In Bing’s Side
Approximation Theorem for 2-spheres in Ea the disks on the approximating
sphere and the disks on the given sphere may intersect in a very complicated
manner. It is shown in this paper that the disks may be chosen so that there
are the same number of disks on the approximating sphere as on the given
sphere and the disks intersect in a one-to-one fashion. Furthermore, the
approximating homeomorphism may be chosen so that it maps each disk
on the given sphere onto the disk on the approximating sphere which it
intersects.
Applications are given to a study of the preservation of tameness of subsets of the
boundary of a crumpled cube under re-embeddings of the crumpled cube in
E8.
