A Selection Theorem for the Carath\'eodory Kernel Convergence of Domains in Cⁿ

We present a selection theorem for domains in Cⁿ, n≥ 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 to the study of normal families of holomorphic maps with variable domains and ranges. In particular, it applies to the rescaling method initiated by Pinchuk.