## Illinois Journal of Mathematics

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

J. W. Cannon

#### 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 $S$.

#### Article information

Source
Illinois J. Math., Volume 18, Issue 1 (1974), 27-36.

Dates
First available in Project Euclid: 20 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1256051346

Digital Object Identifier
doi:10.1215/ijm/1256051346

Mathematical Reviews number (MathSciNet)
MR0334215

Zentralblatt MATH identifier
0273.57002

Subjects
Primary: 57A10

#### Citation

Cannon, J. W. Singular side approximations for $2$-spheres in $E^{3}$. Illinois J. Math. 18 (1974), no. 1, 27--36. doi:10.1215/ijm/1256051346. https://projecteuclid.org/euclid.ijm/1256051346