Article CONNSP_2, MML version 4.99.1005

:: CONNSP_2:modenot 1 => CONNSP_2:mode 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  mode a_neighborhood of A2 -> Element of bool the carrier of a1 means
    a2 in Int it;
end;

:: CONNSP_2:dfs 1
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a3 is a_neighborhood of a2
it is sufficient to prove
  thus a2 in Int a3;

:: CONNSP_2:def 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is a_neighborhood of b2
   iff
      b2 in Int b3;

:: CONNSP_2:modenot 2 => CONNSP_2:mode 2
definition
  let a1 be TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  mode a_neighborhood of A2 -> Element of bool the carrier of a1 means
    a2 c= Int it;
end;

:: CONNSP_2:dfs 2
definiens
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a3 is a_neighborhood of a2
it is sufficient to prove
  thus a2 c= Int a3;

:: CONNSP_2:def 2
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b3 is a_neighborhood of b2
iff
   b2 c= Int b3;

:: CONNSP_2:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
      st b3 is a_neighborhood of b2 & b4 is a_neighborhood of b2
   holds b3 \/ b4 is a_neighborhood of b2;

:: CONNSP_2:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
   b3 is a_neighborhood of b2 & b4 is a_neighborhood of b2
iff
   b3 /\ b4 is a_neighborhood of b2;

:: CONNSP_2: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 the carrier of b1
      st b2 is open(b1) & b3 in b2
   holds b2 is a_neighborhood of b3;

:: CONNSP_2: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 the carrier of b1
      st b2 is a_neighborhood of b3
   holds b3 in b2;

:: CONNSP_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is a_neighborhood of b2
   holds ex b4 being non empty Element of bool the carrier of b1 st
      b4 is a_neighborhood of b2 & b4 is open(b1) & b4 c= b3;

:: CONNSP_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is a_neighborhood of b2
   iff
      ex b4 being Element of bool the carrier of b1 st
         b4 is open(b1) & b4 c= b3 & b2 in b4;

:: CONNSP_2:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      for b3 being Element of the carrier of b1
            st b3 in b2
         holds ex b4 being Element of bool the carrier of b1 st
            b4 is a_neighborhood of b3 & b4 c= b2;

:: CONNSP_2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is a_neighborhood of {b2}
   iff
      b3 is a_neighborhood of b2;

:: CONNSP_2:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of the carrier of b1 | b2
for b4 being Element of bool the carrier of b1 | b2
for b5 being Element of bool the carrier of b1
for b6 being Element of the carrier of b1
      st b2 is open(b1) & b4 is a_neighborhood of b3 & b4 = b5 & b3 = b6
   holds b5 is a_neighborhood of b6;

:: CONNSP_2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of the carrier of b1 | b2
for b4 being Element of bool the carrier of b1 | b2
for b5 being Element of bool the carrier of b1
for b6 being Element of the carrier of b1
      st b5 is a_neighborhood of b6 & b4 = b5 & b3 = b6
   holds b4 is a_neighborhood of b3;

:: CONNSP_2:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is_a_component_of b1 & b2 c= b3
   holds b2 is_a_component_of b3;

:: CONNSP_2:th 14
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 b1
for b4 being Element of the carrier of b2
      st b3 = b4
   holds Component_of b4 c= Component_of b3;

:: CONNSP_2:prednot 1 => CONNSP_2:pred 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  pred A1 is_locally_connected_in A2 means
    for b1 being Element of bool the carrier of a1
          st b1 is a_neighborhood of a2
       holds ex b2 being Element of bool the carrier of a1 st
          b2 is a_neighborhood of a2 & b2 is connected(a1) & b2 c= b1;
end;

:: CONNSP_2:dfs 3
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
To prove
     a1 is_locally_connected_in a2
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 is a_neighborhood of a2
       holds ex b2 being Element of bool the carrier of a1 st
          b2 is a_neighborhood of a2 & b2 is connected(a1) & b2 c= b1;

:: CONNSP_2:def 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
      b1 is_locally_connected_in b2
   iff
      for b3 being Element of bool the carrier of b1
            st b3 is a_neighborhood of b2
         holds ex b4 being Element of bool the carrier of b1 st
            b4 is a_neighborhood of b2 & b4 is connected(b1) & b4 c= b3;

:: CONNSP_2:attrnot 1 => CONNSP_2:attr 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  attr a1 is locally_connected means
    for b1 being Element of the carrier of a1 holds
       a1 is_locally_connected_in b1;
end;

:: CONNSP_2:dfs 4
definiens
  let a1 be non empty TopSpace-like TopStruct;
To prove
     a1 is locally_connected
it is sufficient to prove
  thus for b1 being Element of the carrier of a1 holds
       a1 is_locally_connected_in b1;

:: CONNSP_2:def 4
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally_connected
   iff
      for b2 being Element of the carrier of b1 holds
         b1 is_locally_connected_in b2;

:: CONNSP_2:prednot 2 => CONNSP_2:pred 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
  pred A2 is_locally_connected_in A3 means
    for b1 being non empty Element of bool the carrier of a1
          st a2 = b1
       holds ex b2 being Element of the carrier of a1 | b1 st
          b2 = a3 & a1 | b1 is_locally_connected_in b2;
end;

