Article CONNSP_1, MML version 4.99.1005

:: CONNSP_1:prednot 1 => CONNSP_1:pred 1
definition
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2,A3 are_separated means
    Cl a2 misses a3 & a2 misses Cl a3;
  symmetry;
::  for a1 being TopStruct
::  for a2, a3 being Element of bool the carrier of a1
::        st a2,a3 are_separated
::     holds a3,a2 are_separated;
end;

:: CONNSP_1:dfs 1
definiens
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2,a3 are_separated
it is sufficient to prove
  thus Cl a2 misses a3 & a2 misses Cl a3;

:: CONNSP_1:def 1
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   Cl b2 misses b3 & b2 misses Cl b3;

:: CONNSP_1:th 2
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 are_separated
   holds b2 misses b3;

:: CONNSP_1:th 3
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st [#] b1 = b2 \/ b3 & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3
   holds b2,b3 are_separated;

:: CONNSP_1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st [#] b1 = b2 \/ b3 & b2 is open(b1) & b3 is open(b1) & b2 misses b3
   holds b2,b3 are_separated;

:: CONNSP_1:th 5
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st [#] b1 = b2 \/ b3 & b2,b3 are_separated
   holds b2 is open(b1) & b2 is closed(b1) & b3 is open(b1) & b3 is closed(b1);

:: CONNSP_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
      st b5 = b3 & b6 = b4 & b5,b6 are_separated
   holds b3,b4 are_separated;

:: CONNSP_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5, b6 being Element of bool the carrier of b2
      st b3 = b5 & b4 = b6 & b3 \/ b4 c= [#] b2 & b3,b4 are_separated
   holds b5,b6 are_separated;

:: CONNSP_1:th 8
theorem
for b1 being TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b3 are_separated & b4 c= b2 & b5 c= b3
   holds b4,b5 are_separated;

:: CONNSP_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b3 are_separated & b2,b4 are_separated
   holds b2,b3 \/ b4 are_separated;

:: CONNSP_1:th 10
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st (b2 is closed(b1) & b3 is closed(b1) or b2 is open(b1) & b3 is open(b1))
   holds b2 \ b3,b3 \ b2 are_separated;

:: CONNSP_1:attrnot 1 => CONNSP_1:attr 1
definition
  let a1 be TopStruct;
  attr a1 is connected means
    for b1, b2 being Element of bool the carrier of a1
          st [#] a1 = b1 \/ b2 & b1,b2 are_separated & b1 <> {} a1
       holds b2 = {} a1;
end;

:: CONNSP_1:dfs 2
definiens
  let a1 be TopStruct;
To prove
     a1 is connected
it is sufficient to prove
  thus for b1, b2 being Element of bool the carrier of a1
          st [#] a1 = b1 \/ b2 & b1,b2 are_separated & b1 <> {} a1
       holds b2 = {} a1;

:: CONNSP_1:def 2
theorem
for b1 being TopStruct holds
      b1 is connected
   iff
      for b2, b3 being Element of bool the carrier of b1
            st [#] b1 = b2 \/ b3 & b2,b3 are_separated & b2 <> {} b1
         holds b3 = {} b1;

:: CONNSP_1:th 11
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is connected
   iff
      for b2, b3 being Element of bool the carrier of b1
            st [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is closed(b1) & b3 is closed(b1)
         holds b2 meets b3;

:: CONNSP_1:th 12
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is connected
   iff
      for b2, b3 being Element of bool the carrier of b1
            st [#] b1 = b2 \/ b3 & b2 <> {} b1 & b3 <> {} b1 & b2 is open(b1) & b3 is open(b1)
         holds b2 meets b3;

:: CONNSP_1:th 13
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is connected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 <> {} b1 & b2 <> [#] b1
         holds Cl b2 meets Cl (([#] b1) \ b2);

:: CONNSP_1:th 14
theorem
for b1 being TopSpace-like TopStruct holds
      b1 is connected
   iff
      for b2 being Element of bool the carrier of b1
            st b2 is open(b1) & b2 is closed(b1) & b2 <> {} b1
         holds b2 = [#] b1;

:: CONNSP_1:th 15
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
      st b3 is continuous(b1, b2) & b3 .: [#] b1 = [#] b2 & b1 is connected
   holds b2 is connected;

