Article WAYBEL19, MML version 4.99.1005

:: WAYBEL19:attrnot 1 => WAYBEL19:attr 1
definition
  let a1 be non empty TopRelStr;
  attr a1 is lower means
    {(uparrow b1) ` where b1 is Element of the carrier of a1: TRUE} is prebasis of a1;
end;

:: WAYBEL19:dfs 1
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is lower
it is sufficient to prove
  thus {(uparrow b1) ` where b1 is Element of the carrier of a1: TRUE} is prebasis of a1;

:: WAYBEL19:def 1
theorem
for b1 being non empty TopRelStr holds
      b1 is lower
   iff
      {(uparrow b2) ` where b2 is Element of the carrier of b1: TRUE} is prebasis of b1;

:: WAYBEL19:condreg 1
registration
  cluster non empty trivial TopSpace-like reflexive -> lower (TopRelStr);
end;

:: WAYBEL19:exreg 1
registration
  cluster non empty trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete non void strict lower TopRelStr;
end;

:: WAYBEL19:th 1
theorem
for b1 being non empty RelStr holds
   ex b2 being TopSpace-like strict TopAugmentation of b1 st
      b2 is lower;

:: WAYBEL19:exreg 2
registration
  let a1 be non empty RelStr;
  cluster non empty TopSpace-like strict lower TopAugmentation of a1;
end;

:: WAYBEL19:th 2
theorem
for b1, b2 being non empty TopSpace-like lower TopRelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds the topology of b1 = the topology of b2;

:: WAYBEL19:funcnot 1 => WAYBEL19:func 1
definition
  let a1 be non empty RelStr;
  func omega A1 -> Element of bool bool the carrier of a1 means
    for b1 being TopSpace-like lower TopAugmentation of a1 holds
       it = the topology of b1;
end;

:: WAYBEL19:def 2
theorem
for b1 being non empty RelStr
for b2 being Element of bool bool the carrier of b1 holds
      b2 = omega b1
   iff
      for b3 being TopSpace-like lower TopAugmentation of b1 holds
         b2 = the topology of b3;

:: WAYBEL19:th 3
theorem
for b1, b2 being non empty RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds omega b1 = omega b2;

:: WAYBEL19:th 4
theorem
for b1 being non empty lower TopRelStr
for b2 being Element of the carrier of b1 holds
   (uparrow b2) ` is open(b1) & uparrow b2 is closed(b1);

:: WAYBEL19:th 5
theorem
for b1 being non empty transitive lower TopRelStr
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is lower(b1);

:: WAYBEL19:th 6
theorem
for b1 being non empty transitive lower TopRelStr
for b2 being Element of bool the carrier of b1
      st b2 is closed(b1)
   holds b2 is upper(b1);

:: WAYBEL19:th 7
theorem
for b1 being non empty TopSpace-like TopRelStr holds
      b1 is lower
   iff
      {(uparrow b2) ` where b2 is Element of bool the carrier of b1: b2 is finite} is Basis of b1;

:: WAYBEL19:th 8
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st for b4 being non empty Element of bool the carrier of b1 holds
           b3 preserves_inf_of b4
   holds b3 is continuous(b1, b2);

:: WAYBEL19:th 9
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is infs-preserving(b1, b2)
   holds b3 is continuous(b1, b2);

:: WAYBEL19:th 10
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
for b2 being prebasis of b1
for b3 being non empty filtered Element of bool the carrier of b1
      st for b4 being Element of bool the carrier of b1
              st b4 in b2 & "/\"(b3,b1) in b4
           holds b3 meets b4
   holds "/\"(b3,b1) in Cl b3;

:: WAYBEL19:th 11
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is continuous(b1, b2)
   holds b3 is filtered-infs-preserving(b1, b2);

:: WAYBEL19:th 12
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is continuous(b1, b2) &
         (for b4 being finite Element of bool the carrier of b1 holds
            b3 preserves_inf_of b4)
   holds b3 is infs-preserving(b1, b2);

:: WAYBEL19:th 13
theorem
for b1 being non empty TopSpace-like reflexive transitive lower TopRelStr
for b2 being Element of the carrier of b1 holds
   Cl {b2} = uparrow b2;

:: WAYBEL19:modenot 1
definition
  mode TopPoset is TopSpace-like reflexive transitive antisymmetric TopRelStr;
end;

:: WAYBEL19:condreg 2
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric lower -> discerning (TopRelStr);
end;

:: WAYBEL19:condreg 3
registration
  let a1 be non empty lower-bounded RelStr;
  cluster -> lower-bounded (TopAugmentation of a1);
end;

:: WAYBEL19:th 14
theorem
for b1, b2 being non empty RelStr
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2 holds
   (uparrow [b3,b4]) ` = [:(uparrow b3) `,the carrier of b2:] \/ [:the carrier of b1,(uparrow b4) `:];

