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[tex]H^+(S)=\overline{D^+(S)}-I^-[D^+(S)][/tex]

Where the overline means the closure of the set. D+ is the future domain of dependence (i.e. all points in the manifold which can be connected to S by a past inextendible causal curve), and I- is the chronological past.

It seems to me that the closure of the set D+ does not include some parts of the set I-(D+) since the second term is the entire chronological past including the chronological past of the points on S.

Is there a mistake in this definition? I would have thought (intuitively) that the future Cauchy Horizon of a set S would simply be the boundary of the Future domain of dependence of S minus S itself. In that case, Wald defined the interior of D+ as:

[tex]int(D^+(S))=I^-[D^+(S)]\cap I^+(S)[/tex]

Perhaps he meant this as the second term? Am I missing something here?