\begin{definition}[Definition:Pointwise Supremum]
Let $S$ be a set.
Let $\struct {T, \preceq}$ be an ordered set.
Let $\family {f_i}_{i \mathop \in I}, f_i: S \to T$ be an $I$-indexed collection of mappings.
Suppose that for all $s \in S$, it holds that:
:$\ds \sup_{i \mathop \in I} \map {f_i} s \in T$
where the supremum is taken in $T$.
Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\ds \sup_{i \mathop \in I} f_i: S \to T$, is defined by:
:$\ds \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$
where the latter supremum is again taken in $T$.
By assumption, this supremum is guaranteed to exist.
Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation.
\end{definition}