5;
       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;
     hence (the mapping of commute(N,x,Omega Y)).i
         = (commute the mapping of N).x.i by Def3
        .= ((the mapping of N).i).x by A1,PRALG_2:5;
  end;

theorem Th25:
 for X, Y being non empty TopSpace,
     N being eventually-directed net of ContMaps(X,Omega Y),
     x being Point of X holds
  commute(N,x,Omega Y) is eventually-directed
  proof
    let X, Y be non empty TopSpace,
        N be eventually-directed net of ContMaps(X,Omega Y),
        x be Point of X;
    set M = commute(N,x,Omega Y),
        L = (Omega Y) |^ the carrier of X;
      for i being Element of M ex j being Element of M st
     for k being Element of M st j <= k holds M.i <= M.k
    proof
      let i be Element of M;
A1:   ContMaps(X,Omega Y) is SubRelStr of L 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 A2:   the RelStr of N = the RelStr of M by Def3;
      then reconsider a = i as Element of N;
      consider b being Element of N such that
A3:     for c being Element of N st b <= c holds N.a <= N.c by WAYBEL_0:11;
      reconsider j = b as Element of M by A2;
      take j;
      let k be Element of M;
      reconsider c = k as Element of N by A2;
      reconsider Na = N.a, Nc = N.c as Element of L by A1,YELLOW_0:59;
      reconsider A = Na, C = Nc as Element of product((the carrier of X)
        --> Omega Y) by YELLOW_1:def 5;
      assume j <= k;
      then b <= c by A2,YELLOW_0:1;
      then N.a <= N.c by A3;
      then Na <= Nc by A1,YELLOW_0:60;
      then A <= C by YELLOW_1:def 5;
then A4:   A.x <= C.x by WAYBEL_3:28;
A5:   (the mapping of M).a = (the mapping of N).i.x &
        (the mapping of M).c = (the mapping of N).k.x by Th24;
        M.i = (the mapping of M).a & M.k = (the mapping of M).c &
       A.x = (the mapping of N).i.x & C.x = (the mapping of N).k.x
        by WAYBEL_0:def 8;
      hence M.i <= M.k by A4,A5,FUNCOP_1:13;
    end;
    hence thesis by WAYBEL_0:11;
  end;

definition let X, Y be non empty TopSpace,
               N be eventually-directed net of ContMaps(X,Omega Y),
               x be Point of X;
 cluster commute(N,x,Omega Y) -> eventually-directed;
coherence by Th25;
end;

definition let X, Y be non empty TopSpace;
 cluster -> Function-yielding net of ContMaps(X,Omega Y);
coherence;
end;

Lm6:
now
  let X, Y be non empty TopSpace,
      N be net of ContMaps(X,Omega Y),
      i be Element of N;
  thus
A1: (the mapping of N).i is map of X, Omega Y by WAYBEL24:21;
    the carrier of Y = the carrier of Omega Y by Lm1;
  hence (the mapping of N).i is map of X, Y by A1;
end;

Lm7:
now
  let X, Y be non empty TopSpace,
      N be net of ContMaps(X,Omega Y),
      x be Point of X;
    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:the RelStr of N = the RelStr 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;
end;

theorem Th26:
 for X being non empty TopSpace,
     Y being T_0-TopSpace,
     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
    let X be non empty TopSpace,
        Y be T_0-TopSpace,
        N be net of ContMaps(X,Omega Y) such that
A1:   for x being Point of X holds ex_sup_of commute(N,x,Omega Y);
    set n = the mapping of N,
        L = (Omega Y) |^ the carrier of X;
    deffunc F(Element of X) = sup commute(N,$1,Omega Y);
    consider f being Function of the carrier of X, the carrier of Omega Y
      such that
A2:  for u being Element of X holds f.u = F(u) from LambdaD;
     reconsider a = f as Element of L by WAYBEL24:19;
      ex a being Element of L st rng n is_<=_than a &
      for b being Element of L st rng n is_<=_than b holds a <= b
    proof
A3:   dom n = the carrier of N by FUNCT_2:def 1;
A4:   L = product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
      take a;
      thus rng n is_<=_than a
      proof
        let k be Element of L;
        assume k in rng n;
        then consider i being set such that
A5:       i in dom n and
A6:       k = n.i by FUNCT_1:def 5;
        reconsider i as Point of N by A5;
        reconsider k' = k, a' = a as Element of
          product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
          for u being Element of X holds k'.u <= a'.u
        proof
          let u be Element of X;
A7:       Omega Y = ((the carrier of X) --> Omega Y).u by FUNCOP_1:13;
            ex_sup_of commute(N,u,Omega Y) by A1;
then A8:       ex_sup_of rng the mapping of commute(N,u,Omega Y), Omega Y
            by WAYBEL_9:def 3;
A9:       dom the mapping of commute(N,u,Omega Y) = the carrier of N
            by A3,Lm7;
A10:       a'.u = sup commute(N,u,Omega Y) by A2
              .= Sup the mapping of commute(N,u,Omega Y) by WAYBEL_2:def 1
              .= sup rng the mapping of commute(N,u,Omega Y) by YELLOW_2:def 5;
            k'.u = (the mapping of commute(N,u,Omega Y)).i by A6,Th24;
          then k'.u in rng the mapping of commute(N,u,Omega Y) by A9,FUNCT_1:
def 5;
          hence k'.u <= a'.u by A7,A8,A10,YELLOW_4:1;
        end;
        hence k <= a by A4,WAYBEL_3:28;
      end;
      let b be Element of L such that
A11:     for k being Element of L st k in rng n holds k <= b;
      reconsider a' = a, b' = b as Element of
        product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
        for u being Element of X holds a'.u <= b'.u
      proof
        let u be Element of X;
A12:     Omega Y = ((the carrier of X) --> Omega Y).u by FUNCOP_1:13;
A13:     a'.u = sup commute(N,u,Omega Y) by A2
            .= Sup the mapping of commute(N,u,Omega Y) by WAYBEL_2:def 1
            .= sup rng the mapping of commute(N,u,Omega Y) by YELLOW_2:def 5;
          ex_sup_of commute(N,u,Omega Y) by A1;
then A14:     ex_sup_of rng the mapping of commute(N,u,Omega Y), Omega Y
          by WAYBEL_9:def 3;
          rng the mapping of commute(N,u,Omega Y) is_<=_than b.u
        proof
          let s be Element of Omega Y;
          assume s in rng the mapping of commute(N,u,Omega Y);
          then consider i being set such that
A15:         i in dom the mapping of commute(N,u,Omega Y) and
A16:         (the mapping of commute(N,u,Omega Y)).i = s by FUNCT_1:def 5;
          reconsider i as Point of N by A3,A15,Lm7;
A17:       s = n.i.u by A16,Th24;
A18:       n.i in rng n by A3,FUNCT_1:def 5;
            n.i is map of X, Omega Y by WAYBEL24:21;
          then reconsider k = n.i as Element of L by WAYBEL24:19;
          reconsider k' = k as Element of
            product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
            k <= b by A11,A18;
          then k' <= b' by YELLOW_1:def 5;
          hence s <= b.u by A12,A17,WAYBEL_3:28;
        end;
        hence a'.u <= b'.u by A12,A13,A14,YELLOW_0:30;
      end;
      hence a <= b by A4,WAYBEL_3:28;
    end;
    hence ex_sup_of rng n, L by YELLOW_0:15;
  end;

