Article YELLOW_8, MML version 4.99.1005

:: YELLOW_8:th 1
theorem
for b1, b2, b3 being set
      st b2 in Fin b1 & b3 c= b2
   holds b3 in Fin b1;

:: YELLOW_8:th 2
theorem
for b1 being set
for b2 being Element of bool bool b1
      st b2 c= Fin b1
   holds meet b2 in Fin b1;

:: YELLOW_8:attrnot 1 => REALSET1:attr 1
definition
  let a1 be set;
  attr a1 is trivial means
    for b1, b2 being Element of a1 holds
    b1 = b2;
end;

:: YELLOW_8:dfs 1
definiens
  let a1 be non empty set;
To prove
     a1 is trivial
it is sufficient to prove
  thus for b1, b2 being Element of a1 holds
    b1 = b2;

:: YELLOW_8:def 1
theorem
for b1 being non empty set holds
      b1 is trivial
   iff
      for b2, b3 being Element of b1 holds
      b2 = b3;

:: YELLOW_8:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
   b2,COMPLEMENT b2 are_equipotent;

:: YELLOW_8:th 5
theorem
for b1, b2 being set
      st b1,b2 are_equipotent & b1 is countable
   holds b2 is countable;

:: YELLOW_8:th 14
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 ` in COMPLEMENT b2
   iff
      b3 in b2;

:: YELLOW_8:th 15
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
   Intersect COMPLEMENT b2 = (union b2) `;

:: YELLOW_8:th 16
theorem
for b1 being 1-sorted
for b2 being Element of bool bool the carrier of b1 holds
   union COMPLEMENT b2 = (Intersect b2) `;

:: YELLOW_8:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b3 c= b2 &
         b2 is closed(b1) &
         (for b4 being Element of bool the carrier of b1
               st b3 c= b4 & b4 is closed(b1)
            holds b2 c= b4)
   holds b2 = Cl b3;

:: YELLOW_8:th 18
theorem
for b1 being TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1
      st b3 is open(b1)
   holds b3 = union {b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b4 c= b3};

:: YELLOW_8:th 19
theorem
for b1 being TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1
      st b3 in b2
   holds b3 is open(b1);

:: YELLOW_8:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Basis of b1
for b3 being Element of bool the carrier of b1 holds
   Int b3 = union {b4 where b4 is Element of bool the carrier of b1: b4 in b2 & b4 c= b3};

:: YELLOW_8:modenot 1 => YELLOW_8:mode 1
definition
  let a1 be non empty TopStruct;
  let a2 be Element of the carrier of a1;
  mode Basis of A2 -> Element of bool bool the carrier of a1 means
    it c= the topology of a1 &
     a2 in Intersect it &
     (for b1 being Element of bool the carrier of a1
           st b1 is open(a1) & a2 in b1
        holds ex b2 being Element of bool the carrier of a1 st
           b2 in it & b2 c= b1);
end;

:: YELLOW_8:dfs 2
definiens
  let a1 be non empty TopStruct;
  let a2 be Element of the carrier of a1;
  let a3 be Element of bool bool the carrier of a1;
To prove
     a3 is Basis of a2
it is sufficient to prove
  thus a3 c= the topology of a1 &
     a2 in Intersect a3 &
     (for b1 being Element of bool the carrier of a1
           st b1 is open(a1) & a2 in b1
        holds ex b2 being Element of bool the carrier of a1 st
           b2 in a3 & b2 c= b1);

:: YELLOW_8:def 2
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool bool the carrier of b1 holds
      b3 is Basis of b2
   iff
      b3 c= the topology of b1 &
       b2 in Intersect b3 &
       (for b4 being Element of bool the carrier of b1
             st b4 is open(b1) & b2 in b4
          holds ex b5 being Element of bool the carrier of b1 st
             b5 in b3 & b5 c= b4);

:: YELLOW_8:th 21
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2
for b4 being Element of bool the carrier of b1
      st b4 in b3
   holds b4 is open(b1) & b2 in b4;

:: YELLOW_8:th 22
theorem
for b1 being non empty TopStruct
for b2 being Element of the carrier of b1
for b3 being Basis of b2
for b4 being Element of bool the carrier of b1
      st b2 in b4 & b4 is open(b1)
   holds ex b5 being Element of bool the carrier of b1 st
      b5 in b3 & b5 c= b4;

