I was given this definition of continuity in the distributional sense.
A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ \varphi_j\}_j$ that converges to an element $\varphi$ , the sequence $\{\langle T ,\varphi_j\rangle\}_j $ converges to $\langle T ,\varphi\rangle$.
Where the convergence in the space of test functions is defined to be uniform (that includes the derivatives) and the supports of the functions must subsets of a compact set.
Now, I would like to know if is this equivalent to continuity in the topological sense. That is, that the preimage of every open set is open.
I think that the first think to do would be to define a topology , I'm not sure if the weak is the right thing to do, but Wikipedia's article is way over my head.
Could someone explain if they are both equivalent? If possible, more intuitively than rigorously.