Approximations and decompositions in \(S^ 3\) (Q1083722)

Taming theory for surfaces in Euclidean 3-space has made continued use of tame approximations to those surfaces. The authors investigate the extent to which k, \(k>1\), approximations can be made to intersect the sphere and each other nicely. The conclusion contains a number of technical conditions which we shall not repeat. However, the result is powerful enough to serve as the basis of a new proof of E. Woodruff's 2-sphere decomposition theorem. The result relies heavily on W. T. Eaton's difficult two-sided approximation theorem for 2-spheres.