Let $(M,g)$ be a stably causal spacetime. Then we have a global time function $t:M \rightarrow \mathbb{R}$. A set $S_a:=t^{-1}(a)$ is said to be a Cauchy-hypersurface, if the domain of dependence of $S_a$ equals $M$.
Now if I understand everything right the term Cauchy-hypersurface is only defined for level sets of the time functions. Still, in my book I read something like $(M,g)$ stably causal spacetime, $S \subset M$ a Cauchy hypersurface. I don't understand what $S$ is supposed to be here. Some random level set?