Singular side approximations for $2$-spheres in $E^{3}$

Abstract

Bing has proved that each $2$-sphere in $E^{3}$ can almost be mapped free of itself in the following very nice sense: Suppose that $S$ is a $2$-sphere in $E^{3}$ and $\varepsilon > 0$; then there is an $\varepsilon$-map $$f:S \rightarrow S \cup \mathrm{Int}\,S$$ such that $f(S)\cap S$ and $f^{-1}(f(S)\cap S)$ are $0$-dimensional and $$f|S - f^{-1} (S) \cap S$$ is a homeomorphism. This paper illustrates how Bing's theorem can be used advantageously as a substitute for Bing's original side approximation theorem. The following are the principal results. (1) A $2$-sphere $S$ is tame if it is (singularly) spanned or capped on tame sets.(2) A $2$-sphere $S$ is tame if each of its points is an inaccessible point of a Sierpiński curve in $S$ which can be pushed by a homotopy into each complementary domain of $S$.