Editing Directed set (section)

====Finite subsets====

The set <math>\operatorname{Finite}(I)</math> of all finite subsets of a set <math>I</math> is directed with respect to <math>\,\subseteq\,</math> since given any two <math>A, B \in \operatorname{Finite}(I),</math> their union <math>A \cup B \in \operatorname{Finite}(I)</math> is an upper bound of <math>A</math> and <math>B</math> in <math>\operatorname{Finite}(I).</math> This particular directed set is used to define the sum <math>{\textstyle\sum\limits_{i \in I}} r_i</math> of a [[Generalized series (mathematics)|generalized series]] of an <math>I</math>-indexed collection of numbers <math>\left(r_i\right)_{i \in I}</math> (or more generally, the sum of [[Series (mathematics)#Abelian topological groups|elements in an]] [[abelian topological group]], such as [[Series (mathematics)#Series in topological vector spaces|vectors]] in a [[topological vector space]]) as the [[Limit of a net|limit of the net]] of [[partial sum]]s <math>F \in \operatorname{Finite}(I) \mapsto {\textstyle\sum\limits_{i \in F}} r_i;</math> that is:
<math display=block>\sum_{i \in I} r_i ~:=~ \lim_{F \in \operatorname{Finite}(I)} \ \sum_{i \in F} r_i ~=~ \lim \left\{\sum_{i \in F} r_i \,: F \subseteq I, F \text{ finite }\right\}.</math>