In topological space formula_31 the empty set formula_16 and the set formula_8 itself are always open. These two sets the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see why formula_8 is clopen, begin by recalling that the sets formula_8 and formula_16 are, by definition, always open subsets (of formula_8). Also by definition, a subset formula_24 is called if (and only if) its complement in formula_39 which is the set formula_40 is an open subset. Because the complement (in formula_8) of the entire set formula_42 is the empty set (i.e. formula_43), which is an open subset, this means that formula_44 is a closed subset of formula_8 (by definition of "closed subset"). Hence, no matter what topology is placed on formula_39 the entire space formula_8 is simultaneously both an open subset and also a closed subset of formula_8; said differently, formula_8 is a clopen subset of formula_50 Because the empty set's complement is formula_51 which is an open subset, the same reasoning can be used to conclude that formula_52 is also a clopen subset of formula_50