Definition 5.26.3 (Basic Preimage Function).label Let $X$ be a set, $Y$ be a topological space, $\mathcal{B}$ be a base for $Y$ with $\emptyset \in \mathcal{B}$, and $p: \mathcal{B}\to 2^{X}$, then $p$ is a basic preimage function if:

1. (PF1)

   $P(\emptyset) = \emptyset$.

2. (PF2’)

   For each $\mathcal{U}\subset \mathcal{B}$ and $V \in \mathcal{B}$ with $V \subset \bigcup_{U \in \mathcal{U}}U$, $p(V) \subset \bigcup_{U \in \mathcal{U}}p(U)$.

3. (PF3’)

   For each $U, V \in \mathcal{B}$, $p(U) \cap p(V) \subset \bigcup_{W \in \mathcal{B}, W \subset U \cap V}p(W)$.