Spheres with continuous tangent planes
\textit{C. E. Burgess} [Spheres with a continuous family of tangent planes: A negative answer to Problem 12, in: The Scottish book. Mathematics from the Scottish Café, 76-81 (1981; Zbl 0485.01013)] solved Problem 12 in The Scottish Book by exhibiting a wild 2-sphere in \(E^ 3\) having a continuous family of tangent planes. The wildness of the spheres described by Burgess [loc. cit.] occurs at points of the 2-sphere \(\Sigma\) that belong to its rim. The rim R of \(\Sigma\) is the set of all points q of \(\Sigma\) where the normal to some tangent plane to \(\Sigma\) at q fails to pierce \(\Sigma\) at q. In Burgess' paper [loc. cit.] the rim of \(\Sigma\) is a simple closed curve containing the single wild point of \(\Sigma\). The original motivation for this paper came from a desire to better understand the rim of \(\Sigma\) and its relation to the wild set. A point q of a 2-sphere \(\Sigma\) in \(E^ 3\) is said to belong to the wild set W of \(\Sigma\) if there is no 2-cell K in \(\Sigma\) such that q lies in Int K and K lies on a tame 2-sphere in \(E^ 3\). Example 4.2 describes a 2-sphere \(\Sigma\) in \(E^ 3\) with a continuous family of tangent planes, a 1-dimensional wild set, and a rim that is the union of a countable sequence of disjoint simple closed curves. The close connection between the existence of a tangent plane to \(\Sigma\) at q and the existence of a double cone touching \(\Sigma\) only at its vertex q is explored in the next section where these concepts, when properly stated, are shown to be equivalent. An analysis of the nature of the rim R of \(\Sigma\) when \(\Sigma\) has a family F of tangent planes is given in {\S} 3. Although the wild set W of a 2-sphere \(\Sigma\) with tangent planes must lie in the closure of R (Theorem 4.1), it need not lie in R. From the results of {\S} 2 and {\S} 3 it follows that W has dimension less than two when \(\Sigma\) has a continuous family F of tangent planes (Theorem 4.3). Furthermore, the set Q of all points of W that are components of W must be dense in W according to Theorem 4.5. This means W cannot be a connected set when F is continuous and W is nondegenerate.
scientific article; zbMATH DE number 4133122