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
  f is being_homeomorphism iff dom f = [#]T & rng f = [#]S & f is
  one-to-one & for P holds P is closed iff f.:P is closed
proof
  hereby
    assume
A1: f is being_homeomorphism;
    hence
A2: dom f = [#]T & rng f = [#]S & f is one-to-one;
    let P;
A3: f" is continuous by A1;
    hereby
      assume
A4:   P is closed;
      f is onto by A2,FUNCT_2:def 3;
      then (f")"P = ((f qua Function)")"P by A2,Def4
        .= f.:P by A2,FUNCT_1:84;
      hence f.:P is closed by A3,A4;
    end;
    assume
A5: f.:P is closed;
    f is continuous by A1;
    then
A6: f"(f.:P) is closed by A5;
    dom f = [#]T by FUNCT_2:def 1;
    then
A7: P c= f"(f.:P) by FUNCT_1:76;
    f"(f.:P) c= P by A2,FUNCT_1:82;
    hence P is closed by A6,A7,XBOOLE_0:def 10;
  end;
  assume that
A8: dom f = [#]T and
A9: rng f = [#]S and
A10: f is one-to-one;
  assume
A11: for P being Subset of T holds P is closed iff f.:P is closed;
A12: f is continuous
  proof
    let B be Subset of S such that
A13: B is closed;
    set D = f"B;
    B = f.:D by A9,FUNCT_1:77;
    hence thesis by A11,A13;
  end;
  f" is continuous
  proof
    let B be Subset of T such that
A14: B is closed;
    f is onto by A9,FUNCT_2:def 3;
    then (f")"B = ((f qua Function)")"B by A10,Def4
      .= f.:B by A10,FUNCT_1:84;
    hence thesis by A11,A14;
  end;
  hence thesis by A8,A9,A10,A12;
end;