begin  :: Monotone convergence topological spaces

::p.125 definition 3.9
definition let T be non empty TopSpace;
 attr T is monotone-convergence means :Def4:
  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;

theorem Th27:
 for S, T being non empty TopSpace st
  the TopStruct of S = the TopStruct of T & S is monotone-convergence holds
 T is monotone-convergence
  proof
    let S, T be non empty TopSpace such that
A1:   the TopStruct of S = the TopStruct of T and
A2:  for D being non empty directed Subset of Omega S holds
      ex_sup_of D,Omega S &
      for V being open Subset of S st sup D in V holds D meets V;
    let E be non empty directed Subset of Omega T;
A3: Omega S = Omega T by A1,Th13;
    hence ex_sup_of E,Omega T by A2;
    let V be open Subset of T such that
A4:   sup E in V;
    reconsider W = V as Subset of S by A1;
    reconsider D = E as Subset of Omega S by A3;
      W is open & sup D in W by A1,A4,Th13,TOPS_3:76;
    hence E meets V by A2,A3;
  end;

Lm8:
now
  let T be non empty 1-sorted,
      D be non empty Subset of T,
      d be Element of T such that
A1: the carrier of T = {d};
  thus D ={d}
  proof
    thus D c= {d} by A1;
    let a be set;
    assume a in {d};
then A2: a = d by TARSKI:def 1;
    consider x being Element of D;
      x in D;
    hence thesis by A1,A2,TARSKI:def 1;
  end;
end;

definition
 cluster trivial -> monotone-convergence T_0-TopSpace;
coherence
  proof
    let T be T_0-TopSpace;
    assume T is trivial;
    then consider d being Element of T such that
A1:   the carrier of T = {d} by TEX_1:def 1;
    let D be non empty directed Subset of Omega T;
A2: the carrier of T = the carrier of Omega T by Lm1;
    then reconsider d as Element of Omega T;
      D = {d} by A1,A2,Lm8;
    hence ex_sup_of D,Omega T by YELLOW_0:38;
    let V be open Subset of T;
    assume sup D in V;
then A3: V = {d} by A1,Lm8;
      {d} meets {d};
    hence D meets V by A1,A2,A3,Lm8;
  end;
end;

definition
 cluster strict trivial non empty TopSpace;
existence
  proof
    consider A being strict trivial non empty TopSpace;
    take A;
    thus thesis;
  end;
end;

