Let $K$ be a compact subset of a Euclidean space of very large dimension $N$. Assume that any point $x\in K$ has a neighborhood $U\subset K$ which is contained in a smooth $l$-dimensional submanifold of $\mathbb R ^N$, for some fixed not very large $l$.
Is the whole set $K$ contained in a smooth $l$-dimensional submanifold?
In the case that I have encountered, I had the additional assumption that $K$ is topologically an $l$-dimensional manifold with boundary, in which case the statement is not very difficult to prove. I was wondering if the statement is true without any additional assumptions on $K$.