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
  W is closed & W is finite implies union W is closed
proof
  reconsider C = COMPLEMENT(W) as Subset-Family of GX;
  assume W is closed & W is finite;
  then COMPLEMENT(W) is open & COMPLEMENT(W) is finite by Th8,Th9;
  then
A1: meet C is open by Th20;
  now
    assume W <> {};
    then meet COMPLEMENT(W) = (union W)` by Th6;
    hence thesis by A1;
  end;
  hence thesis by ZFMISC_1:2;
end;