theorem
   for S being monotone-convergence T_0-TopSpace,
     T being T_0-TopSpace st S, T are_homeomorphic holds
   T is monotone-convergence
  proof
    let S be monotone-convergence T_0-TopSpace,
        T be T_0-TopSpace;
    given h being map of S, T such that
A1:   h is_homeomorphism;
    let D be non empty directed Subset of Omega T;
A2: the carrier of S = the carrier of Omega S by Lm1;
   the carrier of T = the carrier of Omega T by Lm1;
    then reconsider f = h as map of Omega S, Omega T by A2;
A3: f is isomorphic by A1,Th18;
then A4: f" is isomorphic by YELLOW14:11;
A5: rng f = the carrier of Omega T by A3,WAYBEL_0:66;
A6: rng h = [#]T & h is one-to-one by A1,TOPS_2:def 5;
A7: rng f = [#]Omega T by A5,PRE_TOPC:12;
      f" is monotone by A4,WAYBEL_0:def 38;
    then f".:D is directed by YELLOW_2:17;
then A8: f"D is non empty directed Subset of Omega S by A6,A7,TOPS_2:68;
    then ex_sup_of f"D,Omega S by Def4;
    then ex_sup_of f.:f"D,Omega T by A3,YELLOW14:17;
    hence
A9:   ex_sup_of D,Omega T by A5,FUNCT_1:147;
    let V be open Subset of T;
A10: h"V is open by A1,TOPGRP_1:26;
      f" is sups-preserving by A4,WAYBEL13:20;
then A11: f" preserves_sup_of D by WAYBEL_0:def 33;
    assume sup D in V;
    then h".sup D in h".:V by FUNCT_2:43;
    then h".sup D in h"V by A6,TOPS_2:68;
    then h qua Function".sup D in h"V by A6,TOPS_2:def 4;
    then f".sup D in h"V by A6,A7,TOPS_2:def 4;
    then sup (f".:D) in h"V by A9,A11,WAYBEL_0:def 31;
    then sup (f"D) in h"V by A6,A7,TOPS_2:68;
    then f"D meets h"V by A8,A10,Def4;
    then consider a being set such that
A12:   a in f"D & a in h"V by XBOOLE_0:3;
    reconsider a as Element of S by A12;
      now take b = h.a;
     thus b in D & b in V by A12,FUNCT_2:46;
    end; hence thesis by XBOOLE_0:3;
  end;

theorem Th29:
 for S being Scott complete TopLattice holds S is monotone-convergence
  proof
    let S be Scott complete TopLattice;
    let D be non empty directed Subset of Omega S;
    thus
A1:   ex_sup_of D,Omega S by YELLOW_0:17;
    let V be open Subset of S such that
A2:   sup D in V;
A3: Omega S = the TopRelStr of S by Th15;
then A4: the RelStr of Omega S = the RelStr of S;
    reconsider E = D as Subset of S by A3;
A5: E is non empty directed Subset of S by A4,WAYBEL_0:3;
      sup E in V by A1,A2,A4,YELLOW_0:26;
    hence D meets V by A5,WAYBEL11:def 1;
  end;

definition let L be complete LATTICE;
 cluster -> monotone-convergence (Scott TopAugmentation of L);
coherence by Th29;
end;

definition let L be complete LATTICE,
               S be Scott TopAugmentation of L;
 cluster the TopStruct of S -> monotone-convergence;
coherence by Th27;
end;

theorem Th30:
 for X being monotone-convergence T_0-TopSpace holds Omega X is up-complete
  proof
    let X be monotone-convergence T_0-TopSpace;
      for A being non empty directed Subset of Omega X holds
     ex_sup_of A,Omega X by Def4;
    hence thesis by WAYBEL_0:75;
  end;

definition let X be monotone-convergence T_0-TopSpace;
 cluster Omega X -> up-complete;
coherence by Th30;
end;

theorem Th31:
 for X being monotone-convergence (non empty TopSpace),
     N being eventually-directed prenet of Omega X holds ex_sup_of N
  proof
    let X be monotone-convergence (non empty TopSpace),
        N be eventually-directed prenet of Omega X;
      rng netmap (N,Omega X) is directed by WAYBEL_2:18;
    then rng the mapping of N is non empty directed by WAYBEL_0:def 7;
    hence ex_sup_of rng the mapping of N,Omega X by Def4;
  end;

theorem Th32:
 for X being monotone-convergence (non empty TopSpace),
     N being eventually-directed net of Omega X holds sup N in Lim N
  proof
    let X be monotone-convergence (non empty TopSpace),
        N be eventually-directed net of Omega X;
      rng netmap (N,Omega X) is directed by WAYBEL_2:18;
    then reconsider D = rng the mapping of N as non empty directed
      Subset of Omega X by WAYBEL_0:def 7;
      for V being a_neighborhood of sup N holds N is_eventually_in V
    proof
      let V be a_neighborhood of sup N;
A1:   the TopStruct of X = the TopStruct of Omega X by Def2;
      then reconsider I = Int V as Subset of X;
A2:   I is open by A1,TOPS_3:76;
        sup N in I by CONNSP_2:def 1;
      then Sup the mapping of N in I by WAYBEL_2:def 1;
      then sup D in I by YELLOW_2:def 5;
      then D meets I by A2,Def4;
      then consider y being set such that
A3:     y in D and
A4:     y in I by XBOOLE_0:3;
      reconsider y as Point of X by A4;
      consider x being set such that
A5:     x in dom the mapping of N and
A6:     (the mapping of N).x = y by A3,FUNCT_1:def 5;
      reconsider x as Element of N by A5;
      consider j being Element of N such that
A7:     for k being Element of N st j <= k holds N.x <= N.k by WAYBEL_0:11;
      take j;
      let k be Element of N;
      assume j <= k;
      then N.x <= N.k by A7;
      then consider Y being Subset of X such that
A8:     Y = {N.k} and
A9:     N.x in Cl Y by Def2;
        y in Cl Y by A6,A9,WAYBEL_0:def 8;
      then Y meets I by A2,A4,PRE_TOPC:def 13;
      then consider m being set such that
A10:     m in Y /\ I by XBOOLE_0:4;
A11:   m in Y & m in I by A10,XBOOLE_0:def 3;
then A12:   m = N.k by A8,TARSKI:def 1;
        Int V c= V by TOPS_1:44;
      hence N.k in V by A11,A12;
    end;
    hence thesis by YELLOW_6:def 18;
  end;

