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

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

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

Moreover, can one demand that $\phi_{i}(x) = \phi_{j}(x)$ for all $x \in W_{i} \cap W_{j}$?

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