reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;

theorem
  f is continuous iff for O being open Subset of NTY holds
  f"O is open Subset of NTX
  proof
    hereby
      assume f is continuous;
      then for A being Subset of NTX holds f.:(Cl A) c= Cl(f.:A)
        by Lm13;
      then for S being closed Subset of NTY holds f"S is closed Subset of NTX
        by Lm23;
      hence for O being open Subset of NTY holds f"O is open Subset of NTX
        by Lm24;
    end;
    assume for O being open Subset of NTY holds f"O is open Subset of NTX;
    hence f is continuous by Lm26;
  end;
