A Selection Theorem for the Carath\'eodory 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 Carath\'eodory. 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.