How are spacetimes which present compactly generated Cauchy horizons different from the ones with compactly generated cronologic horizons? I am comfused because I mix the meanings of compactness of horizons and of S when studying CTCs. As I understand it, compactly generated horizons are the ones whose null geodesics do not come from the infinity or a singularity. Whereas S is , as I understand it, just a region of constant time from which we study the causality of its domain of dependence. S can be compact or non-compact depending if the CTC would be created in a finite region of time (then I get S is non-compact) or not (then S is compact). Can someone provide an elucidation on the statements above if they are wrong? Also note I don't have a really advanced knowledge of topology, so try to answer the clearest way possible. Thanks.
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I think in context there is no difference between the two. Chronological horizons are a specific type of Cauchy horizons, and since as far as I know compact generation only matters for chronology horizons, it is likely that both usage just refer to chronology horizons.
