By Solomon Lefschetz
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PROPOSITION. The set of points at which M is strongly causal is open. Proof. 12. 14. PROPOSITION. Let A <= M and suppose that strong causality holds at every point of A. Then A can be covered by a locally finite (countable) system of local causality neighborhoods. If A is compact, then a finite number of such neighborhoods will suffice. Proof. 12 and the paracompactness of M . 15. PROPOSITION. No local causality neighborhood can contain a future- or past-endless causal trip. Proof. Suppose a local causality neighborhood L contains a future-endless causal trip y.
Is not past-endless it has a past endpoint y on B (since B is a closed set). 19 again to obtain another geodesic £ on B with future endpoint y and which does not continue t]. 19 this would lead to chronologically related points on B, contradicting the achronality of B. 21. Remark. 20 have been illustrated in our examples: in Fig. 19, a past endless null geodesic exists on B; whereas in Fig. 20, all null geodesies which are maximally extended on B, have past endpoints on S = r\ ( = S). Note also that in Fig.
Suppose M is not future-distinguishing at p (£V) and consider dl + (p). 4, p e d l + ( p ) . Furthermore, l + (p) = I + (q) for some q ^ p. 19 to B = dl+(p) = dl + (q) to obtain a null geodesic y on dl+(p) which extends indefinitely into the past from its future endpoint p. ) Now if r £ y we have r -< p, so I + (r) •=> I+(p)', also we have r e dl + (p) so I+(r) c I+(p) (and r$I + (p)). Thus / + (r) = I + (p), so future-distinction fails at each point of y. Furthermore r$I + (p) = I + (r), so r ^ F, whence 7 c ~ V.