Article WAYBEL18, MML version 4.99.1005

:: WAYBEL18:th 2
theorem
for b1 being set
for b2, b3 being Element of bool bool b1
      st (b3 = b2 \ {{}} or b2 = b3 \/ {{}})
   holds UniCl b2 = UniCl b3;

:: WAYBEL18:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1 holds
      b2 is Basis of b1
   iff
      b2 \ {{}} is Basis of b1;

:: WAYBEL18:attrnot 1 => WAYBEL18:attr 1
definition
  let a1 be Relation-like set;
  attr a1 is TopSpace-yielding means
    for b1 being set
          st b1 in proj2 a1
       holds b1 is TopStruct;
end;

:: WAYBEL18:dfs 1
definiens
  let a1 be Relation-like set;
To prove
     a1 is TopSpace-yielding
it is sufficient to prove
  thus for b1 being set
          st b1 in proj2 a1
       holds b1 is TopStruct;

:: WAYBEL18:def 1
theorem
for b1 being Relation-like set holds
      b1 is TopSpace-yielding
   iff
      for b2 being set
            st b2 in proj2 b1
         holds b2 is TopStruct;

:: WAYBEL18:condreg 1
registration
  cluster Relation-like Function-like TopSpace-yielding -> 1-sorted-yielding (set);
end;

:: WAYBEL18:exreg 1
registration
  let a1 be set;
  cluster Relation-like Function-like TopSpace-yielding ManySortedSet of a1;
end;

:: WAYBEL18:exreg 2
registration
  let a1 be set;
  cluster Relation-like Function-like non-Empty TopSpace-yielding ManySortedSet of a1;
end;

:: WAYBEL18:funcnot 1 => WAYBEL18:func 1
definition
  let a1 be non empty set;
  let a2 be TopSpace-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  redefine func a2 . a3 -> TopStruct;
end;

:: WAYBEL18:funcnot 2 => WAYBEL18:func 2
definition
  let a1 be set;
  let a2 be TopSpace-yielding ManySortedSet of a1;
  func product_prebasis A2 -> Element of bool bool product Carrier a2 means
    for b1 being Element of bool product Carrier a2 holds
          b1 in it
       iff
          ex b2 being set st
             ex b3 being TopStruct st
                ex b4 being Element of bool the carrier of b3 st
                   b2 in a1 & b4 is open(b3) & b3 = a2 . b2 & b1 = product ((Carrier a2) +*(b2,b4));
end;

:: WAYBEL18:def 2
theorem
for b1 being set
for b2 being TopSpace-yielding ManySortedSet of b1
for b3 being Element of bool bool product Carrier b2 holds
      b3 = product_prebasis b2
   iff
      for b4 being Element of bool product Carrier b2 holds
            b4 in b3
         iff
            ex b5 being set st
               ex b6 being TopStruct st
                  ex b7 being Element of bool the carrier of b6 st
                     b5 in b1 & b7 is open(b6) & b6 = b2 . b5 & b4 = product ((Carrier b2) +*(b5,b7));

:: WAYBEL18:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
   TopStruct(#b1,UniCl FinMeetCl b2#) is TopSpace-like;

:: WAYBEL18:funcnot 3 => WAYBEL18:func 3
definition
  let a1 be set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  func product A2 -> strict TopSpace-like TopStruct means
    the carrier of it = product Carrier a2 & product_prebasis a2 is prebasis of it;
end;

:: WAYBEL18:def 3
theorem
for b1 being set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being strict TopSpace-like TopStruct holds
      b3 = product b2
   iff
      the carrier of b3 = product Carrier b2 & product_prebasis b2 is prebasis of b3;

:: WAYBEL18:funcreg 1
registration
  let a1 be set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  cluster product a2 -> non empty strict TopSpace-like;
end;

:: WAYBEL18:funcnot 4 => WAYBEL18:func 4
definition
  let a1 be non empty set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  redefine func a2 . a3 -> non empty TopStruct;
end;

:: WAYBEL18:funcreg 2
registration
  let a1 be set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  cluster product a2 -> strict TopSpace-like constituted-Functions;
end;

:: WAYBEL18:funcnot 5 => WAYBEL18:func 5
definition
  let a1 be non empty set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  let a3 be Element of the carrier of product a2;
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of the carrier of a2 . a4;
end;

