Article YELLOW19, MML version 4.99.1005

:: YELLOW19:th 2
theorem
for b1 being non empty set
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset b1
for b3 being set
      st b3 in b2
   holds b3 is not empty;

:: YELLOW19:funcnot 1 => YELLOW19:func 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  func NeighborhoodSystem A2 -> Element of bool the carrier of BoolePoset [#] a1 equals
    {b1 where b1 is a_neighborhood of a2: TRUE};
end;

:: YELLOW19:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   NeighborhoodSystem b2 = {b3 where b3 is a_neighborhood of b2: TRUE};

:: YELLOW19:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being set holds
      b3 in NeighborhoodSystem b2
   iff
      b3 is a_neighborhood of b2;

:: YELLOW19:funcreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  cluster NeighborhoodSystem a2 -> non empty proper filtered upper;
end;

:: YELLOW19:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being upper Element of bool the carrier of BoolePoset [#] b1 holds
      b2 is_a_convergence_point_of b3,b1
   iff
      NeighborhoodSystem b2 c= b3;

:: YELLOW19:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   b2 is_a_convergence_point_of NeighborhoodSystem b2,b1;

:: YELLOW19:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      for b3 being Element of the carrier of b1
         st b3 in b2
      for b4 being non empty filtered upper Element of bool the carrier of BoolePoset [#] b1
            st b3 is_a_convergence_point_of b4,b1
         holds b2 in b4;

:: YELLOW19:modenot 1 => YELLOW19:mode 1
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  mode Subset of A1,A2 -> Element of bool the carrier of a1 means
    ex b1 being Element of the carrier of a2 st
       it = rng the mapping of a2 | b1;
end;

:: YELLOW19:dfs 2
definiens
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a3 is Subset of a1,a2
it is sufficient to prove
  thus ex b1 being Element of the carrier of a2 st
       a3 = rng the mapping of a2 | b1;

:: YELLOW19:def 2
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of bool the carrier of b1 holds
      b3 is Subset of b1,b2
   iff
      ex b4 being Element of the carrier of b2 st
         b3 = rng the mapping of b2 | b4;

:: YELLOW19:th 7
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being Element of the carrier of b2 holds
   rng the mapping of b2 | b3 is Subset of b1,b2;

:: YELLOW19:condreg 1
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty reflexive NetStr over a1;
  cluster -> non empty (Subset of a1,a2);
end;

:: YELLOW19:th 8
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b2
for b4 being set holds
      b4 in rng the mapping of b2 | b3
   iff
      ex b5 being Element of the carrier of b2 st
         b3 <= b5 & b4 = b2 . b5;

:: YELLOW19:th 9
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being Subset of b1,b2 holds
   b2 is_eventually_in b3;

:: YELLOW19:th 10
theorem
for b1 being non empty 1-sorted
for b2 being non empty transitive directed NetStr over b1
for b3 being non empty finite set
      st for b4 being Element of b3 holds
           b4 is Subset of b1,b2
   holds ex b4 being Subset of b1,b2 st
      b4 c= meet b3;

:: YELLOW19:funcnot 2 => YELLOW19:func 2
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  func a_filter A2 -> Element of bool the carrier of BoolePoset [#] a1 equals
    {b1 where b1 is Element of bool the carrier of a1: a2 is_eventually_in b1};
end;

:: YELLOW19:def 3
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1 holds
   a_filter b2 = {b3 where b3 is Element of bool the carrier of b1: b2 is_eventually_in b3};

:: YELLOW19:th 11
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1
for b3 being set holds
      b3 in a_filter b2
   iff
      b2 is_eventually_in b3 & b3 is Element of bool the carrier of b1;

:: YELLOW19:funcreg 2
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  cluster a_filter a2 -> non empty upper;
end;

:: YELLOW19:funcreg 3
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty transitive directed NetStr over a1;
  cluster a_filter a2 -> proper filtered;
end;

:: YELLOW19:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1 holds
      b3 is_a_cluster_point_of b2
   iff
      b3 is_a_cluster_point_of a_filter b2,b1;

:: YELLOW19:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1 holds
      b3 in Lim b2
   iff
      b3 is_a_convergence_point_of a_filter b2,b1;

:: YELLOW19:funcnot 3 => YELLOW19:func 3
definition
  let a1 be non empty 1-sorted;
  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 a_net A3 -> non empty strict NetStr over a1 means
    the carrier of it = {[b1,b2] where b1 is Element of the carrier of a1, b2 is Element of a3: b1 in b2} &
     (for b1, b2 being Element of the carrier of it holds
        b1 <= b2
     iff
        b2 `2 c= b1 `2) &
     (for b1 being Element of the carrier of it holds
        it . b1 = b1 `1);
end;

:: YELLOW19:def 4
theorem
for b1 being non empty 1-sorted
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
for b4 being non empty strict NetStr over b1 holds
      b4 = a_net b3
   iff
      the carrier of b4 = {[b5,b6] where b5 is Element of the carrier of b1, b6 is Element of b3: b5 in b6} &
       (for b5, b6 being Element of the carrier of b4 holds
          b5 <= b6
       iff
          b6 `2 c= b5 `2) &
       (for b5 being Element of the carrier of b4 holds
          b4 . b5 = b5 `1);

:: YELLOW19:funcreg 4
registration
  let a1 be non empty 1-sorted;
  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;
  cluster a_net a3 -> non empty reflexive transitive strict;
end;

:: YELLOW19:funcreg 5
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty Element of bool the carrier of a1;
  let a3 be non empty proper filtered upper Element of bool the carrier of BoolePoset a2;
  cluster a_net a3 -> non empty strict directed;
end;

