I've often seen it repeated that for any convergent sequence of test functions $\phi_i$ in $C_0^\infty(\Omega)$, there must exist a compact set $K$ such that for all $i$, the support of $\phi_i$ is in $K$. I'm having trouble proving this, and in fact it seems false to me.
Let $K_n$ be an increasing sequence of compact sets whose union is $\Omega$, then define $\phi_i$ to be some smooth function which is zero on $K_i$, but has a little bump of height $1$ somewhere in $\Omega\backslash K_i$. Does this sequence not converge to $0$ in the test function topology?
The topology induced by the seminorms $|| . ||_{K_n,N}$ is the topology f uniform convergence on compact sets (with all its derivatives). The "commonly-used" topology on the space f test functions is strictly finer.
A distribution is continuous with respect to the topology induced by the seminorms if (and only if!) the distribution has compact support.