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?