Say I want a set $\mathcal{S}$ to represent the people present in a room at any given time. Person $k$ enters the room at time $t_{\mathrm{in},k}$ and exits at time $t_{\mathrm{out},k}$. To give a simple definition, I assume that people who have left the room do not re-enter.
I want to write something like:
$$ \mathcal{S} \text{ at time $t$}= \left\{k: t_{\mathrm{in},k}\leq t\leq t_{\mathrm{out},k} \right\} $$
Somehow it feels strange to write $\mathcal{S}(t)$ on the LHS because it implies a function, and I am used to functions mapping onto an element of a set, not onto a set. Of course, we can consider the set of all possible combinations of people in the room at time $t$, of which $\mathcal{S}$ is an element, but this feels clunky. Would $\mathcal{S}_t$ or something else be preferable?