In the book Distribution theory by Duistermmat/Kolk convergence in the space of test functions is defined as:
$\phi_j \to \phi$ in $C_0^{\infty}(\Omega) $ (where $\Omega$ is an open subset of $\mathbb{R}^n$ )if:
$i)$ There is a compact set $K: \text{supp} \phi_j \subset K$ for every $j$.
$ ii)$ For every multi-index $\alpha$ the sequence $\partial^\alpha \phi_j \to \partial \phi$ uniformly.
I'm not really sure how to interpret this. By "uniformly" in criterion $ii)$ is it meant uniformly on every compact subset of $\Omega$?
It is not a criterion that for every multi-index $\alpha$ there is a compact $K \subset \Omega$ such that $\text{supp} \partial^\alpha \phi_j$? If no is it possible to $\phi_j \to \phi$ in $C_0^{\infty}(\Omega)$ but the support of its derivatives "diverging"?