Remark 2

However, the uniformity of $\mathscr{S}$-convergence is unaltered by replacing $\mathscr{S}$ by the set $\mathscr{S}'$ of all subsets of $X$ which are contained in finite unions of sets of $\mathscr{S}$. In the study of $\mathscr{S}$-convergence we may therefore always restrict ourselves to the case where the set $\mathscr{S}$ satisfies the following two conditions:

(F'$_1$) Every subset of a set of $\mathscr{S}$ belongs to $\mathscr{S}$.

(F'$_2$) Every finite union of sets of $\mathscr{S}$ belongs to $\mathscr{S}$.

If (F'$_2$) is satisfied, we obtain a fundamental system of entourages of $\mathscr{F}_\mathscr{S}(X; Y)$ by taking all the sets $W(A, V)$, where $A$ runs through $\mathscr{S}$ and $V$ runs through a fundamental system of entourages of $Y$.