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?



Smoothness as a topological property
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.