theorem Th33:
 for X being monotone-convergence (non empty TopSpace),
     N being eventually-directed net of Omega X holds
  N is convergent
  proof
    let X be monotone-convergence (non empty TopSpace),
        N be eventually-directed net of Omega X;
    thus Lim N <> {} by Th32;
  end;

definition let X be monotone-convergence (non empty TopSpace);
 cluster -> convergent (eventually-directed net of Omega X);
coherence by Th33;
end;

theorem
   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
    let X be non empty TopSpace such that
A1:  for N being eventually-directed net of Omega X holds
      ex_sup_of N & sup N in Lim N;
    let D be non empty directed Subset of Omega X;
A2: ex_sup_of Net-Str D by A1;
      Net-Str D = NetStr (#D, (the InternalRel of (Omega X))|_2 D,
     (id the carrier of (Omega X))|D#) by WAYBEL17:def 4;
then A3: rng the mapping of Net-Str D = D by YELLOW14:2;
    hence ex_sup_of D,Omega X by A2,WAYBEL_9:def 3;
    let V be open Subset of X such that
A4:   sup D in V;
A5: sup Net-Str D in Lim Net-Str D by A1;
    reconsider Z = V as Subset of Omega X by Lm2;
      Z is a_neighborhood of sup Net-Str D
    proof
A6:   sup Net-Str D = Sup the mapping of Net-Str D by WAYBEL_2:def 1
       .= sup rng the mapping of Net-Str D by YELLOW_2:def 5;
        the TopStruct of X = the TopStruct of Omega X by Def2;
      then Int Z = Int V by TOPS_3:77;
      hence sup Net-Str D in Int Z by A3,A4,A6,TOPS_1:55;
    end;
    then Net-Str D is_eventually_in V by A5,YELLOW_6:def 18;
    then consider i being Element of Net-Str D such that
A7:   for j being Element of Net-Str D st i <= j holds (Net-Str D).j in V
       by WAYBEL_0:def 11;
      now take a = (Net-Str D).i;
      dom the mapping of Net-Str D = the carrier of Net-Str D by FUNCT_2:def 1
;
    then (the mapping of Net-Str D).i in rng the mapping of Net-Str D
      by FUNCT_1:def 5;
     hence a in D by A3,WAYBEL_0:def 8;
     thus a in V by A7;
    end; hence thesis by XBOOLE_0:3;
  end;

::p.125 lemma 3.10
theorem Th35:
 for X being monotone-convergence (non empty TopSpace),
     Y being T_0-TopSpace,
     f being continuous map of Omega X, Omega Y holds
  f is directed-sups-preserving
  proof
    let X be monotone-convergence (non empty TopSpace),
        Y be T_0-TopSpace,
        f be continuous map of Omega X, Omega Y;
    let D be non empty directed Subset of Omega X;
    assume
A1:   ex_sup_of D,Omega X;
A2: the TopStruct of X = the TopStruct of Omega X by Def2;
A3: the TopStruct of Y = the TopStruct of Omega Y by Def2;
      ex a being Element of Omega Y st f.:D is_<=_than a &
      for b being Element of Omega Y st f.:D is_<=_than b holds a <= b
    proof
      take a = f.sup D;
        D is_<=_than sup D by A1,YELLOW_0:def 9;
      hence f.:D is_<=_than a by YELLOW_2:16;
      let b be Element of Omega Y such that
A4:     for c being Element of Omega Y st c in f.:D holds c <= b;
      reconsider Z = {b} as Subset of Y by Lm2;
        for G being Subset of Y st G is open holds
       a in G implies Z meets G
      proof
        let G be Subset of Y such that
A5:       G is open and
A6:       a in G;
A7:     b in {b} by TARSKI:def 1;
        reconsider H = G as open Subset of Omega Y by A3,A5,TOPS_3:76;
          f"H is open by TOPS_2:55;
then A8:     f"H is open Subset of X by A2,TOPS_3:76;
          sup D in f"H by A6,FUNCT_2:46;
        then D meets f"H by A8,Def4;
        then consider c being set such that
A9:       c in D and
A10:       c in f"H by XBOOLE_0:3;
        reconsider c as Point of Omega X by A9;
          f.c in f.:D by A9,FUNCT_2:43;
then A11:     f.c <= b by A4;
          f.c in H by A10,FUNCT_2:46;
        then b in G by A11,WAYBEL_0:def 20;
        hence Z meets G by A7,XBOOLE_0:3;
      end;
      then a in Cl Z by A3,PRE_TOPC:def 13;
      hence a <= b by Def2;
    end;
    hence
A12:   ex_sup_of f.:D,Omega Y by YELLOW_0:15;
    assume
A13:   sup (f.:D) <> f.sup D;
    set V = (downarrow sup (f.:D))`;
A14: f"V is open by TOPS_2:55;
    reconsider fV = f"V as Subset of X by A2;
    reconsider fV as open Subset of X by A2,A14,TOPS_3:76;
      sup (f.:D) <= sup (f.:D);
then A15: sup (f.:D) in downarrow sup (f.:D) by WAYBEL_0:17;
      sup (f.:D) <= f.sup D by A1,A12,WAYBEL17:15;
    then not f.sup D <= sup (f.:D) by A13,ORDERS_1:25;
    then not f.sup D in downarrow sup (f.:D) by WAYBEL_0:17;
    then f.sup D in V by YELLOW_6:10;
    then sup D in fV by FUNCT_2:46;
    then D meets fV by Def4;
    then consider d being set such that
A16:   d in D and
A17:   d in fV by XBOOLE_0:3;
    reconsider d as Element of Omega X by A16;
A18: f.d in V by A17,FUNCT_2:46;
      f.d in f.:D by A16,FUNCT_2:43;
    then f.d <= sup (f.:D) by A12,YELLOW_4:1;
    then sup (f.:D) in V by A18,WAYBEL_0:def 20;
    hence contradiction by A15,YELLOW_6:10;
  end;

