Article TOPMETR, MML version 4.99.1005

:: TOPMETR:th 1
theorem
for b1 being TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is_a_cover_of b1
   iff
      the carrier of b1 c= union b2;

:: TOPMETR:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3 being Element of the carrier of b2 holds
   b3 is Element of the carrier of b1;

:: TOPMETR:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
      st b1 is being_T2
   holds b2 is being_T2;

:: TOPMETR:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
      st the carrier of b2 c= the carrier of b3
   holds b2 is SubSpace of b3;

:: TOPMETR:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b1 | b2 is SubSpace of b1 | (b2 \/ b3) & b1 | b3 is SubSpace of b1 | (b2 \/ b3);

:: TOPMETR:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being non empty Element of bool the carrier of b1
   st b2 in b3
for b4 being a_neighborhood of b2
for b5 being Element of the carrier of b1 | b3
for b6 being Element of bool the carrier of b1 | b3
      st b6 = b4 /\ b3 & b5 = b2
   holds b6 is a_neighborhood of b5;

:: TOPMETR:th 7
theorem
for b1, b2, b3 being TopSpace-like TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
   st b4 is continuous(b1, b3) & b3 is SubSpace of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 = b4
   holds b5 is continuous(b1, b2);

:: TOPMETR:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being SubSpace of b2
   st b3 is continuous(b1, b2)
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b4
      st b5 = b3
   holds b5 is continuous(b1, b4);

:: TOPMETR:th 9
theorem
for b1, b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2
   st b3 is continuous(b1, b2)
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 | b4
      st b5 = b3
   holds b5 is continuous(b1, b2 | b4);

:: TOPMETR:th 10
theorem
for b1 being TopStruct
for b2 being non empty TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1 | b3,the carrier of b2
      st b4 is continuous(b1, b2) & b5 = b4 | b3
   holds b5 is continuous(b1 | b3, b2);

:: TOPMETR:modenot 1 => TOPMETR:mode 1
definition
  let a1 be Reflexive discerning symmetric triangle MetrStruct;
  mode SubSpace of A1 -> Reflexive discerning symmetric triangle MetrStruct means
    the carrier of it c= the carrier of a1 &
     (for b1, b2 being Element of the carrier of it holds
     (the distance of it) .(b1,b2) = (the distance of a1) .(b1,b2));
end;

:: TOPMETR:dfs 1
definiens
  let a1, a2 be Reflexive discerning symmetric triangle MetrStruct;
To prove
     a2 is SubSpace of a1
it is sufficient to prove
  thus the carrier of a2 c= the carrier of a1 &
     (for b1, b2 being Element of the carrier of a2 holds
     (the distance of a2) .(b1,b2) = (the distance of a1) .(b1,b2));

:: TOPMETR:def 1
theorem
for b1, b2 being Reflexive discerning symmetric triangle MetrStruct holds
   b2 is SubSpace of b1
iff
   the carrier of b2 c= the carrier of b1 &
    (for b3, b4 being Element of the carrier of b2 holds
    (the distance of b2) .(b3,b4) = (the distance of b1) .(b3,b4));

:: TOPMETR:exreg 1
registration
  let a1 be Reflexive discerning symmetric triangle MetrStruct;
  cluster strict Reflexive discerning symmetric triangle SubSpace of a1;
end;

:: TOPMETR:exreg 2
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  cluster non empty strict Reflexive discerning symmetric triangle SubSpace of a1;
end;

:: TOPMETR:th 12
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty SubSpace of b1
for b3 being Element of the carrier of b2 holds
   b3 is Element of the carrier of b1;

:: TOPMETR:th 13
theorem
for b1 being real set
for b2 being Reflexive discerning symmetric triangle MetrStruct
for b3 being SubSpace of b2
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b3
      st b4 = b5
   holds Ball(b5,b1) = (Ball(b4,b1)) /\ the carrier of b3;

:: TOPMETR:funcnot 1 => TOPMETR:func 1
definition
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be non empty Element of bool the carrier of a1;
  func A1 | A2 -> strict SubSpace of a1 means
    the carrier of it = a2;
end;

