Article TEX_1, MML version 4.99.1005

:: TEX_1:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 modified_with_respect_to b2
      st b3 = b4 & b3 is open(b1)
   holds b4 is open(b1 modified_with_respect_to b2);

:: TEX_1:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 modified_with_respect_to b2
      st b3 = b4 & b3 is closed(b1)
   holds b4 is closed(b1 modified_with_respect_to b2);

:: TEX_1:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 modified_with_respect_to (b2 `)
      st b3 = b2
   holds b3 is closed(b1 modified_with_respect_to (b2 `));

:: TEX_1:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 modified_with_respect_to b2
      st b3 = b2 & b2 is dense(b1)
   holds b3 is dense(b1 modified_with_respect_to b2) & b3 is open(b1 modified_with_respect_to b2);

:: TEX_1:th 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 bool the carrier of b1 modified_with_respect_to b2
      st b2 c= b3 & b2 is dense(b1)
   holds b3 is everywhere_dense(b1 modified_with_respect_to b2);

:: TEX_1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 modified_with_respect_to (b2 `)
      st b3 = b2 & b2 is boundary(b1)
   holds b3 is boundary(b1 modified_with_respect_to (b2 `)) & b3 is closed(b1 modified_with_respect_to (b2 `));

:: TEX_1:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of b1 modified_with_respect_to (b2 `)
      st b3 c= b2 & b2 is boundary(b1)
   holds b3 is nowhere_dense(b1 modified_with_respect_to (b2 `));

:: TEX_1:attrnot 1 => STRUCT_0:attr 7
definition
  let a1 be 1-sorted;
  attr a1 is trivial means
    ex b1 being Element of the carrier of a1 st
       the carrier of a1 = {b1};
end;

:: TEX_1:dfs 1
definiens
  let a1 be non empty 1-sorted;
To prove
     a1 is trivial
it is sufficient to prove
  thus ex b1 being Element of the carrier of a1 st
       the carrier of a1 = {b1};

:: TEX_1:def 1
theorem
for b1 being non empty 1-sorted holds
      b1 is trivial
   iff
      ex b2 being Element of the carrier of b1 st
         the carrier of b1 = {b2};

:: TEX_1:exreg 1
registration
  cluster non empty trivial strict TopStruct;
end;

:: TEX_1:exreg 2
registration
  cluster non empty non trivial strict TopStruct;
end;

:: TEX_1:th 8
theorem
for b1 being non empty trivial TopStruct
      st the topology of b1 is not empty & b1 is almost_discrete
   holds b1 is TopSpace-like;

:: TEX_1:exreg 3
registration
  cluster non empty trivial strict TopSpace-like TopStruct;
end;

:: TEX_1:condreg 1
registration
  cluster non empty trivial TopSpace-like -> discrete anti-discrete (TopStruct);
end;

:: TEX_1:condreg 2
registration
  cluster non empty TopSpace-like discrete anti-discrete -> trivial (TopStruct);
end;

:: TEX_1:exreg 4
registration
  cluster non empty non trivial strict TopSpace-like TopStruct;
end;

:: TEX_1:condreg 3
registration
  cluster non empty TopSpace-like non discrete -> non trivial (TopStruct);
end;

:: TEX_1:condreg 4
registration
  cluster non empty TopSpace-like non anti-discrete -> non trivial (TopStruct);
end;

:: TEX_1:funcnot 1 => TEX_1:func 1
definition
  let a1 be set;
  func cobool A1 -> Element of bool bool a1 equals
    {{},a1};
end;

:: TEX_1:def 2
theorem
for b1 being set holds
   cobool b1 = {{},b1};

:: TEX_1:funcreg 1
registration
  let a1 be set;
  cluster cobool a1 -> non empty;
end;

:: TEX_1:funcnot 2 => TEX_1:func 2
definition
  let a1 be set;
  func ADTS A1 -> TopStruct equals
    TopStruct(#a1,cobool a1#);
end;

:: TEX_1:def 3
theorem
for b1 being set holds
   ADTS b1 = TopStruct(#b1,cobool b1#);

:: TEX_1:funcreg 2
registration
  let a1 be set;
  cluster ADTS a1 -> strict TopSpace-like anti-discrete;
end;

:: TEX_1:funcreg 3
registration
  let a1 be non empty set;
  cluster ADTS a1 -> non empty;
end;

:: TEX_1:th 11
theorem
for b1 being non empty TopSpace-like anti-discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 is empty implies Cl b2 = {}) & (b2 is empty or Cl b2 = the carrier of b1);

