Because of the quantifier position thing, I doubt it is possible to write $\{ f_{k} \to f\ \text{uniformly} \}$. If it is possible, how so? If not, how so?
This question is out of this: I used to write "$f_{k} \to f$ uniformly on E", say; in an attempt to write it as "$\{ f_{k} \to f\ \text{uniformly} \} \supset E$", I came across this problem.
On the other hand, we can write "$\{ f_{k} \to f\ \text{pointwisely} \} \supset E$" to express "$f_{k} \to f$ pointwisely on $E$"; for $$ \{ f_{k} \to f\ \text{pointwisely} \} = \cap_{m}\cup_{N}\cap_{k \geq N} \{ |f_{k} - f| < 1/m \}. $$