:: YELLOW_8:th 23
theorem
for b1 being non empty TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 c= the topology of b1 &
         (for b3 being Element of the carrier of b1 holds
            ex b4 being Basis of b3 st
               b4 c= b2)
   holds b2 is Basis of b1;

:: YELLOW_8:attrnot 2 => YELLOW_8:attr 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is Baire means
    for b1 being Element of bool bool the carrier of a1
          st b1 is countable &
             (for b2 being Element of bool the carrier of a1
                   st b2 in b1
                holds b2 is everywhere_dense(a1))
       holds ex b2 being Element of bool the carrier of a1 st
          b2 = Intersect b1 & b2 is dense(a1);
end;

:: YELLOW_8:dfs 3
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is Baire
it is sufficient to prove
  thus for b1 being Element of bool bool the carrier of a1
          st b1 is countable &
             (for b2 being Element of bool the carrier of a1
                   st b2 in b1
                holds b2 is everywhere_dense(a1))
       holds ex b2 being Element of bool the carrier of a1 st
          b2 = Intersect b1 & b2 is dense(a1);

:: YELLOW_8:def 3
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is Baire
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is countable &
               (for b3 being Element of bool the carrier of b1
                     st b3 in b2
                  holds b3 is everywhere_dense(b1))
         holds ex b3 being Element of bool the carrier of b1 st
            b3 = Intersect b2 & b3 is dense(b1);

:: YELLOW_8:th 24
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is Baire
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is countable &
               (for b3 being Element of bool the carrier of b1
                     st b3 in b2
                  holds b3 is nowhere_dense(b1))
         holds union b2 is boundary(b1);

:: YELLOW_8:attrnot 3 => YELLOW_8:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is irreducible means
    a2 is not empty &
     a2 is closed(a1) &
     (for b1, b2 being Element of bool the carrier of a1
           st b1 is closed(a1) & b2 is closed(a1) & a2 = b1 \/ b2 & b1 <> a2
        holds b2 = a2);
end;

:: YELLOW_8:dfs 4
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is irreducible
it is sufficient to prove
  thus a2 is not empty &
     a2 is closed(a1) &
     (for b1, b2 being Element of bool the carrier of a1
           st b1 is closed(a1) & b2 is closed(a1) & a2 = b1 \/ b2 & b1 <> a2
        holds b2 = a2);

:: YELLOW_8:def 4
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is irreducible(b1)
   iff
      b2 is not empty &
       b2 is closed(b1) &
       (for b3, b4 being Element of bool the carrier of b1
             st b3 is closed(b1) & b4 is closed(b1) & b2 = b3 \/ b4 & b3 <> b2
          holds b4 = b2);

:: YELLOW_8:condreg 1
registration
  let a1 be TopStruct;
  cluster irreducible -> non empty (Element of bool the carrier of a1);
end;

:: YELLOW_8:prednot 1 => YELLOW_8:pred 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  pred A3 is_dense_point_of A2 means
    a3 in a2 & a2 c= Cl {a3};
end;

:: YELLOW_8:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
To prove
     a3 is_dense_point_of a2
it is sufficient to prove
  thus a3 in a2 & a2 c= Cl {a3};

:: YELLOW_8:def 5
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 is_dense_point_of b2
   iff
      b3 in b2 & b2 c= Cl {b3};

:: YELLOW_8:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
   st b2 is closed(b1)
for b3 being Element of the carrier of b1
      st b3 is_dense_point_of b2
   holds b2 = Cl {b3};

:: YELLOW_8:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   Cl {b2} is irreducible(b1);

:: YELLOW_8:exreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster irreducible Element of bool the carrier of a1;
end;

:: YELLOW_8:attrnot 4 => YELLOW_8:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is sober means
    for b1 being irreducible Element of bool the carrier of a1 holds
       ex b2 being Element of the carrier of a1 st
          b2 is_dense_point_of b1 &
           (for b3 being Element of the carrier of a1
                 st b3 is_dense_point_of b1
              holds b2 = b3);
end;

