let $M$ be a smooth manifold, $N$ be a closed embedded submanifold of $M$ with $dim N< dim M$ and $U$ be an open neighborhood of $N$ in $M$. I wonder when it´s possible to find a smaller open neighborhood $V$ of $N$ such that one has $N \subseteq V \subseteq \bar{V}\subseteq U$. (Here, $\bar{V}$ is the closure of $V$ in $M$.) Probably this statement isn´t true in general, but maybe under some (hopefully weak) extra assumptions? Please note I cannot assume $N$ to be compact.
Many thanks for your help!