
theorem Th33:
  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 closed
proof
  let S be TopSpace, T be non empty TopSpace,
  BB be Basis of T, f be Function of S,T;
A1: BB c= the topology of T by TOPS_2:64;
  hereby
    assume
A2: f is continuous;
    let A be Subset of T;
    assume A in BB;
    then A is open by A1;
    then A` is closed by TOPS_1:4;
    hence f"A` is closed by A2;
  end;
  assume
A3: for A being Subset of T st A in BB holds f"A` is closed;
  let A be Subset of T;
  assume A is closed;
  then A` is open by TOPS_1:3;
  then
A4: A` = union {G where G is Subset of T: G in BB & G c= A`} by YELLOW_8:9;
  set F = {f"G where G is Subset of T: G in BB & G c= A`};
  F c= bool the carrier of S
  proof
    let a be object;
    assume a in F;
    then ex G being Subset of T st a = f"G & G in BB & G c= A`;
    hence thesis;
  end;
  then reconsider F as Subset-Family of S;
  reconsider F as Subset-Family of S;
  F c= the topology of S
  proof
    let a be object;
    assume a in F;
    then consider G being Subset of T such that
A5: a = f"G and
A6: G in BB and G c= A`;
A7: f"G` is closed by A3,A6;
    (f"G)` = f"G` by Th2;
    then f"G is open by A7,TOPS_1:4;
    hence thesis by A5;
  end;
  then
A8: union F in the topology of S by PRE_TOPC:def 1;
  defpred P[Subset of T] means $1 in BB & $1 c= A`;
  deffunc F(Subset of T) = $1;
  f"union {F(G) where G is Subset of T: P[G]}
  = union {f"F(G) where G is Subset of T: P[G]} from ABC;
  then f"A` is open by A4,A8;
  then (f"A)` is open by Th2;
  hence thesis by TOPS_1:3;
end;
