Definition 25.6.1 (Admissible Approximant Function).label Let $X$ be a topological space and $\mathcal{A}: X \to 2^{X}$, then $\mathcal{A}$ is an admissible approximant function on $X$ if:

1. (AA1)

   For each $x \in X$, $x \in \overline{\mathcal{A}(x)^o}$.

2. (AA2)

   $\bigcap_{x \in X}\mathcal{A}(x) \ne \emptyset$.

and $\mathcal{A}$ is Borel measurable if:

1. (B)

   For any $x_{0} \in X$, $\bracs{x \in X|x_0 \in \mathcal{A}(x)}\in \cb_{X}$.