Reading about the space $\mathcal{C}^\infty_0(\Omega)$ of all compactly supported functions, I've came across a claim that this space is not complete with respect to the family of seminorms
$$ \|\varphi\|_j = \max_{|\alpha|\leq j}\sup_{x\in \Omega} |\partial^\alpha \varphi(x)| \ , \forall \varphi \in \mathcal{C}^\infty_0(\Omega) \ , $$
but I'm not quite sure how to produce a counterexample for this. Anyway, because of this, we have to produce a topology that is not quite as simple as the one defined by those seminorms (namely, adapt the subspace topology for $\mathcal{C}^\infty_0(K)$ for each compact subspace $K \subset \Omega$). But then, if you take a covering by an increasing sequence of compact subsets $(K_n)$ in $\Omega$, it can be shown that the family of seminorms
$$ p_{j,n} (\varphi) = \|\varphi\|_{j,n} = \max_{|\alpha|\leq j} \sup_{x\in K_n} |\partial^\alpha \varphi(x)| \ , \forall \varphi \in \mathcal{C}^\infty_0 (\Omega) \ , n\in \mathbb{N} \ $$
induces a Fréchet space structure on $\mathcal{C}^\infty_0 (\Omega)$, so what is achieved by this family that is not by the first one?
As mentioned in a comment above, none of these systems of seminorms $||\cdot||_j$ or $p_{j,n}$ is good as far as defining the topology of $C_0^{\infty}(\Omega)=\mathscr{D}(\Omega)$. For an explanation of how to correctly define the topology see my answer
Doubt in understanding Space $D(\Omega)$
A useful rule of thumb to see if a system of seminorms is good or not is the following test. The seminorms typically make sense for arbitrary smooth function $\varphi$ on $\Omega$. The caveat is that these would then take values in $[0,\infty]$ instead $[0,\infty)$. One must have property that if all seminorms evaluated on $\varphi$ are finite then $\varphi$ has to be of compact support. If this property does not hold, one is pretty much guaranteed that the proposed system of seminorms is not the right one.