Article NAGATA_1, MML version 4.99.1005

:: NAGATA_1:attrnot 1 => NAGATA_1:attr 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is discrete means
    for b1 being Element of the carrier of a1 holds
       ex b2 being open Element of bool the carrier of a1 st
          b1 in b2 &
           (for b3, b4 being Element of bool the carrier of a1
                 st b3 in a2 & b4 in a2 & b2 meets b3 & b2 meets b4
              holds b3 = b4);
end;

:: NAGATA_1:dfs 1
definiens
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is discrete
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       ex b2 being open Element of bool the carrier of a1 st
          b1 in b2 &
           (for b3, b4 being Element of bool the carrier of a1
                 st b3 in a2 & b4 in a2 & b2 meets b3 & b2 meets b4
              holds b3 = b4);

:: NAGATA_1:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is discrete(b1)
   iff
      for b3 being Element of the carrier of b1 holds
         ex b4 being open Element of bool the carrier of b1 st
            b3 in b4 &
             (for b5, b6 being Element of bool the carrier of b1
                   st b5 in b2 & b6 in b2 & b4 meets b5 & b4 meets b6
                holds b5 = b6);

:: NAGATA_1:exreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster discrete Element of bool bool the carrier of a1;
end;

:: NAGATA_1:exreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster empty discrete Element of bool bool the carrier of a1;
end;

:: NAGATA_1:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st ex b3 being Element of bool the carrier of b1 st
           b2 = {b3}
   holds b2 is discrete(b1);

:: NAGATA_1:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 c= b3 & b3 is discrete(b1)
   holds b2 is discrete(b1);

:: NAGATA_1:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is discrete(b1)
   holds b2 /\ b3 is discrete(b1);

:: NAGATA_1:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool bool the carrier of b1
      st b2 is discrete(b1)
   holds b2 \ b3 is discrete(b1);

:: NAGATA_1:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool bool the carrier of b1
      st b2 is discrete(b1) & b3 is discrete(b1) & INTERSECTION(b2,b3) = b4
   holds b4 is discrete(b1);

:: NAGATA_1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
for b3, b4 being Element of bool the carrier of b1
      st b2 is discrete(b1) & b3 in b2 & b4 in b2 & b3 <> b4
   holds b3 misses b4;

:: NAGATA_1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is discrete(b1)
for b3 being Element of the carrier of b1 holds
   ex b4 being open Element of bool the carrier of b1 st
      b3 in b4 &
       (INTERSECTION({b4},b2)) \ {{}} is trivial;

:: NAGATA_1:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is discrete(b1)
   iff
      (for b3 being Element of the carrier of b1 holds
          ex b4 being open Element of bool the carrier of b1 st
             b3 in b4 &
              (INTERSECTION({b4},b2)) \ {{}} is trivial) &
       (for b3, b4 being Element of bool the carrier of b1
             st b3 in b2 & b4 in b2 & b3 <> b4
          holds b3 misses b4);

:: NAGATA_1:funcreg 1
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be discrete Element of bool bool the carrier of a1;
  cluster clf a2 -> discrete;
end;

:: NAGATA_1:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
   st b2 is discrete(b1)
for b3, b4 being Element of bool the carrier of b1
      st b3 in b2 & b4 in b2
   holds Cl (b3 /\ b4) = (Cl b3) /\ Cl b4;

:: NAGATA_1:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is discrete(b1)
   holds Cl union b2 = union clf b2;

:: NAGATA_1:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st b2 is discrete(b1)
   holds b2 is locally_finite(b1);

:: NAGATA_1:modenot 1
definition
  let a1 be TopSpace-like TopStruct;
  mode FamilySequence of a1 is Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
end;

:: NAGATA_1:funcnot 1 => NAGATA_1:func 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  let a3 be Element of NAT;
  redefine func a2 . a3 -> Element of bool bool the carrier of a1;
end;

:: NAGATA_1:funcnot 2 => NAGATA_1:func 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  redefine func Union a2 -> Element of bool bool the carrier of a1;
end;

:: NAGATA_1:attrnot 2 => NAGATA_1:attr 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  attr a2 is sigma_discrete means
    for b1 being Element of NAT holds
       a2 . b1 is discrete(a1);
end;

:: NAGATA_1:dfs 2
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
     a2 is sigma_discrete
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a2 . b1 is discrete(a1);

:: NAGATA_1:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
      b2 is sigma_discrete(b1)
   iff
      for b3 being Element of NAT holds
         b2 . b3 is discrete(b1);

:: NAGATA_1:exreg 3
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty Relation-like Function-like quasi_total total sigma_discrete Relation of NAT,bool bool the carrier of a1;
end;

:: NAGATA_1:attrnot 3 => NAGATA_1:attr 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  attr a2 is sigma_locally_finite means
    for b1 being Element of NAT holds
       a2 . b1 is locally_finite(a1);
end;

:: NAGATA_1:dfs 3
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
     a2 is sigma_locally_finite
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a2 . b1 is locally_finite(a1);