:: CONNSP_2:dfs 5
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of bool the carrier of a1;
  let a3 be Element of the carrier of a1;
To prove
     a2 is_locally_connected_in a3
it is sufficient to prove
  thus for b1 being non empty Element of bool the carrier of a1
          st a2 = b1
       holds ex b2 being Element of the carrier of a1 | b1 st
          b2 = a3 & a1 | b1 is_locally_connected_in b2;

:: CONNSP_2:def 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 the carrier of b1 holds
      b2 is_locally_connected_in b3
   iff
      for b4 being non empty Element of bool the carrier of b1
            st b2 = b4
         holds ex b5 being Element of the carrier of b1 | b4 st
            b5 = b3 & b1 | b4 is_locally_connected_in b5;

:: CONNSP_2:attrnot 2 => CONNSP_2:attr 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty Element of bool the carrier of a1;
  attr a2 is locally_connected means
    a1 | a2 is locally_connected;
end;

:: CONNSP_2:dfs 6
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty Element of bool the carrier of a1;
To prove
     a2 is locally_connected
it is sufficient to prove
  thus a1 | a2 is locally_connected;

:: CONNSP_2:def 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1 holds
      b2 is locally_connected(b1)
   iff
      b1 | b2 is locally_connected;

:: CONNSP_2:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
      b1 is_locally_connected_in b2
   iff
      for b3, b4 being Element of bool the carrier of b1
            st b3 is a_neighborhood of b2 & b4 is_a_component_of b3 & b2 in b4
         holds b4 is a_neighborhood of b2;

:: CONNSP_2:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
      b1 is_locally_connected_in b2
   iff
      for b3 being non empty Element of bool the carrier of b1
            st b3 is open(b1) & b2 in b3
         holds ex b4 being Element of the carrier of b1 | b3 st
            b4 = b2 & b2 in Int Component_of b4;

:: CONNSP_2:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally_connected
for b2 being Element of bool the carrier of b1
      st b2 is_a_component_of b1
   holds b2 is open(b1);

:: CONNSP_2:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
   st b1 is_locally_connected_in b2
for b3 being non empty Element of bool the carrier of b1
      st b3 is open(b1) & b2 in b3
   holds b3 is_locally_connected_in b2;

:: CONNSP_2:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally_connected
for b2 being non empty Element of bool the carrier of b1
      st b2 is open(b1)
   holds b2 is locally_connected(b1);

:: CONNSP_2:th 24
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is locally_connected
   iff
      for b2 being non empty Element of bool the carrier of b1
      for b3 being Element of bool the carrier of b1
            st b2 is open(b1) & b3 is_a_component_of b2
         holds b3 is open(b1);

:: CONNSP_2:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally_connected
for b2 being non empty Element of bool the carrier of b1
for b3 being non empty Element of bool the carrier of b1 | b2
      st b3 is connected(b1 | b2) & b3 is open(b1 | b2)
   holds ex b4 being Element of bool the carrier of b1 st
      b4 is open(b1) & b4 is connected(b1) & b3 = b2 /\ b4;

:: CONNSP_2:th 26
theorem
for b1 being non empty TopSpace-like TopStruct holds
      b1 is being_T4
   iff
      for b2, b3 being Element of bool the carrier of b1
            st b2 <> {} & b3 <> [#] b1 & b2 c= b3 & b2 is closed(b1) & b3 is open(b1)
         holds ex b4 being Element of bool the carrier of b1 st
            b4 is open(b1) & b2 c= b4 & Cl b4 c= b3;

:: CONNSP_2:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
   st b1 is locally_connected & b1 is being_T4
for b2, b3 being Element of bool the carrier of b1
      st b2 <> {} & b3 <> {} & b2 is closed(b1) & b3 is closed(b1) & b2 misses b3 & b2 is connected(b1) & b3 is connected(b1)
   holds ex b4, b5 being Element of bool the carrier of b1 st
      b4 is connected(b1) & b5 is connected(b1) & b4 is open(b1) & b5 is open(b1) & b2 c= b4 & b3 c= b5 & b4 misses b5;

:: CONNSP_2:th 28
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 open(b1) & b4 is closed(b1) & b2 in b4
   holds b3 <> {};

:: CONNSP_2:funcnot 1 => CONNSP_2:func 1
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be Element of the carrier of a1;
  func qComponent_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 open(a1) & b2 is closed(a1) & a2 in b2) &
        meet b1 = it;
end;

:: CONNSP_2:def 7
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 = qComponent_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 open(b1) & b5 is closed(b1) & b2 in b5) &
          meet b4 = b3;

:: CONNSP_2:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   b2 in qComponent_of b2;

:: CONNSP_2:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
      st b3 is open(b1) & b3 is closed(b1) & b2 in b3 & b3 c= qComponent_of b2
   holds b3 = qComponent_of b2;

:: CONNSP_2:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   qComponent_of b2 is closed(b1);

:: CONNSP_2:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
   Component_of b2 c= qComponent_of b2;

:: CONNSP_2:th 34
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 holds
      b3 in Cl b2
   iff
      for b4 being a_neighborhood of b3 holds
         b4 meets b2;

:: CONNSP_2:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   [#] b1 is a_neighborhood of b2;

:: CONNSP_2:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being a_neighborhood of b2 holds
   b2 c= b3;