:: CONNSP_1:attrnot 2 => CONNSP_1:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is connected means
    a1 | a2 is connected;
end;

:: CONNSP_1:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is connected
it is sufficient to prove
  thus a1 | a2 is connected;

:: CONNSP_1:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is connected(b1)
   iff
      b1 | b2 is connected;

:: CONNSP_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is connected(b1)
   iff
      for b3, b4 being Element of bool the carrier of b1
            st b2 = b3 \/ b4 & b3,b4 are_separated & b3 <> {} b1
         holds b4 = {} b1;

:: CONNSP_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 is connected(b1) & b2 c= b3 \/ b4 & b3,b4 are_separated & not b2 c= b3
   holds b2 c= b4;

:: CONNSP_1:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is connected(b1) & b3 is connected(b1) & not b2,b3 are_separated
   holds b2 \/ b3 is connected(b1);

:: CONNSP_1:th 19
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is connected(b1) & b2 c= b3 & b3 c= Cl b2
   holds b3 is connected(b1);

:: CONNSP_1:th 20
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is connected(b1)
   holds Cl b2 is connected(b1);

:: CONNSP_1:th 21
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b1 is connected & b2 is connected(b1) & ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated
   holds b2 \/ b3 is connected(b1) & b2 \/ b4 is connected(b1);

:: CONNSP_1:th 22
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st ([#] b1) \ b2 = b3 \/ b4 & b3,b4 are_separated & b2 is closed(b1)
   holds b2 \/ b3 is closed(b1) & b2 \/ b4 is closed(b1);

:: CONNSP_1:th 23
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is connected(b1) & b2 meets b3 & b2 \ b3 <> {} b1
   holds b2 meets Fr b3;

:: CONNSP_1:th 24
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b3 is connected(b1)
   iff
      b4 is connected(b2);

:: CONNSP_1:th 25
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1) & b2 \/ b3 is connected(b1) & b2 /\ b3 is connected(b1)
   holds b2 is connected(b1) & b3 is connected(b1);

:: CONNSP_1:th 26
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st (for b3 being Element of bool the carrier of b1
               st b3 in b2
            holds b3 is connected(b1)) &
         (ex b3 being Element of bool the carrier of b1 st
            b3 <> {} b1 &
             b3 in b2 &
             (for b4 being Element of bool the carrier of b1
                   st b4 in b2 & b4 <> b3
                holds not b3,b4 are_separated))
   holds union b2 is connected(b1);

:: CONNSP_1:th 27
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st (for b3 being Element of bool the carrier of b1
               st b3 in b2
            holds b3 is connected(b1)) &
         meet b2 <> {} b1
   holds union b2 is connected(b1);

:: CONNSP_1:th 28
theorem
for b1 being TopSpace-like TopStruct holds
      [#] b1 is connected(b1)
   iff
      b1 is connected;

:: CONNSP_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   {b2} is connected(b1);

:: CONNSP_1:prednot 2 => CONNSP_1:pred 2
definition
  let a1 be TopStruct;
  let a2, a3 be Element of the carrier of a1;
  pred A2,A3 are_joined means
    ex b1 being Element of bool the carrier of a1 st
       b1 is connected(a1) & a2 in b1 & a3 in b1;
end;

:: CONNSP_1:dfs 4
definiens
  let a1 be TopStruct;
  let a2, a3 be Element of the carrier of a1;
To prove
     a2,a3 are_joined
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a1 st
       b1 is connected(a1) & a2 in b1 & a3 in b1;

:: CONNSP_1:def 4
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1 holds
   b2,b3 are_joined
iff
   ex b4 being Element of bool the carrier of b1 st
      b4 is connected(b1) & b2 in b4 & b3 in b4;

:: CONNSP_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
      st ex b2 being Element of the carrier of b1 st
           for b3 being Element of the carrier of b1 holds
              b2,b3 are_joined
   holds b1 is connected;

:: CONNSP_1:th 31
theorem
for b1 being TopSpace-like TopStruct holds
      ex b2 being Element of the carrier of b1 st
         for b3 being Element of the carrier of b1 holds
            b2,b3 are_joined
   iff
      for b2, b3 being Element of the carrier of b1 holds
      b2,b3 are_joined;

:: CONNSP_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
      st for b2, b3 being Element of the carrier of b1 holds
        b2,b3 are_joined
   holds b1 is connected;