:: NAGATA_1:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
      b2 is sigma_locally_finite(b1)
   iff
      for b3 being Element of NAT holds
         b2 . b3 is locally_finite(b1);

:: NAGATA_1:attrnot 4 => NAGATA_1:attr 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is sigma_discrete means
    ex b1 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of a1 st
       a2 = Union b1;
end;

:: NAGATA_1:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is sigma_discrete
it is sufficient to prove
  thus ex b1 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of a1 st
       a2 = Union b1;

:: NAGATA_1:def 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is sigma_discrete(b1)
   iff
      ex b3 being Function-like quasi_total sigma_discrete Relation of NAT,bool bool the carrier of b1 st
         b2 = Union b3;

:: NAGATA_1:attrnot 5 => CARD_4:attr 1
notation
  let a1 be set;
  antonym uncountable for countable;
end;

:: NAGATA_1:condreg 1
registration
  cluster non countable -> non empty (set);
end;

:: NAGATA_1:exreg 4
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty Relation-like Function-like quasi_total total sigma_locally_finite Relation of NAT,bool bool the carrier of a1;
end;

:: NAGATA_1:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1
      st b2 is sigma_discrete(b1)
   holds b2 is sigma_locally_finite(b1);

:: NAGATA_1:th 13
theorem
for b1 being non countable set holds
   ex b2 being Element of bool bool the carrier of 1TopSp [:b1,b1:] st
      b2 is locally_finite(1TopSp [:b1,b1:]) & b2 is sigma_discrete(not 1TopSp [:b1,b1:]);

:: NAGATA_1:attrnot 6 => NAGATA_1:attr 5
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  attr a2 is Basis_sigma_discrete means
    a2 is sigma_discrete(a1) & Union a2 is Basis of a1;
end;

:: NAGATA_1:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
     a2 is Basis_sigma_discrete
it is sufficient to prove
  thus a2 is sigma_discrete(a1) & Union a2 is Basis of a1;

:: NAGATA_1:def 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
      b2 is Basis_sigma_discrete(b1)
   iff
      b2 is sigma_discrete(b1) & Union b2 is Basis of b1;

:: NAGATA_1:attrnot 7 => NAGATA_1:attr 6
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
  attr a2 is Basis_sigma_locally_finite means
    a2 is sigma_locally_finite(a1) & Union a2 is Basis of a1;
end;

:: NAGATA_1:dfs 6
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Function-like quasi_total Relation of NAT,bool bool the carrier of a1;
To prove
     a2 is Basis_sigma_locally_finite
it is sufficient to prove
  thus a2 is sigma_locally_finite(a1) & Union a2 is Basis of a1;

:: NAGATA_1:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 holds
      b2 is Basis_sigma_locally_finite(b1)
   iff
      b2 is sigma_locally_finite(b1) & Union b2 is Basis of b1;

:: NAGATA_1:th 14
theorem
for b1 being non empty MetrStruct
for b2 being real set
   st b1 is non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Element of the carrier of b1 holds
   ([#] b1) \ cl_Ball(b3,b2) in Family_open_set b1;

:: NAGATA_1:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is metrizable
   holds b1 is being_T3 & b1 is being_T1;

:: NAGATA_1:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is metrizable
   holds ex b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 st
      b2 is Basis_sigma_locally_finite(b1);

:: NAGATA_1:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st for b3 being Element of NAT holds
           b2 . b3 is open(b1)
   holds Union b2 is open(b1);

:: NAGATA_1:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
        for b3 being open Element of bool the carrier of b1
              st b2 is closed(b1) & b3 is open(b1) & b2 c= b3
           holds ex b4 being Function-like quasi_total Relation of NAT,bool the carrier of b1 st
              b2 c= Union b4 &
               Union b4 c= b3 &
               (for b5 being Element of NAT holds
                  Cl (b4 . b5) c= b3 & b4 . b5 is open(b1))
   holds b1 is being_T4;

:: NAGATA_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is being_T3
for b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1
   st Union b2 is Basis of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b3 is open(b1) & b4 in b3
   holds ex b5 being Element of bool the carrier of b1 st
      b4 in b5 & Cl b5 c= b3 & b5 in Union b2;

:: NAGATA_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
      st b1 is being_T3 &
         b1 is being_T1 &
         (ex b2 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 st
            b2 is Basis_sigma_locally_finite(b1))
   holds b1 is being_T4;

:: NAGATA_1:funcnot 3 => NAGATA_1:func 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Function-like quasi_total Relation of the carrier of a1,REAL;
  redefine func A2 + A3 -> Function-like quasi_total Relation of the carrier of a1,REAL means
    for b1 being Element of the carrier of a1 holds
       it . b1 = (a2 . b1) + (a3 . b1);
  commutativity;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2, a3 being Function-like quasi_total Relation of the carrier of a1,REAL holds
::  a2 + a3 = a3 + a2;
end;

:: NAGATA_1:def 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
   b4 = b2 + b3
iff
   for b5 being Element of the carrier of b1 holds
      b4 . b5 = (b2 . b5) + (b3 . b5);

