CNED sets: countably negligible for extremal distances

Abstract

The author has recently introduced the class of \( CNED \) sets in Euclidean space, generalizing the classical notion of \( NED \) sets, and shown that they are quasiconformally removable. A set E is \( CNED \) if the conformal modulus of a curve family is not affected when one restricts to the subfamily intersecting E at countably many points. We prove that several classes of sets that were known to be removable are also \( CNED \), including sets of \(\sigma \)-finite Hausdorff \((n-1)\)-measure and boundaries of domains with n-integrable quasihyperbolic distance. Thus, this work puts in common framework many known results on the problem of quasiconformal removability and suggests that the \( CNED \) condition should also be necessary for removability. We give a new necessary and sufficient criterion for closed sets to be (C)NED. Applying this criterion, we show that countable unions of closed (C)NED sets are (C)NED. Therefore we enlarge significantly the known classes of quasiconformally removable sets.