Mapping cylinder neighborhoods in the plane

Abstract

We characterize those compact subsets of the plane which have mapping cylinder neighborhoods, describe the neighborhood closures, and show that such neighborhood closures are topologically unique. The proofs employ the notion of prime ends. We also show that if U U is a mapping cylinder neighborhood of a pointlike continuum in S 3 {S^3} , then U ¯ \overline U is a 3 3 -cell.