:: TOPMETR:def 2
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being strict SubSpace of b1 holds
      b3 = b1 | b2
   iff
      the carrier of b3 = b2;

:: TOPMETR:funcreg 1
registration
  let a1 be non empty Reflexive discerning symmetric triangle MetrStruct;
  let a2 be non empty Element of bool the carrier of a1;
  cluster a1 | a2 -> non empty strict;
end;

:: TOPMETR:funcnot 2 => TOPMETR:func 2
definition
  let a1, a2 be real set;
  assume a1 <= a2;
  func Closed-Interval-MSpace(A1,A2) -> non empty strict SubSpace of RealSpace means
    for b1 being non empty Element of bool the carrier of RealSpace
          st b1 = [.a1,a2.]
       holds it = RealSpace | b1;
end;

:: TOPMETR:def 3
theorem
for b1, b2 being real set
   st b1 <= b2
for b3 being non empty strict SubSpace of RealSpace holds
      b3 = Closed-Interval-MSpace(b1,b2)
   iff
      for b4 being non empty Element of bool the carrier of RealSpace
            st b4 = [.b1,b2.]
         holds b3 = RealSpace | b4;

:: TOPMETR:th 14
theorem
for b1, b2 being real set
      st b1 <= b2
   holds the carrier of Closed-Interval-MSpace(b1,b2) = [.b1,b2.];

:: TOPMETR:attrnot 1 => TOPMETR:attr 1
definition
  let a1 be MetrStruct;
  let a2 be Element of bool bool the carrier of a1;
  attr a2 is being_ball-family means
    for b1 being set
          st b1 in a2
       holds ex b2 being Element of the carrier of a1 st
          ex b3 being Element of REAL st
             b1 = Ball(b2,b3);
end;

:: TOPMETR:dfs 4
definiens
  let a1 be MetrStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is being_ball-family
it is sufficient to prove
  thus for b1 being set
          st b1 in a2
       holds ex b2 being Element of the carrier of a1 st
          ex b3 being Element of REAL st
             b1 = Ball(b2,b3);

:: TOPMETR:def 4
theorem
for b1 being MetrStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is being_ball-family(b1)
   iff
      for b3 being set
            st b3 in b2
         holds ex b4 being Element of the carrier of b1 st
            ex b5 being Element of REAL st
               b3 = Ball(b4,b5);

:: TOPMETR:prednot 1 => TOPMETR:pred 1
definition
  let a1 be MetrStruct;
  let a2 be Element of bool bool the carrier of a1;
  pred A2 is_a_cover_of A1 means
    the carrier of a1 c= union a2;
end;

:: TOPMETR:dfs 5
definiens
  let a1 be MetrStruct;
  let a2 be Element of bool bool the carrier of a1;
To prove
     a2 is_a_cover_of a1
it is sufficient to prove
  thus the carrier of a1 c= union a2;

:: TOPMETR:def 5
theorem
for b1 being MetrStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is_a_cover_of b1
   iff
      the carrier of b1 c= union b2;

:: TOPMETR:prednot 2 => TOPMETR:attr 1
notation
  let a1 be MetrStruct;
  let a2 be Element of bool bool the carrier of a1;
  synonym a2 is_ball-family for being_ball-family;
end;

:: TOPMETR:th 15
theorem
for b1, b2 being Element of the carrier of RealSpace
for b3, b4 being real set
      st b3 = b1 & b4 = b2
   holds dist(b1,b2) = abs (b3 - b4);

:: TOPMETR:th 16
theorem
for b1 being MetrStruct holds
   the carrier of b1 = the carrier of TopSpaceMetr b1 & the topology of TopSpaceMetr b1 = Family_open_set b1;

:: TOPMETR:th 19
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being non empty SubSpace of b1 holds
   TopSpaceMetr b2 is SubSpace of TopSpaceMetr b1;

:: TOPMETR:th 20
theorem
for b1 being natural set
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being non empty Element of bool the carrier of Euclid b1
      st b2 = b3
   holds (TOP-REAL b1) | b2 = TopSpaceMetr ((Euclid b1) | b3);

