Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of degree $k$, can we control $m$ in term of $n$ and $k$? 



The quotient of $\mathbb{R}^{n}$ by a closed subset
I shall prove the case for $S^{n}$ rather than $\mathbb{R}^{n}$ since $S^{n}$ works better being compact (for closed non-compact sets of $\mathbb{R}^{n}$ it does not work; see Daniele Zuddas's answer). Suppose that $A$ is a closed subset of $S^{n}\subseteq\mathbb{R}^{n+1}$. Then define a mapping $f:S^{n}\rightarrow\mathbb{R}^{n+1}$ by letting $f(x)=d(x,A)\cdot x$ where $d(x,A)$ denotes the distance from $x$ to $A$. Then clearly $f$ is a continuous map. Since $S^{n}$ is compact, the mapping $f$ is a quotient map and $f(x)=f(y)$ if and only if $x=y$ or $x,y\in A$. Therefore $S^{n}/A\simeq f[S^{n}]$.


