Definition 5.26.1 (Open Preimage Function).label Let $X$ be a set, $(Y, \topo)$ be a topological space, and $P: \topo \to 2^{X}$, then $P$ is an open preimage function if

1. (PF1)

   $P(\emptyset) = \emptyset$.

2. (PF2’)

   For each $\mathcal{U}\subset \topo$, $\bigcup_{U \in \mathcal{U}}P(U) = P\paren{\bigcup_{U \in \mathcal{U}}U}$.

3. (PF3’)

   For each $U, V \in \topo$, $P(U \cap V) = P(U) \cap P(V)$.