Embeddings of finite-dimensional compacta in Euclidean spaces

TL;DR

This paper investigates the conditions under which finite-dimensional compact spaces can be embedded into Euclidean spaces, establishing density and genericity results for certain classes of embeddings based on the dimension and intersection properties.

Contribution

It introduces new density and $G_\delta$-set results for classes of embeddings of finite-dimensional compacta into Euclidean spaces, extending previous embedding theorems.

Findings

Certain classes of embeddings are dense and $G_\delta$ in the space of continuous maps.

Conditions on the dimension and intersection properties ensure the existence of embeddings.

Results apply to various dimensions and intersection constraints, including low-dimensional cases.

Abstract

If $g$ is a map from a space $X$ into $\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\Pi^d\subset\mathbb R^m$ such that $|g^{-1}(\Pi^d)|\geq q$. Let also $\mathcal H(q,d,m,k)$ denote the maps $g\colon X\to\mathbb R^m$ such that $\dim B_{q,d,m}(g)\leq k$. We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\mathcal H(3,1,m,3n+1-m)$ and $\mathcal H(2,1,m,2n)$ is dense and $G_\delta$ in the function space $C(X,\mathbb R^m)$ provided $m\geq 2n+1$ (in this case $\mathcal H(3,1,m,3n+1-m)$ and $\mathcal H(2,1,m,2n)$ can consist of embeddings). The same is true for the sets $\mathcal H(1,d,m,n+d(m-d))\subset C(X,\mathbb R^m)$ if $m\geq n+d$, and $\mathcal H(4,1,3,0)\subset C(X,\mathbb R^3)$ if $\dim X\leq 1$.