I have problems understanding the first part of the proof for the Penrose singularity theorem in the book "Leonor Godinho José Natário An Introduction to Riemannian Geometry":
I know that $\langle\operatorname{grad} t, \operatorname{grad} t\rangle<0 $ and that an integral curve of $\operatorname{grad} t$ is some $c:(- \epsilon, \epsilon) \rightarrow M$, s.t. $c'(t)=\operatorname{grad} t _{c(t)}$.
Now the first thing I don't understand is, do they want to say that the union of all integral curves intersects.... or each curve intersects ....?
But still, I don't understand the statement for neither of the options..
Obviously, if we have a curve, s.t. $\exists t$ s.t. $c(t) \in S$ then the curve intersects $S$ but this does not hold for each curve..