# Collection of local defining maps for smooth Euclidean submanifolds

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

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 $$S$$.