Is a diffeomorphism's image automatically open?

Question

Sorry if this question is trivial, I am new to smooth manifold theory.

Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in $\mathbb R^n$, and $X$ is a subset of $\mathbb R^n$. (The image is a sort of annulus.)

Is $X$ automatically open in $\mathbb R^n$? Is there a simple reason why it should be so?

Answer 1

If $U\subseteq \mathbb{R}^n$ is homeomorphic to an open set in $\mathbb{R}^n$, then $U$ is itself open in $\mathbb{R}^n$. This is Brouwer's theorem on invariance of domain.