Let $M$ be a $n$-dimensional smooth non-compact manifold such that the singular cohomology groups $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a sufficiently big integer $N$, a sufficiently small $\varepsilon>0$ and a smooth embedding $$i_{\varepsilon}:M\hookrightarrow \mathbb{R}^{N}$$ with the following features:

$i_{\varepsilon}(M)$ is closed in $\mathbb{R}^{N}$

for every $x\in i_{\varepsilon}(M)$ and for every $0<\delta<\varepsilon$ the intersection $B_{\delta}(x)\cap i_{\varepsilon}(M)$ is connected. With $B_{\delta}(x)$ I mean the euclidean ball of $\mathbb{R}^{N}$ centered at the point $x$ and of radius $\delta$.

stilldo not understand how the result would necessarily verify the required conditions. Is that written somewhere? – Ricardo Andrade May 25 '13 at 21:11