Definition 6.1.1 (Set-Open Topology). Let $T$ be a set, $\mathfrak{S}\subset 2^{T}$ be a non-empty family of sets, and $(X, \topo)$ be a topological space. For each $S \in \mathfrak{S}$ and $U \subset X$ open, let
\[M(S, U) = \bracs{f \in X^T| f(S) \subset U}\]
and
\[\ce(\mathfrak{S}, \topo) = \bracs{M(S, U)| S \in \mathfrak{S}, U \in \topo}\]
then the topology generated by $\ce$ is the $\mathfrak{S}$-open topology on $T^{X}$.
If $\cb \subset \topo$ generates $\topo$, then $\ce(\mathfrak{S}, \cb)$ generates the $\mathfrak{S}$-open topology.