## Zy ZFy

Clearly Z>+(y) => D+(£f).l£qeJK - D+(£f) there is a neighbourhood ^ of q which does not intersect y. From q there is a past-inextendible curve A which does not intersect Sf. If re Afl I~{q, <%) then I+(r, <%) is an open neighbourhood of q in — D+{£P). Thus JK — D+(£f) is open and the set D+(£f) is closed. Suppose there were a point peD+(£P) which had a neighbourhood "f which did not intersect D+[£P). Choose a point xel~(p, "V"). From x there would be a past-inextendible non-spacelike curve y which did not intersect y. Let yn be a sequence of points on y which did not converge to any point and which were such that yn+i was to the past of yn. Let be convex normal neighbourhoods of the corresponding points yn such that did not intersect Wn. Let zn be a sequence of points such that

There would be an inextendible timelike curve from p which passed through each point zn and which did not intersect £f. This would contradict p e D+(Sf). Thus D+(Sf) is contained in the closure of Z)+(y), and so D+(£P) = D+(£f). □

The future boundary of Z)+(y), that is Z)+(y) - /-(Z>+(y)), marks the limit of the region that can be predicted from knowledge of data on y. We call this closed achronal set the future Cauchy horizon of y and denote it by H+(£f). As figure 43 shows, it will intersect y if y is null or if y has an 'edge'. To make this precise we define edge(y) for an achronal set y as the set of all points qeSP such that in every neighbourhood °U ot q there are points pel~(q,%) and rel+(q, which can be joined by a timelike curve in which does not intersect y. By an argument similar to that in proposition 6.3.1 it follows that if edge (y) is empty for a non-empty achronal set y, then y is a three-dimensional imbedded C1- submanifold.

Proposition 6.5.2

For a closed achronal set y,