Special embeddings of finite-dimensional compacta in Euclidean spaces

TL;DR

This paper demonstrates that for any n-dimensional compact metric space, most embeddings into Euclidean space of dimension at least 2n+1 have the property that the set of lines through points outside the image intersecting the image in at least two points is zero-dimensional, and provides a parametric extension.

Contribution

It establishes the density and genericity of embeddings with controlled line intersection properties in Euclidean spaces for finite-dimensional compacta.

Findings

Most embeddings have zero-dimensional sets of lines through external points intersecting the image in multiple points.

The result holds for embeddings into spaces of dimension at least 2n+1.

A parametric version of the theorem is also proved.

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.