:: WAYBEL19:th 15
theorem
for b1, b2 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b3 being TopSpace-like lower TopAugmentation of b1
for b4 being TopSpace-like lower TopAugmentation of b2 holds
   omega [:b1,b2:] = the topology of [:b3,b4:];

:: WAYBEL19:th 16
theorem
for b1, b2 being non empty TopSpace-like reflexive transitive antisymmetric lower-bounded lower TopRelStr holds
omega [:b1,b2:] = the topology of [:b1,b2:];

:: WAYBEL19:th 17
theorem
for b1, b2 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower TopRelStr
   st b2 is TopAugmentation of [:b1,b1:]
for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
      st b3 = inf_op b1
   holds b3 is continuous(b2, b1);

:: WAYBEL19:sch 1
scheme WAYBEL19:sch 1
{F1 -> TopSpace-like reflexive transitive antisymmetric with_suprema with_infima TopRelStr}:
for b1 being Element of bool the carrier of F1()
      st b1 is open(F1())
   holds P1[b1]
provided
   ex b1 being prebasis of F1() st
      for b2 being Element of bool the carrier of F1()
            st b2 in b1
         holds P1[b2]
and
   for b1 being Element of bool bool the carrier of F1()
         st for b2 being Element of bool the carrier of F1()
                 st b2 in b1
              holds P1[b2]
      holds P1[union b1]
and
   for b1, b2 being Element of bool the carrier of F1()
         st P1[b1] & P1[b2]
      holds P1[b1 /\ b2]
and
   P1[[#] F1()];


:: WAYBEL19:th 18
theorem
for b1, b2 being non empty reflexive antisymmetric up-complete RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#) &
         (for b3 being Element of the carrier of b1 holds
            waybelow b3 is directed(b1) & waybelow b3 is not empty) &
         b1 is satisfying_axiom_of_approximation
   holds b2 is satisfying_axiom_of_approximation;

:: WAYBEL19:condreg 4
registration
  let a1 be non empty reflexive transitive antisymmetric continuous RelStr;
  cluster -> continuous (TopAugmentation of a1);
end;

:: WAYBEL19:th 19
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Refinement of b1,b2
for b4 being Element of bool the carrier of b3
      st (b4 in the topology of b1 or b4 in the topology of b2)
   holds b4 is open(b3);

:: WAYBEL19:th 20
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Refinement of b1,b2
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b3
      st b5 = b4 & b4 is open(b1)
   holds b5 is open(b3);

:: WAYBEL19:th 21
theorem
for b1, b2 being TopSpace-like TopStruct
   st the carrier of b1 = the carrier of b2
for b3 being Refinement of b1,b2
for b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b3
      st b5 = b4 & b4 is closed(b1)
   holds b5 is closed(b3);

:: WAYBEL19:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being set
      st b2 c= b3 & b3 c= the topology of b1
   holds (b2 is Basis of b1 implies b3 is Basis of b1) & (b2 is prebasis of b1 implies b3 is prebasis of b1);

:: WAYBEL19:th 23
theorem
for b1, b2 being non empty TopSpace-like TopStruct
   st the carrier of b1 = the carrier of b2
for b3 being Refinement of b1,b2
for b4 being prebasis of b1
for b5 being prebasis of b2 holds
   b4 \/ b5 is prebasis of b3;

:: WAYBEL19:th 24
theorem
for b1, b2, b3, b4 being non empty TopSpace-like TopStruct
for b5 being Refinement of b1,b2
for b6 being Refinement of b3,b4
for b7 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
for b8 being Function-like quasi_total Relation of the carrier of b2,the carrier of b4
for b9 being Function-like quasi_total Relation of the carrier of b5,the carrier of b6
      st b9 = b7 & b9 = b8 & b7 is continuous(b1, b3) & b8 is continuous(b2, b4)
   holds b9 is continuous(b5, b6);

:: WAYBEL19:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1
for b3 being non empty transitive directed NetStr over b1
for b4 being Element of the carrier of b1
      st for b5 being Element of bool the carrier of b1
              st b4 in b5 & b5 in b2
           holds b3 is_eventually_in b5
   holds b4 in Lim b3;

:: WAYBEL19:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of bool the carrier of b1
      st b2 is_eventually_in b3
   holds Lim b2 c= Cl b3;

:: WAYBEL19:th 27
theorem
for b1 being non empty RelStr
for b2 being non empty Element of bool the carrier of b1 holds
   the mapping of b2 +id = id b2 & the mapping of b2 opp+id = id b2;

:: WAYBEL19:th 28
theorem
for b1 being non empty reflexive antisymmetric RelStr
for b2 being Element of the carrier of b1 holds
   (uparrow b2) /\ downarrow b2 = {b2};

