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;
reserve T for non empty TopSpace,
  S for TopSpace,
  P1 for Subset of S,
  f for Function of T, S;
reserve T for TopSpace,
  S for non empty TopSpace,
  P for Subset of T,
  f for Function of T, S;

theorem
  f is being_homeomorphism iff dom f = [#]T & rng f = [#]S & f is
  one-to-one & for P holds f.:(Cl P) = Cl(f.:P)
proof
  hereby
    assume
A1: f is being_homeomorphism;
    hence
A2: dom f = [#]T & rng f = [#]S & f is one-to-one;
    let P;
    f is continuous by A1;
    then
A3: f.:(Cl P) c= Cl(f.:P) by Th45;
    f" is continuous by A1;
    then
A4: Cl((f")"P) c= (f")"(Cl P) by Th44;
    (f")"P = f.:P & (f")"(Cl P) = f.:(Cl P) by A2,Th54;
    hence f.:(Cl P) = Cl(f.:P) by A3,A4,XBOOLE_0:def 10;
  end;
  assume that
A5: dom f = [#]T and
A6: rng f = [#]S & f is one-to-one;
  assume
A7: for P holds f.:(Cl P) = Cl(f.:P);
  thus dom f = [#]T & rng f = [#]S & f is one-to-one by A5,A6;
  for P holds f.:(Cl P) c= Cl(f.:P) by A7;
  hence f is continuous by Th45;
  now
    let P;
    (f")"P = f.:P & (f")"(Cl P) = f.:(Cl P) by A6,Th54;
    hence Cl((f")"P) c= (f")"(Cl P) by A7;
  end;
  hence thesis by Th44;
end;