:: YELLOW_8:dfs 6
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is sober
it is sufficient to prove
  thus for b1 being irreducible Element of bool the carrier of a1 holds
       ex b2 being Element of the carrier of a1 st
          b2 is_dense_point_of b1 &
           (for b3 being Element of the carrier of a1
                 st b3 is_dense_point_of b1
              holds b2 = b3);

:: YELLOW_8:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is sober
   iff
      for b2 being irreducible Element of bool the carrier of b1 holds
         ex b3 being Element of the carrier of b1 st
            b3 is_dense_point_of b2 &
             (for b4 being Element of the carrier of b1
                   st b4 is_dense_point_of b2
                holds b3 = b4);

:: YELLOW_8:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   b2 is_dense_point_of Cl {b2};

:: YELLOW_8:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   b2 is_dense_point_of {b2};

:: YELLOW_8:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is closed(b1)
   holds b3 \ b2 is closed(b1);

:: YELLOW_8:th 30
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct
for b2 being irreducible Element of bool the carrier of b1 holds
   b2 is trivial;

:: YELLOW_8:condreg 2
registration
  let a1 be non empty TopSpace-like being_T2 TopStruct;
  cluster irreducible -> trivial (Element of bool the carrier of a1);
end;

:: YELLOW_8:th 31
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct holds
   b1 is sober;

:: YELLOW_8:condreg 3
registration
  cluster non empty TopSpace-like being_T2 -> sober (TopStruct);
end;

:: YELLOW_8:exreg 2
registration
  cluster non empty TopSpace-like sober TopStruct;
end;

:: YELLOW_8:th 32
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is discerning
   iff
      for b2, b3 being Element of the carrier of b1
            st Cl {b2} = Cl {b3}
         holds b2 = b3;

:: YELLOW_8:th 33
theorem
for b1 being non empty TopSpace-like sober TopStruct holds
   b1 is discerning;

:: YELLOW_8:condreg 4
registration
  cluster non empty TopSpace-like sober -> discerning (TopStruct);
end;

:: YELLOW_8:funcnot 1 => YELLOW_8:func 1
definition
  let a1 be set;
  func CofinTop A1 -> strict TopStruct means
    the carrier of it = a1 &
     COMPLEMENT the topology of it = {a1} \/ Fin a1;
end;

:: YELLOW_8:def 7
theorem
for b1 being set
for b2 being strict TopStruct holds
      b2 = CofinTop b1
   iff
      the carrier of b2 = b1 &
       COMPLEMENT the topology of b2 = {b1} \/ Fin b1;

:: YELLOW_8:funcreg 1
registration
  let a1 be non empty set;
  cluster CofinTop a1 -> non empty strict;
end;

:: YELLOW_8:funcreg 2
registration
  let a1 be set;
  cluster CofinTop a1 -> strict TopSpace-like;
end;

:: YELLOW_8:th 34
theorem
for b1 being non empty set
for b2 being Element of bool the carrier of CofinTop b1 holds
      b2 is closed(CofinTop b1)
   iff
      (b2 = b1 or b2 is finite);

:: YELLOW_8:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is being_T1
for b2 being Element of the carrier of b1 holds
   Cl {b2} = {b2};

:: YELLOW_8:funcreg 3
registration
  let a1 be non empty set;
  cluster CofinTop a1 -> strict being_T1;
end;

:: YELLOW_8:funcreg 4
registration
  let a1 be infinite set;
  cluster CofinTop a1 -> strict non sober;
end;

:: YELLOW_8:exreg 3
registration
  cluster non empty TopSpace-like being_T1 non sober TopStruct;
end;

:: YELLOW_8:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is being_T3
   iff
      for b2 being Element of the carrier of b1
      for b3 being Element of bool the carrier of b1
            st b2 in Int b3
         holds ex b4 being Element of bool the carrier of b1 st
            b4 is closed(b1) & b4 c= b3 & b2 in Int b4;

:: YELLOW_8:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is being_T3
   holds    b1 is locally-compact
   iff
      for b2 being Element of the carrier of b1 holds
         ex b3 being Element of bool the carrier of b1 st
            b2 in Int b3 & b3 is compact(b1);