definition let X be monotone-convergence (non empty TopSpace),
               Y be T_0-TopSpace;
 cluster continuous -> directed-sups-preserving map of Omega X, Omega Y;
coherence by Th35;
end;

theorem Th36:
 for T being monotone-convergence T_0-TopSpace,
     R being T_0-TopSpace st R is_Retract_of T holds
  R is monotone-convergence
  proof
    let T be monotone-convergence T_0-TopSpace,
        R be T_0-TopSpace;
    given c being continuous map of R, T,
          r being continuous map of T, R such that
A1:   r * c = id R;
    let D be non empty directed Subset of Omega R;
A2: the TopStruct of T = the TopStruct of Omega T by Def2;
A3: the TopStruct of R = the TopStruct of Omega R by Def2;
    then reconsider DR = D as non empty Subset of R;
    reconsider f = c*r as map of Omega T, Omega T by A2;
    reconsider cc = c as map of Omega R, Omega T by A2,A3;
    reconsider rr = r as map of Omega T, Omega R by A2,A3;
      cc is continuous by A2,A3,YELLOW12:36;
    then cc is monotone by Th21;
then A4: cc.:D is directed by YELLOW_2:17;
then A5: ex_sup_of cc.:D, Omega T by Def4;
A6: r.:(c.:D) = (r*c).:DR by RELAT_1:159
           .= D by A1,WAYBEL15:3;
      rr is continuous by A2,A3,YELLOW12:36;
    then rr is directed-sups-preserving by Th35;
then A7: rr preserves_sup_of cc.:D by A4,WAYBEL_0:def 37;
    hence ex_sup_of D, Omega R by A5,A6,WAYBEL_0:def 31;
    let V be open Subset of R such that
A8:   sup D in V;
A9: c.:V c= r"V
    proof
      let a be set;
      assume a in c.:V;
      then consider x being set such that
A10:     x in the carrier of R and
A11:     x in V and
A12:     a = c.x by FUNCT_2:115;
      reconsider x as Point of R by A10;
      A13: a = c.x by A12;
        r.a = (r*c).x by A12,FUNCT_2:21
         .= x by A1,GRCAT_1:11;
      hence a in r"V by A11,A13,FUNCT_2:46;
    end;
A14: r"V is open by TOPS_2:55;
      f is continuous by A2,YELLOW12:36;
    then f is directed-sups-preserving by Th35;
then A15: f preserves_sup_of cc.:D by A4,WAYBEL_0:def 37;
A16: c.sup D = c.(r.sup(cc.:D)) by A5,A6,A7,WAYBEL_0:def 31
           .= f.sup(cc.:D) by A2,FUNCT_2:21
           .= sup(f.:(cc.:D)) by A5,A15,WAYBEL_0:def 31
           .= sup (cc.:D) by A6,RELAT_1:159;
      c.sup D in c.:V by A8,FUNCT_2:43;
    then c.:D meets r"V by A4,A9,A14,A16,Def4;
    then consider d being set such that
A17:   d in cc.:D and
A18:   d in r"V by XBOOLE_0:3;
      now take a = r.d;
     thus a in D by A6,A17,FUNCT_2:43;
     thus a in V by A18,FUNCT_2:46;
    end; hence thesis by XBOOLE_0:3;
  end;

