Article NAGATA_2, MML version 4.99.1005

:: NAGATA_2:th 1
theorem
for b1 being Element of NAT
      st 0 < b1
   holds ex b2, b3 being Element of NAT st
      b1 = (2 |^ b2) * ((2 * b3) + 1);

:: NAGATA_2:funcnot 1 => NAGATA_2:func 1
definition
  func PairFunc -> Function-like quasi_total Relation of [:NAT,NAT:],NAT means
    for b1, b2 being Element of NAT holds
    it . [b1,b2] = ((2 |^ b1) * ((2 * b2) + 1)) - 1;
end;

:: NAGATA_2:def 1
theorem
for b1 being Function-like quasi_total Relation of [:NAT,NAT:],NAT holds
      b1 = PairFunc
   iff
      for b2, b3 being Element of NAT holds
      b1 . [b2,b3] = ((2 |^ b2) * ((2 * b3) + 1)) - 1;

:: NAGATA_2:th 2
theorem
PairFunc is bijective([:NAT,NAT:], NAT);

:: NAGATA_2:funcnot 2 => NAGATA_2:func 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  let a3 be Element of a1;
  func dist(A2,A3) -> Function-like quasi_total Relation of a1,REAL means
    for b1 being Element of a1 holds
       it . b1 = a2 .(a3,b1);
end;

:: NAGATA_2:def 2
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
for b3 being Element of b1
for b4 being Function-like quasi_total Relation of b1,REAL holds
      b4 = dist(b2,b3)
   iff
      for b5 being Element of b1 holds
         b4 . b5 = b2 .(b3,b5);

:: NAGATA_2:th 3
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
   st b3 is open([:b1,b2:])
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of bool the carrier of b1
for b7 being Element of bool the carrier of b2 holds
   (b6 = (pr1(the carrier of b1,the carrier of b2)) .: (b3 /\ [:the carrier of b1,{b5}:]) implies b6 is open(b1)) &
    (b7 = (pr2(the carrier of b1,the carrier of b2)) .: (b3 /\ [:{b4},the carrier of b2:]) implies b7 is open(b2));

:: NAGATA_2:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
   st for b3 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
           st b2 = b3
        holds b3 is continuous([:b1,b1:])
for b3 being Element of the carrier of b1 holds
   dist(b2,b3) is continuous(b1);

:: NAGATA_2:funcnot 3 => NAGATA_2:func 3
definition
  let a1 be non empty set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],REAL;
  let a3 be Element of bool a1;
  func inf(A2,A3) -> Function-like quasi_total Relation of a1,REAL means
    for b1 being Element of a1 holds
       it . b1 = inf ((dist(a2,b1)) .: a3);
end;

:: NAGATA_2:def 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
for b3 being Element of bool b1
for b4 being Function-like quasi_total Relation of b1,REAL holds
      b4 = inf(b2,b3)
   iff
      for b5 being Element of b1 holds
         b4 . b5 = inf ((dist(b2,b5)) .: b3);

:: NAGATA_2:th 5
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
   st b2 is_a_pseudometric_of b1
for b3 being non empty Element of bool b1
for b4 being Element of b1 holds
   0 <= (inf(b2,b3)) . b4;

:: NAGATA_2:th 6
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
   st b2 is_a_pseudometric_of b1
for b3 being Element of bool b1
for b4 being Element of b1
      st b4 in b3
   holds (inf(b2,b3)) . b4 = 0;

:: NAGATA_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
   st b2 is_a_pseudometric_of the carrier of b1
for b3, b4 being Element of the carrier of b1
for b5 being non empty Element of bool the carrier of b1 holds
   abs (((inf(b2,b5)) . b3) - ((inf(b2,b5)) . b4)) <= b2 .(b3,b4);

:: NAGATA_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
   st b2 is_a_pseudometric_of the carrier of b1 &
      (for b3 being Element of the carrier of b1 holds
         dist(b2,b3) is continuous(b1))
for b3 being non empty Element of bool the carrier of b1 holds
   inf(b2,b3) is continuous(b1);

:: NAGATA_2:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],REAL
      st b2 is_metric_of b1
   holds b2 is_a_pseudometric_of b1;

:: NAGATA_2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
      st b2 is_metric_of the carrier of b1 &
         (for b3 being non empty Element of bool the carrier of b1 holds
            Cl b3 = {b4 where b4 is Element of the carrier of b1: (inf(b2,b3)) . b4 = 0})
   holds b1 is metrizable;

:: NAGATA_2:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Functional_Sequence of [:the carrier of b1,the carrier of b1:],REAL
   st (for b3 being Element of NAT holds
         ex b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL st
            b2 . b3 = b4 & b4 is_a_pseudometric_of the carrier of b1) &
      (for b3 being Element of [:the carrier of b1,the carrier of b1:] holds
         b2 # b3 is summable)
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
      st for b4 being Element of [:the carrier of b1,the carrier of b1:] holds
           b3 . b4 = Sum (b2 # b4)
   holds b3 is_a_pseudometric_of the carrier of b1;

:: NAGATA_2:th 12
theorem
for b1 being Element of REAL
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL
   st for b4 being Element of NAT
           st b4 <= b2
        holds b3 . b4 <= b1
for b4 being Element of NAT
      st b4 <= b2
   holds (Partial_Sums b3) . b4 <= b1 * (b4 + 1);

