Motivation: Let $G$ be an $\ell$-group (locally profinite group). A map $G\to \mathbb{C}$ is called smooth provided that it is continuous as a map $$G\to \mathbb{C}_{discrete}.$$This gives us the correct notion of smoothness for $\ell$-groups.
Question: Can we characterize smoothness topologically in other interesting cases, or is this just a coincidence?
The question is a bit awkward, as Pete suggests. First, no need to take an $\ell$-group; an $\ell$-space is the right way to start. Second, you've stated the definition of the word "smooth" in this context. Definitions can't be "correct" -- but the word "smooth" is a good choice in this context, because of some parallels between harmonic analysis on p-adic groups and harmonic analysis on real Lie groups.
I think you should put this question aside for a bit.. look at Ralf Meyer's "Smooth Group Representations on Bornological Vector Spaces" to see an answer to something like your question.
We won't be able to do it for varieties, at least, because a cusp is topologically the same as a smooth point (cuspidal cubic and $\mathbb{P}^1$ are homeomorphic, even over $\mathbb{C}$ in the analytic topology).
