Encountered the term parametrizable for the first time:
The support of $\omega$ is contained inside a single parametrizable open subset $W$ of $X$.
So I am just curious, what kind of sets are not parametrizable? The intersection of $\mathbb{R}^2$ and the Weierstrass function?
If the context of this is manifold theory, the open subset $W$ should be a coordinate neighborhood: it admits a diffeomorphism $\phi_W:W\to \mathbb{R}^n$. That is, $W$ can be given a parametrization by $\mathbb{R}^n$.
So sets that aren't parametrizable are sets that aren't diffeomorphic to $\mathbb{R}^n$. For example, a manifold with nontrivial fundamental group is not parametrizable (although it is locally parametrizable). Another example, any neighborhood of the cusp at $0$ in the double-cone given by $x^2 + y^2 = z^2$ is not parametrizable.