Suppose the set $S \subset \mathbb{R}^{n}$ is a smooth submanifold of dimension $k$, that is [Lee, Proposition 5.16] for every $x \in S$ there exist an open set $W \subset \mathbb{R}^{n}$ and a smooth submersion $\phi : W\to \mathbb{R}^{n - k}$ such that $W \cap S$ is a level set of $\phi$.


The submersion $\phi$ is termed a local defining map.


Can one always find a finite a collection of open sets $W\_{1},\ldots, W\_{m}$ and corresponding local defining maps $\phi\_{1},\ldots,\phi\_{m}$ such that $S \subset \bigcup\_{i = 1}^{m} W\_{i}$?



Collection of local defining maps for smooth Euclidean submanifolds
This is a simple consequence of the finiteness of Lebesgue dimension of any manifold, i.e. for any open cover $U\_a$ of a manifold, there is a refinement $V\_{ij}$, so that $j$ runs through a finite set, and $V\_{ij}\cap V\_{ik}$ is empty if $j\ne k$; see Greub, Halperin, Vanstone, **Connections, Curvature and Cohomology**, volume 1, section 1.2.


As Greub et. al. point out, there is a better result in W. Hurewitz and H. Wallman, **Dimension Theory**, Princeton Univ. Press, Princeton, New Jersey, 1941: we can arrange that $j$ runs from $1$ to $n$, where $n$ is the dimension of the manifold. So we only need $k$ open sets to cover a $k$-dimensional manifold $S$.


