Problem:
Suppose that $\Sigma \subset \mathbb{R}^d$ is a smooth manifold of dimension $d$. Is it true that $\Sigma$ is open? How can I prove it?
I tried using directly the definition of manifold but nothing has come.
I define $\Sigma \subset \mathbb{R}^d$ to be a smooth manifold if every point has an open neighborhood $U$ which is diffeomorphic to an open set of $\mathbb{R}^k$ where in this case $k=d$.