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

theorem
  for X,Y being non empty TopSpace, f being continuous Function of X,Y
  st f is one-to-one onto holds f is being_homeomorphism iff f is closed
proof
  let X,Y be non empty TopSpace;
  let f be continuous Function of X,Y such that
A1: f is one-to-one onto;
  thus f is being_homeomorphism implies f is closed by TOPS_2:58;
  assume
A2: for A being Subset of X st A is closed holds f.:A is closed;
A3: [#] X = the carrier of X & [#] Y = the carrier of Y;
A4: dom f = the carrier of X by FUNCT_2:def 1;
A5: now
    let A be Subset of X;
    assume f.:A is closed;
    then f"(f.:A) is closed by PRE_TOPC:def 6;
    hence A is closed by A1,A4,FUNCT_1:94;
  end;
  thus thesis by A1,A2,A4,A3,A5,TOPS_2:58;
end;
