Suppose that $M$ is a smooth manifold and $O$ is an open subset of $M$. A codimension $k$ submanifold of $M$ is locally defined as the common zero locus of $k$ functionally independent real valued smooth functions $f_1,...,f_k$ on $M$. Is $O$ a codimension $0$ submanifold? I know that open subsets of a manifold are manifolds of the same dimension as that of the manifold by restricting the atlas of the manifold. So maybe there is a match between these two notions. In particular, I think that $O$ must be a codimension $0$ submanifold of $M$. Can anyone illustrate this?
I think I found a solution. If there is a functionally independent real valued smooth map on $M$ then it can not take zero on any open subset of $O$ since it would contradict its functional independence.