:: TEX_1:th 12
theorem
for b1 being non empty TopSpace-like anti-discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 = the carrier of b1 or Int b2 = {}) &
    (b2 = the carrier of b1 implies Int b2 = the carrier of b1);

:: TEX_1:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
              st b2 is not empty
           holds Cl b2 = the carrier of b1
   holds b1 is anti-discrete;

:: TEX_1:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
              st b2 <> the carrier of b1
           holds Int b2 = {}
   holds b1 is anti-discrete;

:: TEX_1:th 15
theorem
for b1 being non empty TopSpace-like anti-discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   (b2 = {} or b2 is dense(b1)) & (b2 = the carrier of b1 or b2 is boundary(b1));

:: TEX_1:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
              st b2 <> {}
           holds b2 is dense(b1)
   holds b1 is anti-discrete;

:: TEX_1:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1
              st b2 <> the carrier of b1
           holds b2 is boundary(b1)
   holds b1 is anti-discrete;

:: TEX_1:funcreg 4
registration
  let a1 be set;
  cluster 1TopSp a1 -> discrete;
end;

:: TEX_1:th 20
theorem
for b1 being non empty TopSpace-like discrete TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2 = b2 & Int b2 = b2;

:: TEX_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1 holds
           Cl b2 = b2
   holds b1 is discrete;

:: TEX_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2 being Element of bool the carrier of b1 holds
           Int b2 = b2
   holds b1 is discrete;

:: TEX_1:th 23
theorem
for b1 being non empty set holds
      ADTS b1 = 1TopSp b1
   iff
      ex b2 being Element of b1 st
         b1 = {b2};

:: TEX_1:exreg 5
registration
  cluster non empty strict TopSpace-like discrete non anti-discrete TopStruct;
end;

:: TEX_1:exreg 6
registration
  cluster non empty strict TopSpace-like non discrete anti-discrete TopStruct;
end;

:: TEX_1:funcnot 3 => TEX_1:func 3
definition
  let a1 be set;
  let a2 be Element of a1;
  func STS(A1,A2) -> TopStruct equals
    TopStruct(#a1,(bool a1) \ {b1 where b1 is Element of bool a1: a2 in b1 & b1 <> a1}#);
end;

:: TEX_1:def 5
theorem
for b1 being set
for b2 being Element of b1 holds
   STS(b1,b2) = TopStruct(#b1,(bool b1) \ {b3 where b3 is Element of bool b1: b2 in b3 & b3 <> b1}#);

:: TEX_1:funcreg 5
registration
  let a1 be set;
  let a2 be Element of a1;
  cluster STS(a1,a2) -> strict TopSpace-like;
end;

:: TEX_1:funcreg 6
registration
  let a1 be non empty set;
  let a2 be Element of a1;
  cluster STS(a1,a2) -> non empty;
end;

:: TEX_1:th 24
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Element of bool the carrier of STS(b1,b2) holds
   ({b2} c= b3 implies b3 is closed(STS(b1,b2))) &
    (b3 is not empty & b3 is closed(STS(b1,b2)) implies {b2} c= b3);

:: TEX_1:th 25
theorem
for b1 being non empty set
for b2 being Element of b1
   st b1 \ {b2} is not empty
for b3 being Element of bool the carrier of STS(b1,b2) holds
   (b3 = {b2} implies b3 is closed(STS(b1,b2)) & b3 is boundary(STS(b1,b2))) &
    (b3 is not empty & b3 is closed(STS(b1,b2)) & b3 is boundary(STS(b1,b2)) implies b3 = {b2});

:: TEX_1:th 26
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Element of bool the carrier of STS(b1,b2) holds
   (b3 c= b1 \ {b2} implies b3 is open(STS(b1,b2))) &
    (b3 <> b1 & b3 is open(STS(b1,b2)) implies b3 c= b1 \ {b2});

:: TEX_1:th 27
theorem
for b1 being non empty set
for b2 being Element of b1
   st b1 \ {b2} is not empty
for b3 being Element of bool the carrier of STS(b1,b2) holds
   (b3 = b1 \ {b2} implies b3 is open(STS(b1,b2)) & b3 is dense(STS(b1,b2))) &
    (b3 <> b1 & b3 is open(STS(b1,b2)) & b3 is dense(STS(b1,b2)) implies b3 = b1 \ {b2});

:: TEX_1:exreg 7
registration
  cluster non empty strict TopSpace-like non discrete non anti-discrete TopStruct;
end;

:: TEX_1:th 28
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace-like TopStruct holds
      TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of STS(b1,b2),the topology of STS(b1,b2)#)
   iff
      the carrier of b3 = b1 &
       (for b4 being Element of bool the carrier of b3 holds
          ({b2} c= b4 implies b4 is closed(b3)) &
           (b4 is not empty & b4 is closed(b3) implies {b2} c= b4));