:: NAGATA_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,REAL
   st b2 is continuous(b1)
for b3 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
      st for b4, b5 being Element of the carrier of b1 holds
        b3 .(b4,b5) = abs ((b2 . b4) - (b2 . b5))
   holds b3 is continuous([:b1,b1:]);

:: NAGATA_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b2 is continuous(b1) & b3 is continuous(b1)
   holds b2 + b3 is continuous(b1);

:: NAGATA_1:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
      st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
        b2 .(b3,b4) = b3 + b4
   holds b2 is having_a_unity(Funcs(the carrier of b1,REAL)) & b2 is commutative(Funcs(the carrier of b1,REAL)) & b2 is associative(Funcs(the carrier of b1,REAL));

:: NAGATA_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
   st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
     b2 .(b3,b4) = b3 + b4
for b3 being Element of Funcs(the carrier of b1,REAL)
      st b3 is_a_unity_wrt b2
   holds b3 is continuous(b1);

:: NAGATA_1:funcnot 4 => NAGATA_1:func 4
definition
  let a1, a2 be non empty set;
  let a3 be Function-like quasi_total Relation of a1,Funcs(a1,a2);
  func A3 Toler -> Function-like quasi_total Relation of a1,a2 means
    for b1 being Element of a1 holds
       it . b1 = (a3 . b1) . b1;
end;

:: NAGATA_1:def 8
theorem
for b1, b2 being non empty set
for b3 being Function-like quasi_total Relation of b1,Funcs(b1,b2)
for b4 being Function-like quasi_total Relation of b1,b2 holds
      b4 = b3 Toler
   iff
      for b5 being Element of b1 holds
         b4 . b5 = (b3 . b5) . b5;

:: NAGATA_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:Funcs(the carrier of b1,REAL),Funcs(the carrier of b1,REAL):],Funcs(the carrier of b1,REAL)
   st for b3, b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
     b2 .(b3,b4) = b3 + b4
for b3 being FinSequence of Funcs(the carrier of b1,REAL)
      st for b4 being Element of NAT
              st 0 <> b4 & b4 <= len b3
           holds b3 . b4 is Function-like quasi_total continuous Relation of the carrier of b1,REAL
   holds b2 "**" b3 is Function-like quasi_total continuous Relation of the carrier of b1,REAL;

:: NAGATA_1:th 26
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,Funcs(the carrier of b1,the carrier of b2)
   st for b4 being Element of the carrier of b1 holds
        b3 . b4 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,bool the carrier of b1
      st for b5 being Element of the carrier of b1 holds
           b5 in b4 . b5 &
            b4 . b5 is open(b1) &
            (for b6, b7 being Element of the carrier of b1
                  st b6 in b4 . b7
               holds (b3 . b6) . b6 = (b3 . b7) . b6)
   holds b3 Toler is continuous(b1, b2);

:: NAGATA_1:funcnot 5 => NAGATA_1:func 5
definition
  let a1 be set;
  let a2 be Element of REAL;
  let a3 be Function-like quasi_total Relation of a1,REAL;
  func min(A2,A3) -> Function-like quasi_total Relation of a1,REAL means
    for b1 being set
          st b1 in a1
       holds it . b1 = min(a2,a3 . b1);
end;

:: NAGATA_1:def 9
theorem
for b1 being set
for b2 being Element of REAL
for b3, b4 being Function-like quasi_total Relation of b1,REAL holds
   b4 = min(b2,b3)
iff
   for b5 being set
         st b5 in b1
      holds b4 . b5 = min(b2,b3 . b5);

:: NAGATA_1:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st b3 is continuous(b1)
   holds min(b2,b3) is continuous(b1);

:: NAGATA_1:prednot 1 => NAGATA_1:pred 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  pred A2 is_a_pseudometric_of A1 means
    a2 is Reflexive(a1) & a2 is symmetric(a1) & a2 is triangle(a1);
end;

:: NAGATA_1:dfs 10
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
To prove
     a2 is_a_pseudometric_of a1
it is sufficient to prove
  thus a2 is Reflexive(a1) & a2 is symmetric(a1) & a2 is triangle(a1);

:: NAGATA_1:def 10
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
      b2 is_a_pseudometric_of b1
   iff
      b2 is Reflexive(b1) & b2 is symmetric(b1) & b2 is triangle(b1);

:: NAGATA_1:th 28
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL holds
      b2 is_a_pseudometric_of b1
   iff
      for b3, b4, b5 being Element of b1 holds
      b2 .(b3,b3) = 0 &
       b2 .(b3,b5) <= (b2 .(b3,b4)) + (b2 .(b5,b4));

:: NAGATA_1:th 29
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
   st b2 is_a_pseudometric_of b1
for b3, b4 being Element of b1 holds
0 <= b2 .(b3,b4);

:: NAGATA_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
      st 0 < b2 & b3 is_a_pseudometric_of the carrier of b1
   holds min(b2,b3) is_a_pseudometric_of the carrier of b1;

:: NAGATA_1:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
      st 0 < b2 & b3 is_metric_of the carrier of b1
   holds min(b2,b3) is_metric_of the carrier of b1;