is it possible to define a set $S$ as the argument of the minimum of a function?

for example, to have $S=\arg\min_{x}f(x)$

I saw this notation somewhere and it was very confusing because my impression is that the $\arg$ gives a specific point and not a set of points, is that right?

@johnny09 Well I've only seen it used for a single point, as you say, but if the author wants to use it for the set of points that attain the minimum, then he/she had better define it.