A Selection Theorem for the Carathéodory Kernel Convergence of Pointed Domains

We present a selection theorem for domains in ${\mathbb{C}}^n$, $n\ge 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carathéodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of holomorphic maps with varying domains and ranges.