reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;

theorem Th43:
  for X,Y being TopStruct, f being Function of X,Y st [#]Y = {}
  implies [#]X = {} holds f is continuous iff for P being Subset of Y st P is
  open holds f"P is open
proof
  let X,Y be TopStruct, f be Function of X,Y;
  assume
A1: [#]Y={} implies [#]X={};
  hereby
    assume
A2: f is continuous;
    let P1 be Subset of Y;
    assume P1 is open;
    then P1` is closed by TOPS_1:4;
    then
A3: f"(P1`) is closed by A2;
    f"(P1`) = f"([#]Y) \ f"P1 by FUNCT_1:69
      .= [#]X \ f"P1 by A1,Th41
      .= (f"P1)`;
    hence f"P1 is open by A3,TOPS_1:4;
  end;
  assume
A4: for P1 being Subset of Y st P1 is open holds f"P1 is open;
  let P1 be Subset of Y;
  assume P1 is closed;
  then P1` is open;
  then
A5: f"(P1`) is open by A4;
  f"(P1`) = f"([#]Y) \ f"P1 by FUNCT_1:69
    .= [#]X \ f"P1 by A1,Th41
    .= (f"P1)`;
  hence thesis by A5;
end;