:: CONNSP_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool bool the carrier of b1
      st for b4 being Element of bool the carrier of b1 holds
              b4 in b3
           iff
              b4 is connected(b1) & b2 in b4
   holds b3 <> {};

:: CONNSP_1:prednot 3 => CONNSP_1:pred 3
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  pred A2 is_a_component_of A1 means
    a2 is connected(a1) &
     (for b1 being Element of bool the carrier of a1
           st b1 is connected(a1) & a2 c= b1
        holds a2 = b1);
end;

:: CONNSP_1:dfs 5
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is_a_component_of a1
it is sufficient to prove
  thus a2 is connected(a1) &
     (for b1 being Element of bool the carrier of a1
           st b1 is connected(a1) & a2 c= b1
        holds a2 = b1);

:: CONNSP_1:def 5
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is_a_component_of b1
   iff
      b2 is connected(b1) &
       (for b3 being Element of bool the carrier of b1
             st b3 is connected(b1) & b2 c= b3
          holds b2 = b3);

:: CONNSP_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is_a_component_of b1
   holds b2 <> {} b1;

:: CONNSP_1:th 35
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
      st b2 is_a_component_of b1
   holds b2 is closed(b1);

:: CONNSP_1:th 36
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is_a_component_of b1 & b3 is_a_component_of b1 & b2 <> b3
   holds b2,b3 are_separated;

:: CONNSP_1:th 37
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is_a_component_of b1 & b3 is_a_component_of b1 & b2 <> b3
   holds b2 misses b3;

:: CONNSP_1:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
   st b2 is connected(b1)
for b3 being Element of bool the carrier of b1
      st b3 is_a_component_of b1 & b2 meets b3
   holds b2 c= b3;

:: CONNSP_1:prednot 4 => CONNSP_1:pred 4
definition
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  pred A3 is_a_component_of A2 means
    ex b1 being Element of bool the carrier of a1 | a2 st
       b1 = a3 & b1 is_a_component_of a1 | a2;
end;

:: CONNSP_1:dfs 6
definiens
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a3 is_a_component_of a2
it is sufficient to prove
  thus ex b1 being Element of bool the carrier of a1 | a2 st
       b1 = a3 & b1 is_a_component_of a1 | a2;

:: CONNSP_1:def 6
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b3 is_a_component_of b2
iff
   ex b4 being Element of bool the carrier of b1 | b2 st
      b4 = b3 & b4 is_a_component_of b1 | b2;

:: CONNSP_1:th 39
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b1 is connected & b2 is connected(b1) & b3 is_a_component_of ([#] b1) \ b2
   holds ([#] b1) \ b3 is connected(b1);

:: CONNSP_1:funcnot 1 => CONNSP_1:func 1
definition
  let a1 be TopStruct;
  let a2 be Element of the carrier of a1;
  func Component_of A2 -> Element of bool the carrier of a1 means
    ex b1 being Element of bool bool the carrier of a1 st
       (for b2 being Element of bool the carrier of a1 holds
              b2 in b1
           iff
              b2 is connected(a1) & a2 in b2) &
        union b1 = it;
end;

:: CONNSP_1:def 7
theorem
for b1 being TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 = Component_of b2
   iff
      ex b4 being Element of bool bool the carrier of b1 st
         (for b5 being Element of bool the carrier of b1 holds
                b5 in b4
             iff
                b5 is connected(b1) & b2 in b5) &
          union b4 = b3;

:: CONNSP_1:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   b2 in Component_of b2;

:: CONNSP_1:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   Component_of b2 is connected(b1);

:: CONNSP_1:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b2 is connected(b1) & Component_of b3 c= b2
   holds b2 = Component_of b3;

:: CONNSP_1:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is_a_component_of b1
   iff
      ex b3 being Element of the carrier of b1 st
         b2 = Component_of b3;

:: CONNSP_1:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b2 is_a_component_of b1 & b3 in b2
   holds b2 = Component_of b3;

:: CONNSP_1:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
   st b2 = Component_of b3
for b4 being Element of the carrier of b1
      st b4 in b2
   holds Component_of b4 = b2;

:: CONNSP_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool bool the carrier of b1
      st for b3 being Element of bool the carrier of b1 holds
              b3 in b2
           iff
              b3 is_a_component_of b1
   holds b2 is_a_cover_of b1;