::p.124 theorem 3.8 (ii, part b)
theorem Th37:
 for T being injective T_0-TopSpace, S being Scott TopAugmentation of Omega T
  holds the TopStruct of S = the TopStruct of T
  proof
    let T be injective T_0-TopSpace,
        S be Scott TopAugmentation of Omega T;
    set SS = Sierpinski_Space,
        B = BoolePoset 1;
    consider M being non empty set such that
A1:   T is_Retract_of product (M --> SS) by WAYBEL18:20;
    consider c being continuous map of T, product (M --> SS),
             r being continuous map of product (M --> SS), T such that
A2:   r * c = id T by A1,Def1;
    consider S_2M being Scott TopAugmentation of product (M --> B);
A3: the topology of S_2M = the topology of product (M --> SS) by WAYBEL18:16;
A4: the carrier of S_2M = the carrier of product (M --> SS) by Th3;
    then reconsider rr = r as map of S_2M, T ;
A5: the RelStr of S_2M = the RelStr of product (M --> B)
      by YELLOW_9:def 4;
A6: the TopStruct of T = the TopStruct of Omega T by Def2;
    then reconsider r1 = r as map of product (M --> B), Omega T
      by A4,A5;
    reconsider c1 = c as map of Omega T, product (M --> B)
      by A4,A5,A6;
A7: the TopStruct of product (M --> SS) =
     the TopStruct of Omega product (M --> SS) by Def2;
    then reconsider c1a = c as map of Omega T, Omega product (M --> SS)
      by A6;
A8: the RelStr of S = the RelStr of Omega T by YELLOW_9:def 4;
    then reconsider r2 = r1 as map of S_2M, S by A5;
    reconsider c2 = c1 as map of S, S_2M by A5,A8;
A9: the RelStr of Omega S_2M = the RelStr of product (M --> B) by Th16;
    then reconsider rr1 = r1 as map of Omega S_2M, Omega T ;
A10: the TopStruct of T = the TopStruct of T;
then A11: rr is continuous by A3,A4,YELLOW12:36;
      the TopStruct of Omega S_2M = the TopStruct of S_2M by Def2;
    then rr1 is continuous by A3,A4,A6,YELLOW12:36;
    then rr1 is directed-sups-preserving by Th35;
    then r2 is directed-sups-preserving by A5,A8,A9,WAYBEL21:6;
then A12: r2 is continuous by WAYBEL17:22;
    reconsider r3 = r2 as map of product (M --> SS), S
      by A4;
      the TopStruct of S = the TopStruct of S;
then A13: r3 is continuous by A3,A4,A12,YELLOW12:36;
      T is_Retract_of S_2M by A1,A3,A4,A10,Th8;
    then reconsider W = T as monotone-convergence (non empty TopSpace)
      by Th36;
    reconsider c1a as continuous map of Omega W, Omega product (M --> SS)
      by A6,A7,YELLOW12:36;
      Omega product (M --> SS) = Omega S_2M by A3,A4,Th13;
then A14: the RelStr of Omega product (M --> SS) = the RelStr of product (M -->
B)
        by Th16
     .= the RelStr of S_2M by YELLOW_9:def 4;
      c2 = c1a;
    then c2 is directed-sups-preserving by A8,A14,WAYBEL21:6;
    then c2 is continuous by WAYBEL17:22;
    then r3 * c is continuous & rr * c2 is continuous by A11,A13,TOPS_2:58;
    hence thesis by A2,YELLOW14:34;
  end;

theorem
   for T being injective T_0-TopSpace holds T is compact locally-compact sober
  proof
    let T be injective T_0-TopSpace;
    consider S being Scott TopAugmentation of Omega T;
A1: the TopStruct of S = the TopStruct of T by Th37;
      S is compact locally-compact sober by WAYBEL14:32;
    hence thesis by A1,YELLOW14:27,28,29;
  end;

theorem Th39:
 for T being injective T_0-TopSpace holds T is monotone-convergence
  proof
    let T be injective T_0-TopSpace;
    consider S being Scott TopAugmentation of Omega T;
      the TopStruct of S = the TopStruct of T by Th37;
    hence T is monotone-convergence by Th27;
  end;

definition
 cluster injective -> monotone-convergence T_0-TopSpace;
coherence by Th39;
end;

theorem Th40:
 for X being non empty TopSpace,
     Y being monotone-convergence T_0-TopSpace,
     N being net of ContMaps(X,Omega Y),
     f, g being map 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
    let X be non empty TopSpace,
        Y be monotone-convergence T_0-TopSpace,
        N be net of ContMaps(X,Omega Y),
        f, g be map of X, Omega Y;
    set m = the mapping of N,
        L = (Omega Y) |^ the carrier of X,
        s = "\/"(rng m,L);
    assume that
A1:   f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) and
A2: ex_sup_of rng m,L and
A3:   g in rng the mapping of N;
    reconsider g1 = g as Element of L by WAYBEL24:19;
      rng m is_<=_than s by A2,YELLOW_0:def 9;
    then g1 <= s by A3,LATTICE3:def 9;
    hence g <= f by A1,WAYBEL10:12;
  end;

