Some questions about geometry on pseudo-Riemannian manifolds I will be using relativistic terminology for pseudo-Riemannian manifolds as in the book by Barret O' Neil. 


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*If one can show that the chronal future and the chronal past of some set are disjoint then why is this enough for showing that the set is achronal? 
I mean one can have a set such that there is precisely one time-like curve intersecting it at just two distinct points. Then clearly this set is not achronal (because this one curve exists) but chronal future and past of this set will include one each of these points and will be disjoint. 
Does something rule out this possibility? 

*Let $p$ be a point on the boundary of some future set. Why is showing that chronal future and chronal past of $p$ are disjoint enough to prove that this future set is edgless? 

*How generic is the situation that Cauchy Horizon of a set separates that part of the causal future of a set which is not a part of its domain of dependence? 
Is this something on which intuition can be built or is this a very special scenario?

*Consider the interval $(0,1)$ on the x-axis on the $2-$dimensional Minkowski space where time is the y-axis. For this interval is the Cauchy Horizon the union of the set of points on the lines, $y=-x$ and $y=x-1$ ?
If yes, then I guess this is an example where the Cauchy Horizon is not a part of the domain of dependence of the interval which is the set of points above the given interval and contained in between the above two straight lines. 
Is this right? 
 A: For the first question, I fear that you may have not understood the definition completely. Let $p,q\in U$ be the two points. Let $\gamma$ be the future time-like curve joining $p = \gamma(0)$ and $q= \gamma(1)$. I claim that for any $s\in (0,1)$, $\gamma(s)$ is in both the future and past of $U$. Indeed, $\gamma(s)$ is in the future of $p\in U$, hence is in the chronal future of $U$. And $\gamma(s)$ is in the past of $q\in U$, hence is in the chronal past of $U$. Hence the chronological future and the chronological past of $U$ must intersect non-emptily, i.e. they are not disjoint. 
Indeed, this shows that if a set $U$ is not achronal, then $I^+(U)$ and $I^-(U)$ must intersect. Hence if the chronological future and past are disjoint, the set $U$ must be achronal. 
For question two, I don't think your statement is right. Can you give me a reference (page number in O'Neill will do)? 
For question three, isn't that just the definition? The Cauchy Horizon is generally regarded as the boundary of the domain of dependence, no? If that's not the case, can you provide the definition you are using for the Cauchy horizon?
For question four: not quite. The future Cauchy horizon of the set is the union $ \{ y = x: x \in [0,1/2] \} \cup \{ y + x = 1: x \in [1/2,1]\} $. The past Cauchy horizon is $\{ y = -x: x \in[0,1/2] \} \cup \{ x-y = 1: x\in[1/2,1]\} $. 
