Article WAYBEL33, MML version 4.99.1005

:: WAYBEL33:funcnot 1 => WAYBEL33:func 1
definition
  let a1 be non empty reflexive transitive antisymmetric RelStr;
  let a2 be non empty Element of bool the carrier of a1;
  let a3 be non empty filtered upper Element of bool the carrier of BoolePoset a2;
  func lim_inf A3 -> Element of the carrier of a1 equals
    "\/"({"/\"(b1,a1) where b1 is Element of bool the carrier of a1: b1 in a3},a1);
end;

:: WAYBEL33:def 1
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of BoolePoset b2 holds
   lim_inf b3 = "\/"({"/\"(b4,b1) where b4 is Element of bool the carrier of b1: b4 in b3},b1);

:: WAYBEL33:th 1
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being non empty Element of bool the carrier of b1
for b4 being non empty Element of bool the carrier of b2
for b5 being non empty filtered upper Element of bool the carrier of BoolePoset b3
for b6 being non empty filtered upper Element of bool the carrier of BoolePoset b4
      st b5 = b6
   holds lim_inf b5 = lim_inf b6;

:: WAYBEL33:attrnot 1 => WAYBEL33:attr 1
definition
  let a1 be non empty TopRelStr;
  attr a1 is lim-inf means
    the topology of a1 = xi a1;
end;

:: WAYBEL33:dfs 2
definiens
  let a1 be non empty TopRelStr;
To prove
     a1 is lim-inf
it is sufficient to prove
  thus the topology of a1 = xi a1;

:: WAYBEL33:def 2
theorem
for b1 being non empty TopRelStr holds
      b1 is lim-inf
   iff
      the topology of b1 = xi b1;

:: WAYBEL33:condreg 1
registration
  cluster non empty lim-inf -> TopSpace-like (TopRelStr);
end;

:: WAYBEL33:condreg 2
registration
  cluster trivial TopSpace-like reflexive transitive antisymmetric with_suprema with_infima -> lim-inf (TopRelStr);
end;

:: WAYBEL33:exreg 1
registration
  cluster non empty TopSpace-like reflexive transitive antisymmetric continuous with_suprema with_infima complete lim-inf TopRelStr;
end;

:: WAYBEL33:th 2
theorem
for b1, b2 being non empty 1-sorted
   st the carrier of b1 = the carrier of b2
for b3 being NetStr over b1 holds
   ex b4 being strict NetStr over b2 st
      RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
       the mapping of b3 = the mapping of b4;

:: WAYBEL33:th 3
theorem
for b1, b2 being non empty 1-sorted
   st the carrier of b1 = the carrier of b2
for b3 being NetStr over b1
      st b3 in NetUniv b1
   holds ex b4 being non empty transitive strict directed NetStr over b2 st
      b4 in NetUniv b2 &
       RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
       the mapping of b3 = the mapping of b4;

:: WAYBEL33:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being non empty transitive directed NetStr over b1
for b4 being non empty transitive directed NetStr over b2
      st RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
         the mapping of b3 = the mapping of b4
   holds lim_inf b3 = lim_inf b4;

:: WAYBEL33:th 5
theorem
for b1, b2 being non empty 1-sorted
   st the carrier of b1 = the carrier of b2
for b3 being non empty transitive directed NetStr over b1
for b4 being non empty transitive directed NetStr over b2
   st RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
      the mapping of b3 = the mapping of b4
for b5 being subnet of b3 holds
   ex b6 being strict subnet of b4 st
      RelStr(#the carrier of b5,the InternalRel of b5#) = RelStr(#the carrier of b6,the InternalRel of b6#) &
       the mapping of b5 = the mapping of b6;

:: WAYBEL33:th 6
theorem
for b1, b2 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
   st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being NetStr over b1
for b4 being set
      st [b3,b4] in lim_inf-Convergence b1
   holds ex b5 being non empty transitive strict directed NetStr over b2 st
      [b5,b4] in lim_inf-Convergence b2 &
       RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b5,the InternalRel of b5#) &
       the mapping of b3 = the mapping of b5;

:: WAYBEL33:th 7
theorem
for b1, b2 being non empty 1-sorted
for b3 being non empty NetStr over b1
for b4 being non empty NetStr over b2
   st RelStr(#the carrier of b3,the InternalRel of b3#) = RelStr(#the carrier of b4,the InternalRel of b4#) &
      the mapping of b3 = the mapping of b4
for b5 being set
      st b3 is_eventually_in b5
   holds b4 is_eventually_in b5;

