reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;

theorem
  for T being TopSpace, FX being Subset-Family of T holds clf FX is closed
proof
  let T be TopSpace, FX be Subset-Family of T;
  for V being Subset of T st V in clf FX holds V is closed
  proof
    let V be Subset of T;
    assume V in clf FX;
    then ex W being Subset of T st V = Cl W & W in FX by Def2;
    hence thesis;
  end;
  hence thesis by TOPS_2:def 2;
end;
