Special embeddings of finite-dimensional compacta in Euclidean spaces

Abstract

If $g$ is a map from a space $X$ into ${\mathbb{R}}^m$ and $z\notin g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines ${\Pi }^1\subset {\mathbb{R}}^m$ containing $z$ such that $|g^{-1}({\Pi }^1)|\ge 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $g:X\to {\mathbb{R}}^m$, where $m\ge 2n+1$, with $\dim P_{2,1,m}(g,z)\le 0$ for all $z\notin g(X)$ form a dense $G_{\delta }$-subset of the function space $C(X,{\mathbb{R}}^m)$. A parametric version of the above theorem is also provided.

9 pages