
theorem Th16:
  for T,S being non empty TopSpace st the carrier of T = the
  carrier of S & the topology of T = the topology of S & T is injective holds S
  is injective
proof
  let T,S be non empty TopSpace;
  assume that
A1: the carrier of T = the carrier of S and
A2: the topology of T = the topology of S and
A3: T is injective;
  let X be non empty TopSpace;
  let f be Function of X,S;
  reconsider f9 = f as Function of X,T by A1;
A4: [#]S <> {};
  assume
A5: f is continuous;
A6: for P being Subset of T st P is open holds (f9)"P is open
  proof
    let P be Subset of T;
    reconsider P9 = P as Subset of S by A1;
    assume P is open;
    then P9 in the topology of S by A2;
    then P9 is open;
    hence thesis by A4,A5,TOPS_2:43;
  end;
  let Y be non empty TopSpace;
  assume
A7: X is SubSpace of Y;
A8: [#]T <> {};
  then f9 is continuous by A6,TOPS_2:43;
  then consider h being Function of Y,T such that
A9: h is continuous and
A10: h|(the carrier of X) = f9 by A3,A7;
  reconsider g = h as Function of Y,S by A1;
  take g;
  for P being Subset of S st P is open holds g"P is open
  proof
    let P be Subset of S;
    reconsider P9 = P as Subset of T by A1;
    assume P is open;
    then P9 in the topology of T by A2;
    then P9 is open;
    hence thesis by A8,A9,TOPS_2:43;
  end;
  hence g is continuous by A4,TOPS_2:43;
  thus thesis by A10;
end;
