Infinitary operation

Remarks

Example: on $[0, 1]$, the map $s \colon [0,1]^{\N} \to [0, 1]$, $s((x_n)_{n \in \N}) = \sup_n x_n$, is an infinitary operation of arity $\aleph_0$. Each sequence in $[0, 1]$ is bounded above by $1$, so its supremum exists and lies in $[0, 1]$.

Non-example: the map $(x_n)_{n \in \N} \mapsto \sum_{n} x_n$ is not an infinitary operation on $\R$, because it is not defined on the whole of $\R^{\N}$: the series $\sum_n 1$ diverges.

Some authors index the arguments by an infinite ordinal $\alpha$ instead of an arbitrary set $I$, and call $\alpha$ the arity.