:: TOPMETR:th 21
theorem
for b1 being real set
for b2 being triangle MetrStruct
for b3 being Element of the carrier of b2
for b4 being Element of bool the carrier of TopSpaceMetr b2
      st b4 = Ball(b3,b1)
   holds b4 is open(TopSpaceMetr b2);

:: TOPMETR:th 22
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool the carrier of TopSpaceMetr b1 holds
      b2 is open(TopSpaceMetr b1)
   iff
      for b3 being Element of the carrier of b1
            st b3 in b2
         holds ex b4 being real set st
            0 < b4 & Ball(b3,b4) c= b2;

:: TOPMETR:attrnot 2 => TOPMETR:attr 2
definition
  let a1 be MetrStruct;
  attr a1 is compact means
    TopSpaceMetr a1 is compact;
end;

:: TOPMETR:dfs 6
definiens
  let a1 be MetrStruct;
To prove
     a1 is compact
it is sufficient to prove
  thus TopSpaceMetr a1 is compact;

:: TOPMETR:def 6
theorem
for b1 being MetrStruct holds
      b1 is compact
   iff
      TopSpaceMetr b1 is compact;

:: TOPMETR:th 23
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct holds
      b1 is compact
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is being_ball-family(b1) & b2 is_a_cover_of b1
         holds ex b3 being Element of bool bool the carrier of b1 st
            b3 c= b2 & b3 is_a_cover_of b1 & b3 is finite;

:: TOPMETR:funcnot 3 => TOPMETR:func 3
definition
  func R^1 -> strict TopSpace-like TopStruct equals
    TopSpaceMetr RealSpace;
end;

:: TOPMETR:def 7
theorem
R^1 = TopSpaceMetr RealSpace;

:: TOPMETR:funcreg 2
registration
  cluster R^1 -> non empty strict TopSpace-like;
end;

:: TOPMETR:th 24
theorem
the carrier of R^1 = REAL;

:: TOPMETR:funcreg 3
registration
  let a1 be set;
  let a2 be Function-like Relation of a1,the carrier of R^1;
  let a3 be set;
  cluster a2 . a3 -> real;
end;

:: TOPMETR:funcnot 4 => TOPMETR:func 4
definition
  let a1, a2 be real set;
  func Closed-Interval-TSpace(A1,A2) -> non empty strict SubSpace of R^1 equals
    TopSpaceMetr Closed-Interval-MSpace(a1,a2);
end;

:: TOPMETR:def 8
theorem
for b1, b2 being real set holds
Closed-Interval-TSpace(b1,b2) = TopSpaceMetr Closed-Interval-MSpace(b1,b2);

:: TOPMETR:th 25
theorem
for b1, b2 being real set
      st b1 <= b2
   holds the carrier of Closed-Interval-TSpace(b1,b2) = [.b1,b2.];

:: TOPMETR:th 26
theorem
for b1, b2 being real set
   st b1 <= b2
for b3 being Element of bool the carrier of R^1
      st b3 = [.b1,b2.]
   holds Closed-Interval-TSpace(b1,b2) = R^1 | b3;

:: TOPMETR:th 27
theorem
Closed-Interval-TSpace(0,1) = I[01];

:: TOPMETR:funcnot 5 => TOPMETR:func 5
definition
  redefine func I[01] -> strict SubSpace of R^1;
end;

:: TOPMETR:th 28
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
      st ex b2, b3 being Element of REAL st
           for b4 being Element of REAL holds
              b1 . b4 = (b2 * b4) + b3
   holds b1 is continuous(R^1, R^1);

:: TOPMETR:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds b1 | b2 is SubSpace of b1 | b3;

:: TOPMETR:th 30
theorem
for b1, b2, b3, b4 being real set
for b5 being Element of bool the carrier of Closed-Interval-TSpace(b3,b4)
      st b3 <= b1 & b1 <= b2 & b2 <= b4 & b5 = [.b1,b2.]
   holds Closed-Interval-TSpace(b1,b2) = (Closed-Interval-TSpace(b3,b4)) | b5;

:: TOPMETR:th 31
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of I[01]
      st 0 <= b1 & b1 <= b2 & b2 <= 1 & b3 = [.b1,b2.]
   holds Closed-Interval-TSpace(b1,b2) = I[01] | b3;