H. Whitney proved that any $n$-dimensional smooth manifold $N$ can be imbedded in $(2n + 1)$-euclidean space (without any compactness assumption). If we consider the case of topological $n$-manifolds (without any additional structure), can anyone explain how to build a topological imbedding $f\colon M \to \mathbb{R}^{2n+1}$ of a TOP manifold $M^n$ into $\mathbb{R}^{2n+1}$. Any references ??
Thanks everyone for your help !!
Greetings..
This is covered in Munkres' Topology, in the section titled "Dimension Theory". Munkres proves that every compact metric space of covering dimension $n$ can be embedded into $\mathbb{R}^{2n+1}$. He also proves that any compact $n$-manifold has covering dimension at most $n$, and it follows that every compact $n$-manifold can be embedded in $\mathbb{R}^{2n+1}$. (Note: Though Munkres does not prove it, every compact $n$-manifold has covering dimension exactly $n$.