theorem Th41:
 for X being non empty TopSpace,
     Y being monotone-convergence T_0-TopSpace,
     N being net of ContMaps(X,Omega Y),
     x being Point of X,
     f being map 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
    let X be non empty TopSpace,
        Y be monotone-convergence T_0-TopSpace,
        N be net of ContMaps(X,Omega Y),
        x be Point of X,
        f be map of X, Omega Y such that
A1:   for a being Point of X holds commute(N,a,Omega Y)
        is eventually-directed and
A2:   f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X);
    set n = the mapping of N,
        m = the mapping of commute(N,x,Omega Y),
        L = (Omega Y) |^ the carrier of X;
A3: dom n = the carrier of N by FUNCT_2:def 1;
then A4: dom m = the carrier of N by Lm7;
      for x being Point of X holds ex_sup_of commute(N,x,Omega Y)
    proof
      let x be Point of X;
        commute(N,x,Omega Y) is eventually-directed by A1;
      hence thesis by Th31;
    end;
then A5: ex_sup_of rng n,L by Th26;
A6: rng m is_<=_than f.x
    proof
      let w be Element of Omega Y;
      assume w in rng m;
      then consider i being set such that
A7:     i in dom m and
A8:     m.i = w by FUNCT_1:def 5;
      reconsider i as Point of N by A3,A7,Lm7;
      reconsider g = n.i as map of X, Omega Y by Lm6;
        g in rng n by A3,FUNCT_1:def 5;
      then g <= f by A2,A5,Th40;
      then ex a, b being Element of Omega Y st
        a = g.x & b = f.x & a <= b by YELLOW_2:def 1;
      hence w <= f.x by A8,Th24;
    end;
      for a being Element of Omega Y st rng m is_<=_than a holds f.x <= a
    proof
      let a be Element of Omega Y;
      assume
A9:     for e being Element of Omega Y st e in rng m holds e <= a;
defpred P[set,set] means
 ($1 = x implies $2 = a) &
 ($1 <> x implies ex u being Element of X st $1 = u &
        $2 = sup commute(N,u,Omega Y));
A10:   for e being Element of X
        ex u being Element of Omega Y st P[e,u]
      proof
        let e be Element of X;
        per cases;
        suppose e = x;
        hence thesis;
        suppose
A11:       e <> x;
        reconsider e as Element of X;
        take sup commute(N,e,Omega Y);
        thus thesis by A11;
      end;
      consider t being Function of the carrier of X, the carrier of Omega Y
        such that
A12:    for u being Element of X holds P[u,t.u]
        from FuncExD(A10);
      reconsider t as map of X, Omega Y ;
      reconsider tt = t as Element of L by WAYBEL24:19;
        tt is_>=_than rng n
      proof
        let s be Element of L;
        assume s in rng n;
        then consider i being set such that
A13:       i in dom n and
A14:       s = n.i by FUNCT_1:def 5;
        reconsider i as Point of N by A13;
        reconsider ss = s as map of X, Omega Y by WAYBEL24:19;
A15:     L = product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
        reconsider s' = s, t' = tt as Element of
          product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
          for j being Element of X holds s'.j <= t'.j
        proof
          let j be Element of X;
A16:       ss.j = (the mapping of commute(N,j,Omega Y)).i by A14,Th24;
A17:       Omega Y = ((the carrier of X) --> Omega Y).j by FUNCOP_1:13;
          per cases;
          suppose j <> x;
          then ex u being Element of X st j = u & t.j = sup commute(N,u,Omega
Y)
            by A12;
then A18:       t'.j = Sup the mapping of commute(N,j,Omega Y) by WAYBEL_2:def
1
              .= sup rng the mapping of commute(N,j,Omega Y) by YELLOW_2:def 5;
            commute(N,j,Omega Y) is eventually-directed by A1;
          then ex_sup_of commute(N,j,Omega Y) by Th31;
then A19:       ex_sup_of rng the mapping of commute(N,j,Omega Y),Omega Y
            by WAYBEL_9:def 3;
            i in dom the mapping of commute(N,j,Omega Y) by A13,Lm7;
          then ss.j in rng the mapping of commute(N,j,Omega Y) by A16,FUNCT_1:
def 5;
          hence thesis by A17,A18,A19,YELLOW_4:1;
          suppose
A20:         j = x;
A21:       ss.x = m.i by A14,Th24;
            m.i in rng m by A4,FUNCT_1:def 5;
          then ss.x <= a by A9,A21;
          hence thesis by A12,A17,A20;
        end;
        hence s <= tt by A15,WAYBEL_3:28;
      end;
then A22:   "\/"(rng n,L) <= tt by A5,YELLOW_0:30;
A23:   tt.x = a by A12;
      reconsider p = "\/"(rng n,L), q = tt as
        Element of product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
A24:   Omega Y = ((the carrier of X) --> Omega Y).x by FUNCOP_1:13;
        p <= q by A22,YELLOW_1:def 5;
      hence f.x <= a by A2,A23,A24,WAYBEL_3:28;
    end;
    hence f.x
       = sup rng the mapping of commute(N,x,Omega Y) by A6,YELLOW_0:30
      .= Sup the mapping of commute(N,x,Omega Y) by YELLOW_2:def 5
      .= sup commute(N,x,Omega Y) by WAYBEL_2:def 1;
  end;

