
theorem Th8:
  for T being non empty TopSpace holds T_0-canonical_map(T) is open
proof
  let T be non empty TopSpace;
  set F = T_0-canonical_map(T);
  for A being Subset of T st A is open holds F.:A is open
  proof
    set D = Indiscernible(T);
A1: F = proj D by BORSUK_1:def 8;
    let A be Subset of T such that
A2: A is open;
A3: for C being Subset of T st C in D & C meets A holds C c= A by A2,Th7;
    set A9 = F.:A;
    A9 is Subset of D by BORSUK_1:def 7;
    then F"A9 = union A9 by A1,EQREL_1:67;
    then A = union A9 by A1,A3,EQREL_1:69;
    then union A9 in the topology of T by A2;
    hence thesis by Th2;
  end;
  hence thesis;
end;
