
theorem
  for T being TopSpace st the TopStruct of T is connected holds T is connected
proof
  let T be TopSpace;
  set G = the TopStruct of T;
  assume
A1: G is connected;
  let A, B be Subset of T such that
A2: [#]T = A \/ B & A,B are_separated;
  reconsider A1 = A, B1 = B as Subset of G;
  [#]G = A1 \/ B1 & A1,B1 are_separated by A2,Th6;
  then A1 = {}G or B1 = {}G by A1;
  hence thesis;
end;