:: NAGATA_2:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
   abs ((Partial_Sums b1) . b2) <= (Partial_Sums abs b1) . b2;

:: NAGATA_2:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Functional_Sequence of the carrier of b1,REAL
   st (for b3 being Element of NAT holds
         ex b4 being Function-like quasi_total Relation of the carrier of b1,REAL st
            b2 . b3 = b4 &
             b4 is continuous(b1) &
             (for b5 being Element of the carrier of b1 holds
                0 <= b4 . b5)) &
      (ex b3 being Function-like quasi_total Relation of NAT,REAL st
         b3 is summable &
          (for b4 being Element of NAT
          for b5 being Element of the carrier of b1 holds
             (b2 # b5) . b4 <= b3 . b4))
for b3 being Function-like quasi_total Relation of the carrier of b1,REAL
      st for b4 being Element of the carrier of b1 holds
           b3 . b4 = Sum (b2 # b4)
   holds b3 is continuous(b1);

:: NAGATA_2:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of REAL
for b3 being Functional_Sequence of [:the carrier of b1,the carrier of b1:],REAL
   st for b4 being Element of NAT holds
        ex b5 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL st
           b3 . b4 = b5 &
            b5 is_a_pseudometric_of the carrier of b1 &
            (for b6 being Element of [:the carrier of b1,the carrier of b1:] holds
               b5 . b6 <= b2) &
            (for b6 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
                  st b5 = b6
               holds b6 is continuous([:b1,b1:]))
for b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
      st for b5 being Element of [:the carrier of b1,the carrier of b1:] holds
           b4 . b5 = Sum ((1 / 2) GeoSeq (#) (b3 # b5))
   holds b4 is_a_pseudometric_of the carrier of b1 &
    (for b5 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
          st b4 = b5
       holds b5 is continuous([:b1,b1:]));

:: NAGATA_2:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
   st b2 is_a_pseudometric_of the carrier of b1 &
      (for b3 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
            st b2 = b3
         holds b3 is continuous([:b1,b1:]))
for b3 being non empty Element of bool the carrier of b1
for b4 being Element of the carrier of b1
      st b4 in Cl b3
   holds (inf(b2,b3)) . b4 = 0;

:: NAGATA_2:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is being_T1
for b2 being Element of REAL
for b3 being Functional_Sequence of [:the carrier of b1,the carrier of b1:],REAL
      st (for b4 being Element of NAT holds
            ex b5 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL st
               b3 . b4 = b5 &
                b5 is_a_pseudometric_of the carrier of b1 &
                (for b6 being Element of [:the carrier of b1,the carrier of b1:] holds
                   b5 . b6 <= b2) &
                (for b6 being Function-like quasi_total Relation of the carrier of [:b1,b1:],REAL
                      st b5 = b6
                   holds b6 is continuous([:b1,b1:]))) &
         (for b4 being Element of the carrier of b1
         for b5 being non empty Element of bool the carrier of b1
               st not b4 in b5 & b5 is closed(b1)
            holds ex b6 being Element of NAT st
               for b7 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL
                     st b3 . b6 = b7
                  holds 0 < (inf(b7,b5)) . b4)
   holds (ex b4 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL st
       b4 is_metric_of the carrier of b1 &
        (for b5 being Element of [:the carrier of b1,the carrier of b1:] holds
           b4 . b5 = Sum ((1 / 2) GeoSeq (#) (b3 # b5)))) &
    b1 is metrizable;

:: NAGATA_2:th 18
theorem
for b1 being non empty set
for b2, b3 being FinSequence of b1
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1
      st b2 is one-to-one &
         b3 is one-to-one &
         rng b3 c= rng b2 &
         b4 is commutative(b1) &
         b4 is associative(b1) &
         (b4 is having_a_unity(b1) or 1 <= len b3 & len b3 < len b2)
   holds ex b5 being FinSequence of b1 st
      b5 is one-to-one &
       rng b5 = (rng b2) \ rng b3 &
       b4 "**" b2 = b4 .(b4 "**" b3,b4 "**" b5);

:: NAGATA_2:funcreg 1
registration
  let a1, a2 be TopSpace-like TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of [:a1,a2:],REAL;
  let a4 be Element of the carrier of a1;
  let a5 be Element of the carrier of a2;
  cluster a3 .(a4,a5) -> real;
end;

:: NAGATA_2:th 19
theorem
for b1 being non empty TopSpace-like TopStruct holds
      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))
   iff
      b1 is metrizable;

:: NAGATA_2:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is metrizable
for b2 being Element of bool bool the carrier of b1
      st b2 is_a_cover_of b1 & b2 is open(b1)
   holds ex b3 being Function-like quasi_total Relation of NAT,bool bool the carrier of b1 st
      Union b3 is open(b1) & Union b3 is_a_cover_of b1 & Union b3 is_finer_than b2 & b3 is sigma_discrete(b1);

:: NAGATA_2:th 21
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_discrete(b1);

:: NAGATA_2:th 22
theorem
for b1 being non empty TopSpace-like TopStruct holds
      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_discrete(b1))
   iff
      b1 is metrizable;