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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 $S \subset \bigcup_{i = 1}^{m} W_{i}$?

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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 $S$.

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