An Ideal Boundary for Domains in N-space

The Royden ideal boundary of a domain is the set of all points in the maximal ideal space of the domain's Royden algebra that do not lie in the domain. Elements of the Royden ideal boundary can be characterized as nets convergent in both the weak and Euclidean topologies that have no subnet which is a sequence. As with other function algebras, boundary bers can be deened as subsets of all points in the ideal boundary that project onto a single Euclidean boundary point. Quasiconformal mappings have homeomorphic extensions if and only if the adjoint of the corresponding Royden algebra isomorphism maps boundary bers to boundary bers. For domains with certain homogeneity properties, all boundary bers are homeomorphic; and for domains nitely connected on the boundary, bers are equivalent to prime ends. In any case, each boundary ber has a complicated topology which contains the complement of the natural numbers in the Stone{Cech compactiication of the natural numbers.