Remark

— 1) The $\mathfrak{S}$*-topology* is identical with the $\mathfrak{S}'$*-topology*, where $\mathfrak{S}'$ denotes the bornology generated by $\mathfrak{S}$ (III, p. 1).

2) Let $M \in \mathfrak{S}$ and let V be a neighbourhood of 0 in F; let $T(M, V)$ denote the set of all $f \in F^E$ such that $f(x) \in V$ for every $x \in M$. If $\mathfrak{S}$ is stable under finite unions, the sets $T(M, V)$ form a fundamental system of neighbourhoods of 0 for the $\mathfrak{S}$*-topology* of $F^E$.