reserve T for TopSpace,
  A, B for Subset of T;

theorem Th37: :: Remark 1
  A is 1st_class iff Border A is empty
proof
A1: Border A is empty implies A is 1st_class
  proof
    assume Border A is empty;
    then Int Cl A \ Cl Int A = {} by Th21;
    then Int Cl A c= Cl Int A by XBOOLE_1:37;
    hence thesis;
  end;
  A is 1st_class implies Border A is empty
  proof
    assume A is 1st_class;
    then Int Cl A c= Cl Int A;
    then Int Cl A \ Cl Int A = {} by XBOOLE_1:37;
    hence thesis by Th21;
  end;
  hence thesis by A1;
end;
