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;

theorem Th9:
  F is closed iff COMPLEMENT(F) is open
proof
  thus F is closed implies COMPLEMENT(F) is open
  proof
    assume
A1: F is closed;
    let P;
    assume P in COMPLEMENT(F);
    then P` in F by SETFAM_1:def 7;
    then P` is closed by A1;
    hence thesis by TOPS_1:4;
  end;
  assume
A2: COMPLEMENT(F) is open;
  let P such that
A3: P in F;
  P``=P;
  then P` in COMPLEMENT(F) by A3,SETFAM_1:def 7;
  then P` is open by A2;
  hence thesis;
end;
