How to show that the interior of the causal future is contained in the chronological future? I want to show that $int[J^+(S)\subset I^+(S)$ for an arbitrary set S.  I've done most of it I think but not up to the level of rigor that I would like.  Here's what I have:
Let $p \in int[J^+(S)]$.  Assume that $p \notin I^+(S)$.  (Proof by contradition.)  By the Corollary to Theorem 8.1.2 in Wald it follows that any causal curve from S to p will be a null geodesic.  Since $J^+(S) \subset \overline {I^+(S)}$ (Wald problem 8.2.a), it follows that $p \in \overline {I^+(S)}$.  Thus $p\in\overline {I^+(S)}-\overline I^+(S)$.  But $\overline {I^+(S)} = \overline {J^+(S)}$. (Wald, bottom p. 191)  Thus $p\in\overline {J^+(S)}-\overline I^+(S)$.  Since $p \in int[J^+(S)$, there exists a neighborhood U of p such that $U \subset J^+(S)$.  Let $q \in (M-\overline {J^+(S)})\cap U$.  Since $q \notin \overline {J^+(S)}$, it follows that $q \notin I^+(S)$.  But since $q\in U$, we have $q \in  J^+(S)$.  By the Corollary to Theorem 8.1.2, it follows that any causal curve from S to q is a null geodesic.  Since $(M-\overline {J^+(S)})\cap U$ is open, it contains a neighborhood W of q.  Let $\lambda$ be a null geodesic from S to q.  We can make a small modification of $\lambda$ to get a timelike curve $\lambda'$ from S to $r \in W$.  But this implies $r \in I^+(S)$ which is impossible because $r \in W \subset (M-\overline {J^+(S)})\cap U$ implies  $r\notin I^+(S)$.  So we have our proof.
Except, I don't know how to show that such a q exists!  How do we know that $(M-\overline {J^+(S)})\cap U$ is not empty?  What if S just consisted of a single point and M were just the future light cone?  In that case, $(M-\overline {J^+(S)})\cap U$ would be empty because $J^+(S)$ would be all of $M$.  The problem with that is that in that case $M$ would not be a manifold.  It would be a manifold with boundary.  But still the question remains of how to show that $(M-\overline {J^+(S)})\cap U$ is not empty in general.  I thought it would be simple topology.  If you stick to simple examples, it is fairly obvious.  But I can't quite come up with the required steps and I am not even sure what I would need to do it.  Evidently theorems about manifolds vs. manifolds with boundaries requires cohomology and so that can't be what is required.  Perhaps using a coordinate system with one axis in the $\lambda$ direction and then some handwaving about their being no points on one side of that line?  That doesn't quite seem adequate because the topologies under consideration can get quite unintuitive with remove points, etc.  Maybe that is not insurmountable, but then my question is just is that indeed what is expected/needed?  This is part of problem 8.2.b in Wald and so everything that is needed to prove this should be available to one at that point in the text.
Thoughts?
 A: Sorry, this is not a proper answer since I do not have time to enter into the details of  your attempt. However the proof immediately arises from a fundamental general result, due to Penrose (*), for causal relations in time oriented spacetimes (see O’Neill’s textbook for instance, but it should appear also in Wald’s textbook as it is of utmost relevance):
If $p$ stays in the causal (chronological) future of $q$ and $r$ stays in the chronological (resp. causal) future of $p$, then $r$ stays in the chronological future of $q$.
Now consider $r\in Int[J^+(S)]$. Therefore, there is an open neighbourhood of $r$ whose points can be connected to points in $S$ by future directed causal curves exiting $S$. Take $p$ in that neighbourhood in the chronological past of $r$. There is a future-directed causal curve from $q\in S$ to $r$, and a timelike analogous curve from $p$ to $r$. On account of the quoted general result, $r$ belongs to the chronological future of $q$, i.e., of $S$. That is the thesis.

(*) 2.18 Proposition in R. Penrose: Techniques of Differential Topology in Relativity (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 7) 