:: WAYBEL18:funcnot 6 => WAYBEL18:func 6
definition
  let a1 be non empty set;
  let a2 be non-Empty TopSpace-yielding ManySortedSet of a1;
  let a3 be Element of a1;
  func proj(A2,A3) -> Function-like quasi_total Relation of the carrier of product a2,the carrier of a2 . a3 equals
    proj(Carrier a2,a3);
end;

:: WAYBEL18:def 4
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds
   proj(b2,b3) = proj(Carrier b2,b3);

:: WAYBEL18:th 5
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool the carrier of b2 . b3 holds
   (proj(b2,b3)) " b4 = product ((Carrier b2) +*(b3,b4));

:: WAYBEL18:th 6
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds
   proj(b2,b3) is continuous(product b2, b2 . b3);

:: WAYBEL18:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty set
for b3 being non-Empty TopSpace-yielding ManySortedSet of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of product b3 holds
      b4 is continuous(b1, product b3)
   iff
      for b5 being Element of b2 holds
         (proj(b3,b5)) * b4 is continuous(b1, b3 . b5);

:: WAYBEL18:attrnot 2 => WAYBEL18:attr 2
definition
  let a1 be TopStruct;
  attr a1 is injective means
    for b1 being non empty TopSpace-like TopStruct
    for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of a1
       st b2 is continuous(b1, a1)
    for b3 being non empty TopSpace-like TopStruct
          st b1 is SubSpace of b3
       holds ex b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of a1 st
          b4 is continuous(b3, a1) & b4 | the carrier of b1 = b2;
end;

:: WAYBEL18:dfs 5
definiens
  let a1 be TopStruct;
To prove
     a1 is injective
it is sufficient to prove
  thus for b1 being non empty TopSpace-like TopStruct
    for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of a1
       st b2 is continuous(b1, a1)
    for b3 being non empty TopSpace-like TopStruct
          st b1 is SubSpace of b3
       holds ex b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of a1 st
          b4 is continuous(b3, a1) & b4 | the carrier of b1 = b2;

:: WAYBEL18:def 5
theorem
for b1 being TopStruct holds
      b1 is injective
   iff
      for b2 being non empty TopSpace-like TopStruct
      for b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
         st b3 is continuous(b2, b1)
      for b4 being non empty TopSpace-like TopStruct
            st b2 is SubSpace of b4
         holds ex b5 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1 st
            b5 is continuous(b4, b1) & b5 | the carrier of b2 = b3;

:: WAYBEL18:th 8
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is injective
   holds product b2 is injective;

:: WAYBEL18:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is injective
for b2 being non empty SubSpace of b1
      st b2 is_a_retract_of b1
   holds b2 is injective;

:: WAYBEL18:funcnot 7 => WAYBEL18:func 7
definition
  let a1 be 1-sorted;
  let a2 be TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func Image A3 -> SubSpace of a2 equals
    a2 | rng a3;
end;

:: WAYBEL18:def 6
theorem
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   Image b3 = b2 | rng b3;

:: WAYBEL18:funcreg 3
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  cluster Image a3 -> non empty;
end;

:: WAYBEL18:th 10
theorem
for b1 being 1-sorted
for b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   the carrier of Image b3 = rng b3;

:: WAYBEL18:funcnot 8 => WAYBEL18:func 8
definition
  let a1 be 1-sorted;
  let a2 be non empty TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  func corestr A3 -> Function-like quasi_total Relation of the carrier of a1,the carrier of Image a3 equals
    a3;
end;

:: WAYBEL18:def 7
theorem
for b1 being 1-sorted
for b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   corestr b3 = b3;

:: WAYBEL18:th 11
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is continuous(b1, b2)
   holds corestr b3 is continuous(b1, Image b3);

:: WAYBEL18:funcreg 4
registration
  let a1 be 1-sorted;
  let a2 be non empty TopStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  cluster corestr a3 -> Function-like quasi_total onto;
end;

:: WAYBEL18:prednot 1 => WAYBEL18:pred 1
definition
  let a1, a2 be TopStruct;
  pred A1 is_Retract_of A2 means
    ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a2 st
       b1 is continuous(a2, a2) & b1 * b1 = b1 & Image b1,a1 are_homeomorphic;
end;

:: WAYBEL18:dfs 8
definiens
  let a1, a2 be TopStruct;
To prove
     a1 is_Retract_of a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total Relation of the carrier of a2,the carrier of a2 st
       b1 is continuous(a2, a2) & b1 * b1 = b1 & Image b1,a1 are_homeomorphic;

