Let $M$ be a $n$-manifold, with some maximal atlas $A$, and let $V \subset M$ be an open set. The standard open-submanifold-atlas on $V$ is $A|_V$ defined as
$$A|_V = \big\{ (U \cap V,x|_{U \cap V}) \in A : (U,x) \in A \ \text{ and } \ U \cap V \neq \varnothing \big\}$$
Is this $A|_V$ maximal for $V$?