
theorem
  for S being TopSpace,T being non empty TopSpace, K being Basis of T
  for f being Function of S,T holds
  f is continuous iff for A being Subset of T st A in K holds f"A is open
proof
  let S be TopSpace,T be non empty TopSpace, K be Basis of T;
  let f be Function of S,T;
  hereby
    assume
A1: f is continuous;
    let A be Subset of T;
    assume A in K;
    then f"A` is closed by A1,Th33;
    then (f"A)` is closed by Th2;
    hence f"A is open by TOPS_1:4;
  end;
  assume
A2: for A being Subset of T st A in K holds f"A is open;
  now
    let A be Subset of T;
    assume A in K;
    then f"A is open by A2;
    then (f"A)` is closed by TOPS_1:4;
    hence f"A` is closed by Th2;
  end;
  hence thesis by Th33;
end;
