In the book "Differential forms in Algebraic Topology" by Bott and Tu, the support of a differential form $\omega$ on a manifold $M$ is defined to be "the smallest closed set $Z$ so that $\omega$ restricted to $Z$ is not $0$." (page 24).
I am a little confused, suppose we let $M = \mathbb{R}$ with the trivial atlas $\{ \mathbb{R}, \text{Id} \}$ and consider the $0-$form $\omega = x$. Then any non - zero point would constitute a set on which the restriction of $\omega$ is non - zero. But I expect the authors want the support to be the smallest closed set containing all the points at which $\omega$ is non - zero. Where is my misunderstanding ? Lots of thanks for help!