reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;
reserve X,Y for non empty TopSpace;

theorem
  (for Y being non empty TopSpace, f being Function of X,Y holds f is
  continuous) implies X is discrete
proof
  set Y = 1TopSp(the carrier of X);
  reconsider f = id the carrier of X as Function of X,Y;
  assume for Y being non empty TopSpace, f being Function of X,Y holds f is
  continuous;
  then
A1: f is continuous;
  for A being Subset of X holds A is closed
  proof
    let A be Subset of X;
    reconsider B = A as Subset of Y;
A2: f"B = A by FUNCT_2:94;
    B is closed by TDLAT_3:16;
    hence thesis by A1,A2;
  end;
  hence thesis by TDLAT_3:16;
end;