:: WAYBEL19:attrnot 2 => WAYBEL19:attr 2
definition
  let a1 be non empty reflexive TopRelStr;
  attr a1 is Lawson means
    (omega a1) \/ sigma a1 is prebasis of a1;
end;

:: WAYBEL19:dfs 3
definiens
  let a1 be non empty reflexive TopRelStr;
To prove
     a1 is Lawson
it is sufficient to prove
  thus (omega a1) \/ sigma a1 is prebasis of a1;

:: WAYBEL19:def 3
theorem
for b1 being non empty reflexive TopRelStr holds
      b1 is Lawson
   iff
      (omega b1) \/ sigma b1 is prebasis of b1;

:: WAYBEL19:th 29
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being TopSpace-like lower TopAugmentation of b1
for b3 being Scott TopAugmentation of b1
for b4 being TopSpace-like TopAugmentation of b1 holds
      b4 is Lawson
   iff
      b4 is Refinement of b3,b2;

:: WAYBEL19:exreg 3
registration
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded up-complete /\-complete non void strict Lawson TopAugmentation of a1;
end;

:: WAYBEL19:exreg 4
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete non void strict Scott TopRelStr;
end;

:: WAYBEL19:exreg 5
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded continuous up-complete /\-complete non void strict Lawson TopRelStr;
end;

:: WAYBEL19:th 30
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
   (sigma b1) \/ {(uparrow b2) ` where b2 is Element of the carrier of b1: TRUE} is prebasis of b1;

:: WAYBEL19:th 31
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
   (sigma b1) \/ {b2 \ uparrow b3 where b2 is Element of bool the carrier of b1, b3 is Element of the carrier of b1: b2 in sigma b1} is prebasis of b1;

:: WAYBEL19:th 32
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr holds
   {b2 \ uparrow b3 where b2 is Element of bool the carrier of b1, b3 is Element of bool the carrier of b1: b2 in sigma b1 & b3 is finite} is Basis of b1;

:: WAYBEL19:funcnot 2 => WAYBEL19:func 2
definition
  let a1 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  func lambda A1 -> Element of bool bool the carrier of a1 means
    for b1 being TopSpace-like Lawson TopAugmentation of a1 holds
       it = the topology of b1;
end;

:: WAYBEL19:def 4
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being Element of bool bool the carrier of b1 holds
      b2 = lambda b1
   iff
      for b3 being TopSpace-like Lawson TopAugmentation of b1 holds
         b2 = the topology of b3;

:: WAYBEL19:th 33
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
   lambda b1 = UniCl FinMeetCl ((sigma b1) \/ omega b1);

:: WAYBEL19:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being TopSpace-like lower TopAugmentation of b1
for b3 being TopSpace-like Scott TopAugmentation of b1
for b4 being Refinement of b3,b2 holds
   lambda b1 = the topology of b4;

:: WAYBEL19:th 35
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima up-complete lower TopRelStr
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is property(S)(b1);

:: WAYBEL19:th 36
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is property(S)(b1);

:: WAYBEL19:th 37
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being TopSpace-like Lawson TopAugmentation of b1
for b3 being Element of bool the carrier of b1
   st b3 is open(b1)
for b4 being Element of bool the carrier of b2
      st b4 = b3
   holds b4 is open(b2);

:: WAYBEL19:th 38
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Element of the carrier of b1 holds
   uparrow b2 is closed(b1) & downarrow b2 is closed(b1) & {b2} is closed(b1);

:: WAYBEL19:th 39
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being Element of the carrier of b1 holds
   (uparrow b2) ` is open(b1) & (downarrow b2) ` is open(b1) & {b2} ` is open(b1);

:: WAYBEL19:th 40
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Lawson TopRelStr
for b2 being Element of the carrier of b1 holds
   wayabove b2 is open(b1) & (wayabove b2) ` is closed(b1);

:: WAYBEL19:th 41
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Scott TopRelStr
for b2 being TopSpace-like Lawson TopAugmentation of b1
for b3 being upper Element of bool the carrier of b2
   st b3 is open(b2)
for b4 being Element of bool the carrier of b1
      st b4 = b3
   holds b4 is open(b1);

:: WAYBEL19:th 42
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being lower Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      b2 is closed_under_directed_sups(b1);

:: WAYBEL19:th 43
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson TopRelStr
for b2 being non empty filtered Element of bool the carrier of b1 holds
   Lim (b2 opp+id) = {"/\"(b2,b1)};

:: WAYBEL19:condreg 5
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete Lawson -> being_T1 compact (TopRelStr);
end;

:: WAYBEL19:condreg 6
registration
  cluster TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete continuous Lawson -> being_T2 (TopRelStr);
end;