section_path: Generalizations/Summations over arbitrary index sets/Examples
section_name: Examples
paragraph_text: Examples
  
  a function whose support is a singleton { a }. {\displaystyle \{a\}.} Then
  
  f = ∑ a ∈ X f a {\displaystyle f=\sum _{a\in X}f_{a}}
  
  in the topology of pointwise convergence (that is, the sum is taken in the infinite product group Y X {\displaystyle Y^{X}}).
  
  While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given x, {\displaystyle x,} only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is locally finite, that is, for every x {\displaystyle x} there is a neighborhood of x {\displaystyle x} in which all but a finite number of functions vanish. Any regularity property of the φ i, {\displaystyle \varphi _{i},} such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.
  
  (in other words, ω 1 {\displaystyle \omega _{1}} copies of 1 is ω 1 {\displaystyle \omega _{1}}) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.