output; that is, the system output approaches/approximates a pure tone. Generalized definitions The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions. Definition for a set The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set. That is, Similarly, the limit superior of X is the supremum of all of the limit points of the set. That is, Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set. Definition for filter bases Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by where is the closure of . This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as when that supremum exists. When X has a total order, is a complete lattice and has the order topology, Similarly, the limit inferior of the filter base B is defined as when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point. Specialization for sequences and nets Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space and the net , where is a directed set and for all . The filter base ("of tails") generated by this net is defined by Therefore, the limit inferior and limit superior of the net are equal to the limit superior and