:: WAYBEL33:th 8
theorem
for b1, b2 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds ConvergenceSpace lim_inf-Convergence b1 = ConvergenceSpace lim_inf-Convergence b2;

:: WAYBEL33:th 9
theorem
for b1, b2 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
      st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
   holds xi b1 = xi b2;

:: WAYBEL33:condreg 3
registration
  let a1 be non empty reflexive /\-complete RelStr;
  cluster -> /\-complete (TopAugmentation of a1);
end;

:: WAYBEL33:condreg 4
registration
  let a1 be reflexive transitive antisymmetric with_infima RelStr;
  cluster -> with_infima (TopAugmentation of a1);
end;

:: WAYBEL33:exreg 2
registration
  let a1 be reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr;
  cluster non empty reflexive transitive antisymmetric lower-bounded up-complete /\-complete with_infima strict lim-inf TopAugmentation of a1;
end;

:: WAYBEL33:th 10
theorem
for b1 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
for b2 being lim-inf TopAugmentation of b1 holds
   xi b1 = the topology of b2;

:: WAYBEL33:funcnot 2 => WAYBEL33:func 2
definition
  let a1 be reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr;
  func Xi A1 -> strict TopAugmentation of a1 means
    it is lim-inf;
end;

:: WAYBEL33:def 3
theorem
for b1 being reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr
for b2 being strict TopAugmentation of b1 holds
      b2 = Xi b1
   iff
      b2 is lim-inf;

:: WAYBEL33:funcreg 1
registration
  let a1 be reflexive transitive antisymmetric up-complete /\-complete with_infima RelStr;
  cluster Xi a1 -> strict lim-inf;
end;

:: WAYBEL33:th 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty transitive directed NetStr over b1 holds
   lim_inf b2 = "\/"({inf (b2 | b3) where b3 is Element of the carrier of b2: TRUE},b1);

:: WAYBEL33:th 12
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1
for b3 being Element of bool the carrier of b1
   st b3 in b2
for b4 being Element of the carrier of a_net b2
      st b4 `2 = b3
   holds "/\"(b3,b1) = inf ((a_net b2) | b4);

:: WAYBEL33:th 13
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1 holds
   lim_inf b2 = lim_inf a_net b2;

:: WAYBEL33:th 14
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1 holds
   a_net b2 in NetUniv b1;

:: WAYBEL33:th 15
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1
for b3 being Function-like quasi_total greater_or_equal_to_id Relation of the carrier of a_net b2,the carrier of a_net b2 holds
   inf ((a_net b2) * b3) <= lim_inf b2;

:: WAYBEL33:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1
for b3 being subnet of a_net b2 holds
   lim_inf b2 = lim_inf b3;

:: WAYBEL33:th 17
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being set
      st b2 is_often_in b3
   holds ex b4 being strict subnet of b2 st
      rng the mapping of b4 c= b3 & b4 is SubNetStr of b2;

:: WAYBEL33:th 18
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lim-inf TopRelStr
for b2 being non empty Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1
            st b2 in b3
         holds lim_inf b3 in b2;

:: WAYBEL33:th 19
theorem
for b1 being non empty reflexive RelStr holds
   sigma b1 c= xi b1;

:: WAYBEL33:th 20
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being prebasis of b1
      st b3 c= the topology of b2 & the carrier of b1 in the topology of b2
   holds the topology of b1 c= the topology of b2;

:: WAYBEL33:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
   omega b1 c= xi b1;

:: WAYBEL33:th 22
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being non empty TopSpace-like TopStruct
   st b3 is TopExtension of b1 & b3 is TopExtension of b2
for b4 being Refinement of b1,b2 holds
   b3 is TopExtension of b4;

:: WAYBEL33:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2 being TopExtension of b1
for b3 being Element of bool the carrier of b1 holds
   (b3 is open(b1) implies b3 is open Element of bool the carrier of b2) &
    (b3 is closed(b1) implies b3 is closed Element of bool the carrier of b2);

:: WAYBEL33:th 24
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr holds
   lambda b1 c= xi b1;

:: WAYBEL33:th 25
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2 being lim-inf TopAugmentation of b1
for b3 being TopSpace-like Lawson TopAugmentation of b1 holds
   b2 is TopExtension of b3;

:: WAYBEL33:th 26
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lim-inf TopRelStr
for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1 holds
   lim_inf b2 is_a_convergence_point_of b2,b1;

:: WAYBEL33:th 27
theorem
for b1 being TopSpace-like reflexive transitive antisymmetric with_suprema with_infima complete lim-inf TopRelStr holds
   b1 is compact & b1 is being_T1;