For a more advanced example reminiscent of the discrete topology, suppose that formula_75 is an ultrafilter on a non-empty set formula_50 Then the union formula_77 is a topology on formula_8 with the property that non-empty proper subset formula_24 of formula_8 is an open subset or else a closed subset, but never both; that is, if formula_81 (where formula_82) then of the following two statements is true: either (1) formula_29 or else, (2) formula_30 Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are formula_16 and formula_50