reserve p1, p2 for Point of TOP-REAL 2,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2;

theorem
  for T1,T2 being non empty TopSpace, f being Function of T1,T2 st f is
  being_homeomorphism for A being Subset of T1 st A is connected holds f.:A is
  connected by TOPS_2:61;