:: TEX_1:th 29
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace-like TopStruct holds
      TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of STS(b1,b2),the topology of STS(b1,b2)#)
   iff
      the carrier of b3 = b1 &
       (for b4 being Element of bool the carrier of b3 holds
          (b4 c= b1 \ {b2} implies b4 is open(b3)) &
           (b4 <> b1 & b4 is open(b3) implies b4 c= b1 \ {b2}));

:: TEX_1:th 30
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace-like TopStruct holds
      TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of STS(b1,b2),the topology of STS(b1,b2)#)
   iff
      the carrier of b3 = b1 &
       (for b4 being non empty Element of bool the carrier of b3 holds
          Cl b4 = b4 \/ {b2});

:: TEX_1:th 31
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace-like TopStruct holds
      TopStruct(#the carrier of b3,the topology of b3#) = TopStruct(#the carrier of STS(b1,b2),the topology of STS(b1,b2)#)
   iff
      the carrier of b3 = b1 &
       (for b4 being Element of bool the carrier of b3
             st b4 <> b1
          holds Int b4 = b4 \ {b2});

:: TEX_1:th 32
theorem
for b1 being non empty set
for b2 being Element of b1 holds
      STS(b1,b2) = ADTS b1
   iff
      b1 = {b2};

:: TEX_1:th 33
theorem
for b1 being non empty set
for b2 being Element of b1 holds
      STS(b1,b2) = 1TopSp b1
   iff
      b1 = {b2};

:: TEX_1:th 34
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Element of bool the carrier of STS(b1,b2)
      st b3 = {b2}
   holds 1TopSp b1 = (STS(b1,b2)) modified_with_respect_to b3;

:: TEX_1:attrnot 2 => TDLAT_3:attr 1
definition
  let a1 be TopStruct;
  attr a1 is discrete means
    for b1 being non empty Element of bool the carrier of a1 holds
       b1 is not boundary(a1);
end;

:: TEX_1:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is discrete
it is sufficient to prove
  thus for b1 being non empty Element of bool the carrier of a1 holds
       b1 is not boundary(a1);

:: TEX_1:def 6
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is discrete
   iff
      for b2 being non empty Element of bool the carrier of b1 holds
         b2 is not boundary(b1);

:: TEX_1:th 35
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 <> the carrier of b1
         holds b2 is not dense(b1);

:: TEX_1:condreg 5
registration
  cluster non empty TopSpace-like non almost_discrete -> non discrete non anti-discrete (TopStruct);
end;

:: TEX_1:attrnot 3 => TDLAT_3:attr 3
definition
  let a1 be TopStruct;
  attr a1 is almost_discrete means
    for b1 being non empty Element of bool the carrier of a1 holds
       b1 is not nowhere_dense(a1);
end;

:: TEX_1:dfs 6
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is almost_discrete
it is sufficient to prove
  thus for b1 being non empty Element of bool the carrier of a1 holds
       b1 is not nowhere_dense(a1);

:: TEX_1:def 7
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being non empty Element of bool the carrier of b1 holds
         b2 is not nowhere_dense(b1);

:: TEX_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is almost_discrete
   iff
      for b2 being Element of bool the carrier of b1
            st b2 <> the carrier of b1
         holds b2 is not everywhere_dense(b1);

:: TEX_1:th 37
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is not almost_discrete
   iff
      ex b2 being non empty Element of bool the carrier of b1 st
         b2 is boundary(b1) & b2 is closed(b1);

:: TEX_1:th 38
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is not almost_discrete
   iff
      ex b2 being Element of bool the carrier of b1 st
         b2 <> the carrier of b1 & b2 is dense(b1) & b2 is open(b1);

:: TEX_1:exreg 8
registration
  cluster non empty strict TopSpace-like non discrete non anti-discrete almost_discrete TopStruct;
end;

:: TEX_1:th 39
theorem
for b1 being non empty set
for b2 being Element of b1 holds
      b1 \ {b2} is not empty
   iff
      STS(b1,b2) is not almost_discrete;

:: TEX_1:exreg 9
registration
  cluster non empty strict TopSpace-like non almost_discrete TopStruct;
end;

:: TEX_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
      st b2 is boundary(b1)
   holds b1 modified_with_respect_to (b2 `) is not almost_discrete;

:: TEX_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 <> the carrier of b1 & b2 is dense(b1)
   holds b1 modified_with_respect_to b2 is not almost_discrete;