Wild 2007-03-31 09:12:35 2007-04-28 21:14:09 Definition tamely imbedded triangulable % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $S$ be a set in ${\mathbb{R}}^n$ and suppose that $S$ is triangulable. ($S$ is \emph{triangulable} means that when regarded as a space, it has a triangulation.) If there is a homeomorphism $h: {\mathbb{R}}^n \to {\mathbb{R}}^n$ such that $h(S)$ is a polyhedron, we say that $S$ is \emph{tamely imbedded}. If $S$ is triangulable but no such homeomorphism exists $S$ is said to be \emph{wild}. In ${\mathbb{R}}^2$ every 1-sphere is tamely imbedded. But in ${\mathbb{R}}^3$ there are wild arcs, 1-spheres and 2-spheres.