:: WAYBEL18:def 8
theorem
for b1, b2 being TopStruct holds
   b1 is_Retract_of b2
iff
   ex b3 being Function-like quasi_total Relation of the carrier of b2,the carrier of b2 st
      b3 is continuous(b2, b2) & b3 * b3 = b3 & Image b3,b1 are_homeomorphic;

:: WAYBEL18:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
   st b1 is injective
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st corestr b3 is being_homeomorphism(b1, Image b3)
   holds b1 is_Retract_of b2;

:: WAYBEL18:funcnot 9 => WAYBEL18:func 9
definition
  func Sierpinski_Space -> strict TopStruct means
    the carrier of it = {0,1} &
     the topology of it = {{},{1},{0,1}};
end;

:: WAYBEL18:def 9
theorem
for b1 being strict TopStruct holds
      b1 = Sierpinski_Space
   iff
      the carrier of b1 = {0,1} &
       the topology of b1 = {{},{1},{0,1}};

:: WAYBEL18:funcreg 5
registration
  cluster Sierpinski_Space -> non empty strict TopSpace-like;
end;

:: WAYBEL18:funcreg 6
registration
  cluster Sierpinski_Space -> strict discerning;
end;

:: WAYBEL18:funcreg 7
registration
  cluster Sierpinski_Space -> strict injective;
end;

:: WAYBEL18:funcreg 8
registration
  let a1 be set;
  let a2 be non empty 1-sorted;
  cluster a1 --> a2 -> non-Empty;
end;

:: WAYBEL18:funcreg 9
registration
  let a1 be set;
  let a2 be TopStruct;
  cluster a1 --> a2 -> TopSpace-yielding;
end;

:: WAYBEL18:funcreg 10
registration
  let a1 be set;
  let a2 be reflexive RelStr;
  cluster a1 --> a2 -> reflexive-yielding;
end;

:: WAYBEL18:funcreg 11
registration
  let a1 be non empty set;
  let a2 be non empty antisymmetric RelStr;
  cluster product (a1 --> a2) -> strict antisymmetric;
end;

:: WAYBEL18:funcreg 12
registration
  let a1 be non empty set;
  let a2 be non empty transitive RelStr;
  cluster product (a1 --> a2) -> strict transitive;
end;

:: WAYBEL18:th 13
theorem
for b1 being Scott TopAugmentation of BoolePoset 1 holds
   the topology of b1 = the topology of Sierpinski_Space;

:: WAYBEL18:th 14
theorem
for b1 being non empty set holds
   {product ((Carrier (b1 --> Sierpinski_Space)) +*(b2,{1})) where b2 is Element of b1: TRUE} is prebasis of product (b1 --> Sierpinski_Space);

:: WAYBEL18:funcreg 13
registration
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric with_suprema with_infima complete RelStr;
  cluster product (a1 --> a2) -> strict with_suprema complete;
end;

:: WAYBEL18:funcreg 14
registration
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric with_suprema with_infima lower-bounded algebraic RelStr;
  cluster product (a1 --> a2) -> strict algebraic;
end;

:: WAYBEL18:th 15
theorem
for b1 being non empty set holds
   ex b2 being Function-like quasi_total Relation of the carrier of BoolePoset b1,the carrier of product (b1 --> BoolePoset 1) st
      b2 is isomorphic(BoolePoset b1, product (b1 --> BoolePoset 1)) &
       (for b3 being Element of bool b1 holds
          b2 . b3 = chi(b3,b1));

:: WAYBEL18:th 16
theorem
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> BoolePoset 1) holds
   the topology of b2 = the topology of product (b1 --> Sierpinski_Space);

:: WAYBEL18:th 17
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st the carrier of b1 = the carrier of b2 & the topology of b1 = the topology of b2 & b1 is injective
   holds b2 is injective;

:: WAYBEL18:th 18
theorem
for b1 being non empty set
for b2 being Scott TopAugmentation of product (b1 --> BoolePoset 1) holds
   b2 is injective;

:: WAYBEL18:th 19
theorem
for b1 being non empty TopSpace-like discerning TopStruct holds
   ex b2 being non empty set st
      ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of product (b2 --> Sierpinski_Space) st
         corestr b3 is being_homeomorphism(b1, Image b3);

:: WAYBEL18:th 20
theorem
for b1 being non empty TopSpace-like discerning TopStruct
      st b1 is injective
   holds ex b2 being non empty set st
      b1 is_Retract_of product (b2 --> Sierpinski_Space);