Special embeddings of finite-dimensional compacta in Euclidean spaces

Abstract

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $z\not\in g(X)$, let $P_{2,1,m}(g,z)$ be the set of all lines $\Pi^1\subset\mathbb R^m$ containing $z$ such that $|g^{-1}(\Pi^1)|\geq 2$. We prove that for any $n$-dimensional metric compactum $X$ the functions $g\colon X\to\mathbb R^m$, where $m\geq 2n+1$, with $\dim P_{2,1,m}(g,z)\leq 0$ for all $z\not\in g(X)$ 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.