spaces that are not metric spaces. Special types of open sets Clopen sets and non-open and/or non-closed sets A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset a closed subset. Such subsets are known as . Explicitly, a subset of a topological space is called if both and its complement are open subsets of ; or equivalently, if and In topological space the empty set and the set itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see why is clopen, begin by recalling that the sets and are, by definition, always open subsets (of ). Also by definition, a subset is called if (and only if) its complement in which is the set is an open subset. Because the complement (in ) of the entire set is the empty set (i.e. ), which is an open subset, this means that is a closed subset of (by definition of "closed subset"). Hence, no matter what topology is placed on the entire space is simultaneously both an open subset and also a closed subset of ; said differently, is a clopen subset of Because the empty set's complement is which is an open subset, the same reasoning can be used to conclude that is also a clopen subset of Consider the real line endowed with its usual Euclidean topology, whose open sets are defined as follows: every interval of real numbers belongs to the topology, every union of such intervals, e.g. belongs to the topology, and as always, both and belong to the topology. The interval is open in because it belongs to the Euclidean topology. If were to have an open complement, it would mean by definition that were closed. But does not have an open complement; its complement is which does belong to the Euclidean topology since it is not a union of open intervals of the form Hence, is an example of a set that is open but not closed. By a similar argument, the interval is a closed subset but not an open subset. Finally, since neither nor its complement belongs to the Euclidean topology (because it can not