In my analysis lecture I am given a topology on the space of distributions as follows:
Let $u_k$ be a sequence in $\mathcal D'(u)$, $u \in \mathcal D'(u)$. We say $u_k \rightarrow u$, if $\forall \phi \in \mathcal D(u) : u_k(\phi) \rightarrow u(\phi)$.
This is the weak-$*$-topology on $\mathcal D'(u)$. It seems lecturers don't care too much about the topology of $\mathcal D'(u)$, hence I wonder whether there are stronger topologies on $\mathcal D'(u)$.