:: YELLOW19:th 14
theorem
for b1 being non empty 1-sorted
for b2 being non empty filtered upper Element of bool the carrier of BoolePoset [#] b1 holds
   b2 \ {{}} = a_filter a_net b2;

:: YELLOW19:th 15
theorem
for b1 being non empty 1-sorted
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1 holds
   b2 = a_filter a_net b2;

:: YELLOW19:th 16
theorem
for b1 being non empty 1-sorted
for b2 being non empty filtered upper Element of bool the carrier of BoolePoset [#] b1
for b3 being non empty Element of bool the carrier of b1 holds
      b3 in b2
   iff
      a_net b2 is_eventually_in b3;

:: YELLOW19:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1
for b3 being Element of the carrier of b1 holds
      b3 is_a_cluster_point_of a_net b2
   iff
      b3 is_a_cluster_point_of b2,b1;

:: YELLOW19:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1
for b3 being Element of the carrier of b1 holds
      b3 in Lim a_net b2
   iff
      b3 is_a_convergence_point_of b2,b1;

:: YELLOW19:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
   st b2 in Cl b3
for b4 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1
      st b4 = NeighborhoodSystem b2
   holds a_net b4 is_often_in b3;

:: YELLOW19:th 21
theorem
for b1 being non empty 1-sorted
for b2 being set
for b3 being non empty transitive directed NetStr over b1
   st b3 is_eventually_in b2
for b4 being subnet of b3 holds
   b4 is_eventually_in b2;

:: YELLOW19:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being set
      st b2 c= b3 & b4 is_a_convergence_point_of b2,b1
   holds b4 is_a_convergence_point_of b3,b1;

:: YELLOW19:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Cl b2
   iff
      ex b4 being non empty transitive directed NetStr over b1 st
         b4 is_eventually_in b2 & b3 is_a_cluster_point_of b4;

:: YELLOW19:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Cl b2
   iff
      ex b4 being non empty transitive directed convergent NetStr over b1 st
         b4 is_eventually_in b2 & b3 in Lim b4;

:: YELLOW19:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being non empty transitive directed NetStr over b1
         st b3 is_eventually_in b2
      for b4 being Element of the carrier of b1
            st b4 is_a_cluster_point_of b3
         holds b4 in b2;

:: YELLOW19:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being non empty transitive directed convergent NetStr over b1
         st b3 is_eventually_in b2
      for b4 being Element of the carrier of b1
            st b4 in Lim b3
         holds b4 in b2;

:: YELLOW19:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Cl b2
   iff
      ex b4 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1 st
         b2 in b4 & b3 is_a_cluster_point_of b4,b1;

:: YELLOW19:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
      b3 in Cl b2
   iff
      ex b4 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1 st
         b2 in b4 & b3 is_a_convergence_point_of b4,b1;

:: YELLOW19:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1
         st b2 in b3
      for b4 being Element of the carrier of b1
            st b4 is_a_cluster_point_of b3,b1
         holds b4 in b2;

:: YELLOW19:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being 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
      for b4 being Element of the carrier of b1
            st b4 is_a_convergence_point_of b3,b1
         holds b4 in b2;

:: YELLOW19:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty transitive directed NetStr over b1
for b3 being Element of the carrier of b1 holds
      b3 is_a_cluster_point_of b2
   iff
      for b4 being Subset of b1,b2 holds
         b3 in Cl b4;

:: YELLOW19:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is closed(b1)
   holds FinMeetCl b2 is closed(b1);

:: YELLOW19:th 33
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] b1 holds
         ex b3 being Element of the carrier of b1 st
            b3 is_a_convergence_point_of b2,b1;

:: YELLOW19:th 34
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty proper filtered upper Element of bool the carrier of BoolePoset [#] b1 holds
         ex b3 being Element of the carrier of b1 st
            b3 is_a_cluster_point_of b2,b1;

:: YELLOW19:th 35
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty transitive directed NetStr over b1 holds
         ex b3 being Element of the carrier of b1 st
            b3 is_a_cluster_point_of b2;

:: YELLOW19:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty transitive directed NetStr over b1
            st b2 in NetUniv b1
         holds ex b3 being Element of the carrier of b1 st
            b3 is_a_cluster_point_of b2;

:: YELLOW19:condreg 2
registration
  let a1 be non empty 1-sorted;
  let a2 be transitive NetStr over a1;
  cluster full -> transitive (SubNetStr of a2);
end;

:: YELLOW19:exreg 1
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty directed NetStr over a1;
  cluster non empty strict directed full SubNetStr of a2;
end;

:: YELLOW19:th 37
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty transitive directed NetStr over b1 holds
         ex b3 being subnet of b2 st
            b3 is convergent(b1);

:: YELLOW19:attrnot 1 => YELLOW19:attr 1
definition
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  attr a2 is Cauchy means
    for b1 being Element of bool the carrier of a1
          st not a2 is_eventually_in b1
       holds a2 is_eventually_in b1 `;
end;

:: YELLOW19:dfs 5
definiens
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
To prove
     a2 is Cauchy
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st not a2 is_eventually_in b1
       holds a2 is_eventually_in b1 `;

:: YELLOW19:def 5
theorem
for b1 being non empty 1-sorted
for b2 being non empty NetStr over b1 holds
      b2 is Cauchy(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st not b2 is_eventually_in b3
         holds b2 is_eventually_in b3 `;

:: YELLOW19:funcreg 6
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty filtered upper ultra Element of bool the carrier of BoolePoset [#] a1;
  cluster a_net a2 -> non empty strict Cauchy;
end;

:: YELLOW19:th 38
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is compact
   iff
      for b2 being non empty transitive directed NetStr over b1
            st b2 is Cauchy(b1)
         holds b2 is convergent(b1);