::p.125 lemma 3.11
theorem Th42:
 for X being non empty TopSpace,
     Y being monotone-convergence T_0-TopSpace,
     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 map of X, Y
  proof
    let X be non empty TopSpace,
        Y be monotone-convergence T_0-TopSpace,
        N be net of ContMaps(X,Omega Y) such that
A1:   for x being Point of X holds commute(N,x,Omega Y) is eventually-directed;
    set m = the mapping of N,
        L = (Omega Y) |^ the carrier of X;
A2: the TopStruct of Y = the TopStruct of Omega Y by Def2;
    reconsider fo = "\/"(rng m,L) as map of X, Omega Y by WAYBEL24:19;
    reconsider f = fo as map of X, Y by A2;
A3: dom m = the carrier of N by FUNCT_2:def 1;
      for V being Subset of Y st V is open holds f"V is open
    proof
      let V be Subset of Y such that
A4:     V is open;
      set M = {A where A is Subset of X:
        ex i being Element of N, g being map of X, Omega Y st
         g = N.i & A = g"V};
        for x being set holds x in f"V iff x in union M
      proof
        let x be set;
        thus x in f"V implies x in union M
        proof
          assume
A5:         x in f"V;
then A6:       f.x in V by FUNCT_2:46;
          reconsider x as Point of X by A5;
          set NET = commute(N,x,Omega Y);
A7:       m in Funcs(the carrier of N,
                    Funcs(the carrier of X, the carrier of Y)) by Lm5;
            ContMaps(X,Omega Y) is SubRelStr of (Omega Y) |^ the carrier of X
            by WAYBEL24:def 3;
then A8:       the carrier of ContMaps(X,Omega Y) c=
           the carrier of (Omega Y) |^ the carrier of X by YELLOW_0:def 13;
          then A9:       the RelStr of NET = the RelStr of N by Def3;
            NET is eventually-directed by A1;
          then reconsider W = rng netmap(NET,Omega Y)
          as non empty directed Subset of Omega Y by WAYBEL_2:18;
            f.x = sup NET by A1,Th41
             .= Sup the mapping of NET by WAYBEL_2:def 1
             .= sup rng the mapping of NET by YELLOW_2:def 5
             .= sup W by WAYBEL_0:def 7;
          then W meets V by A4,A6,Def4;
          then consider k being set such that
A10:         k in W and
A11:         k in V by XBOOLE_0:3;
          consider i being set such that
A12:         i in dom netmap(NET,Omega Y) and
A13:         k = netmap(NET,Omega Y).i by A10,FUNCT_1:def 5;
          reconsider i as Element of N by A9,A12;
A14:       netmap(NET,Omega Y).i
              = (the mapping of NET).i by WAYBEL_0:def 7
             .= (commute the mapping of N).x.i by A8,Def3
             .= ((the mapping of N).i).x by A7,PRALG_2:5;
          reconsider g = N.i as map of X, Omega Y by WAYBEL24:21;
A15:       k = g.x by A13,A14,WAYBEL_0:def 8;
            dom g = the carrier of X by FUNCT_2:def 1;
then A16:       x in g"V by A11,A15,FUNCT_1:def 13;
            g"V in M;
          hence thesis by A16,TARSKI:def 4;
        end;
        assume x in union M;
        then consider Z being set such that
A17:       x in Z and
A18:       Z in M by TARSKI:def 4;
        consider A being Subset of X such that
A19:       Z = A and
A20:       ex i being Element of N, g being map of X, Omega Y st
           g = N.i & A = g"V by A18;
        consider i being Element of N,
                 g being map of X, Omega Y such that
A21:       g = N.i and
A22:       A = g"V by A20;
A23:     g.x in V by A17,A19,A22,FUNCT_1:def 13;
A24:     dom f = the carrier of X by FUNCT_2:def 1;
        reconsider x as Element of X by A17,A19;
          for x being Point of X holds ex_sup_of commute(N,x,Omega Y)
        proof
          let x be Point of X;
            commute(N,x,Omega Y) is eventually-directed by A1;
          hence thesis by Th31;
        end;
then A25:     ex_sup_of rng m,L by Th26;
          m.i in rng m by A3,FUNCT_1:def 5;
        then N.i in rng m by WAYBEL_0:def 8;
        then g <= fo by A21,A25,Th40;
        then ex a, b being Element of Omega Y st a = g.x & b = fo.x & a <= b
         by YELLOW_2:def 1;
        then consider O being Subset of Y such that
A26:       O = {f.x} and
A27:       g.x in Cl O by Def2;
          V meets O by A4,A23,A27,PRE_TOPC:def 13;
        then consider w being set such that
A28:       w in V /\ O by XBOOLE_0:4;
A29:     w in V & w in O by A28,XBOOLE_0:def 3;
        then w = f.x by A26,TARSKI:def 1;
        hence thesis by A24,A29,FUNCT_1:def 13;
      end;
then A30:   f"V = union M by TARSKI:2;
        M c= bool the carrier of X
      proof
        let m be set;
        assume m in M;
        then ex A being Subset of X st m = A &
         ex i being Element of N, g being map of X, Omega Y st
          g = N.i & A = g"V;
        hence thesis;
      end;
      then reconsider M as Subset-Family of X by SETFAM_1:def 7;
      reconsider M as Subset-Family of X;
        M is open
      proof
        let P be Subset of X;
        assume P in M;
        then consider A being Subset of X such that
A31:       P = A and
A32:       ex i being Element of N, g being map of X, Omega Y st
            g = N.i & A = g"V;
        consider i being Element of N,
                 g being map of X, Omega Y such that
A33:       g = N.i and
A34:       A = g"V by A32;
A35:     the TopStruct of X = the TopStruct of X;
          g in the carrier of ContMaps(X,Omega Y) by A33;
        then consider g' being map of X, Omega Y such that
A36:       g = g' & g' is continuous by WAYBEL24:def 3;
        reconsider g'' = g' as continuous map of X, Y
          by A2,A35,A36,YELLOW12:36;
          g''"V is open by A4,TOPS_2:55;
        hence P is open by A31,A34,A36;
      end;
      hence f"V is open by A30,TOPS_2:26;
    end;
    hence thesis by TOPS_2:55;
  end;

::p.126 lemma 3.12 (i)
theorem
   for X being non empty TopSpace,
     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
    let X be non empty TopSpace,
        Y be monotone-convergence T_0-TopSpace;
    set L = (Omega Y) |^ the carrier of X;
    reconsider C = ContMaps(X,Omega Y)
      as non empty full SubRelStr of L by WAYBEL24:def 3;
      C is directed-sups-inheriting
    proof
      let D be directed Subset of C such that
A1:     D <> {} and ex_sup_of D,L;
      reconsider D as non empty directed Subset of C by A1;
      set N = Net-Str D;
A2:   the TopStruct of Y = the TopStruct of Omega Y by Def2;
A3:   the TopStruct of X = the TopStruct of X;
        for x being Point of X holds commute(N,x,Omega Y) is
eventually-directed
        by Th25;
      then "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X)
        is continuous map of X, Y by Th42;
      then "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X)
        is continuous map of X, Omega Y by A2,A3,YELLOW12:36;
then A4:   "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X) in
        the carrier of C by WAYBEL24:def 3;
        Net-Str D = NetStr (#D, (the InternalRel of C)|_2 D,
       (id the carrier of C)|D#) by WAYBEL17:def 4;
      hence thesis by A4,YELLOW14:2;
    end;
    hence thesis;
  end;

Back to top