Definition 4

*Let $(f_i)_{i \in I}$ be a family of real-valued functions defined on a set $X$. The real-valued function on $X$ whose value at each point $x \in X$ is $\sup_{i \in I} (f_i(x))$ [resp. $\inf_{i \in I} (f_i(x))$] is called the upper (resp. lower) envelope of the family $(f_i)$, and is denoted by $\sup_{i \in I} f_i$ or $\sup_i f_i$ (resp. $\inf_{i \in I} f_i$ or $\inf_i f_i$).

The upper envelope of the family $(f_i)$ is thus the least upper bound of this family in the lattice $\overline{\mathbf{R}}$ of real-valued functions defined on $X$, and this justifies the notation $\sup_{i \in I} f_i$.

Furthermore, if we endow $\overline{\mathbf{R}}^X$ with the topology which is the product of the topologies of its factors (all identical with $\overline{\mathbf{R}}$), we have the following proposition: