Example

If $f$ and $g$ are two functions in $\mathscr{H}(\mathfrak{F}, \mathbf{R})$, the relation "there is an $X \in \mathfrak{F}$ such that $f$ and $g$ are defined on $X$ and $f(t) \leq g(t)$ for every $t \in X$" is of local character relative to $f$ and $g$. We denote by $\tilde{f} \leq \tilde{g}$ the relation obtained by passing to the quotient (for $f$ and $g$); we remark that if $\tilde{f} \leq \tilde{g}$ then there exist a function $f_1 \in \tilde{f}$ and a function $g_1 \in \tilde{g}$, defined on all of E, such that $f_1(t) \leq g_1(t)$ for all $t \in E$.