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Extra resources for Almost-periodic functions in abstract spaces

Example text

Trip also, so q e W n dQ c W — Q. 19 is satisfied and we have pas the future endpoint of a null geodesic rj onH+(S). We could repeat the argument at any past endpoint of rj. So (by the achronality of H + (S)) we can extend tj into the past along dH + (S) either indefinitely or until it meets edge S. 13. DEFINITION. A Cauchy hypersurface for M (sometimes called a global Cauchy hypersurface) is an achronal set S for which D(S) = M. 14. PROPOSITION [9]. If S is achronal and intersects every endless null geodesic in M in a nonempty compact set, then S is a Cauchy hypersurface for M.

DEFINITION. Let S be achronal. The edge of S is defined by: edge 5 = {x\every neighborhood Q of x contains points y and z and two trips from y to z just one of which meets S}. Clearly S — S c edge 5 <= S, so if we require S to be closed, we have edge S a S. If edge S = 0 we call S edgeless. If S is cdgeless it must be closed. FIG. 36. 7. Remark. In the case when S is closed, a slightly different formulation of the definition has been given elsewhere [6], namely: x e edge S if and only if x e S and if 7 is a trip from y to z containing x, then every neighborhood of 7 contains trips from y to z not meeting S.

In the case when S is closed, a slightly different formulation of the definition has been given elsewhere [6], namely: x e edge S if and only if x e S and if 7 is a trip from y to z containing x, then every neighborhood of 7 contains trips from y to z not meeting S. 6 (S closed and achronal) is evident. 6 is a corrected version of that given in [9]: a relation r « p « q on page 191 of that reference should be pe <>, q)Q. Otherwise difficulty arises with examples such as Fig. 23. ) The intuitive meaning of edge S is illustrated in Fig.