reserve x, y for set,
  T for TopStruct,
  GX for TopSpace,
  P, Q, M, N for Subset of T,
  F, G for Subset-Family of T,
  W, Z for Subset-Family of GX,
  A for SubSpace of T;
reserve S for non empty TopStruct,
  f for Function of T, S,
  H for Subset-Family of S;
reserve T for non empty TopSpace,
  S for TopSpace,
  P1 for Subset of S,
  f for Function of T, S;
reserve T for TopSpace,
  S for non empty TopSpace,
  P for Subset of T,
  f for Function of T, S;
reserve GX,GY for non empty TopSpace;

theorem
  for X being TopStruct, F being Subset-Family of X holds
  F is closed iff F c= COMPLEMENT the topology of X
  proof
    let X be TopStruct, F be Subset-Family of X;
    thus F is closed implies F c= COMPLEMENT the topology of X
    proof
      assume
A1:   F is closed;
      let A be object;
      assume
A2:   A in F;
      then reconsider A as Subset of X;
      A is closed by A1,A2;
      then A` is open;
      then A` in the topology of X;
      hence thesis by SETFAM_1:def 7;
    end;
    assume
A3: F c= COMPLEMENT the topology of X;
    let A be Subset of X;
    assume A in F;
    then A` in the topology of X by A3,SETFAM_1:def 7;
    then A` is open;
    hence thesis;
  end;
