reserve X for set,
  Y for non empty set;

theorem Th6:
  for T being TopSpace for A be Subset of T holds A
  is_a_component_of {}T iff A is empty
proof
  let T be TopSpace;
  let A be Subset of T;
  thus A is_a_component_of {}T implies A is empty by SPRECT_1:5,XBOOLE_1:3;
  assume
A1: A is empty;
  then reconsider B = A as Subset of T|{}T by XBOOLE_1:2;
  for C being Subset of T|{}T st C is connected holds B c= C implies B = C
  by A1;
  then B is a_component by A1,CONNSP_1:def 5;
  hence thesis by CONNSP_1:def 6;
end;
