Definition 25.6.3 (Approximation of the Identity).label Let $X$ be a topological space and $\net{I}\subset X^{X}$ be a net, then $\net{I}$ is an approximation of the identity if:

1. (AI1)

   For each $x \in X$, $I_{\alpha}(x) \to x$.

For any admissible approximant function $\mathcal{A}: X \to 2^{X}$, $\net{I}$ is $\mathcal{A}$-admissible if:

1. (AI2)

   For each $x \in X$ and $\alpha \in A$, $I_{\alpha}(x) \in \mathcal{A}(x)$.

The approximation $\net{I}$ is simple if $I_{\alpha}$ is finitely-valued for all $\alpha \in A$, and Borel measurable if $I_{\alpha}$ is Borel measurable for all $\alpha \in A$.