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13.1. THE GENERAL CONCEPT OF A FILTER 7 antichain in (P, ) if it is not properly contained in any other antichain of the same p.o. EXAMPLE 13.10. Let A = {{n} : n e u}. Then A is an antichain both of (P(w)\{0},Q and (P(wi)\{0},C), but not of (P(u),C). The antichain A is maximal in (P(u)\{9},C), but not in (P(a i)\{0},C). A subset D C P i s dense in the p.o. (P, ) if \fp G P 3q G D (q p). Note that the set A of Example 13.10 is dense in (P(ou)\{®}, C), but is not dense in (P(UJ), Q). EXERCISE 13.17 (G). Show that if D C P is a dense subset in a p.o. (P, ), then there exists a maximal antichain A in this p.o. such that A C D. A subset C C P i s predense in a p.o. (P, if \/p G F3q G C (p / q). Note that both dense subsets and maximal antichains of (P, ) are examples of predense subsets of (P,). EXERCISE 13.18 (G). (a) Show that if C is a finite predense subset of a p.o. (P, ) and F is an ultrafilter in (P, ), then F n C + 0. (b) Show that if (P, ) has the property that for all p G P there exist q,r p such that g l r , then for every ultrafilter F in (P, ), the set P\F is dense in (P, ). By point (b) of the last exercise, as long as the partial order (P, ) has enough incompatible elements, no ultrafilter in (P, ) intersects every predense subset of P. However, it is often useful to know that there exist ultrafilters that intersect every set in a given large family of predense subsets of P. Sufficient conditions for the existence of such ultrafilters will be investigated in Chapters 18 and 19. Exercise 13.18(a) can be generalized a little if the filter in question has certain completeness properties. EXERCISE 13.19 (G). Show that if K is a regular infinite cardinal, C is a pre- dense subset of a p.o. (P, ) such that \C\ ft, and F is a ^-complete ultrafilter in (P,), t h e n F H C ^ 0 . A p.o. (P, ) satisfies the K-chain condition (abbreviated K-C.C), if every an- tichain in (P, ) has cardinality less than K. The Hi-c.c. is also called the countable chain condition and abbreviated c.c.c? By Exercise 13.17, if (P, ) has the K -C.C., then every predense subset of P contains a predense subset of size less than K. EXAMPLE 13.11. If K, 0 is a cardinal, then (P(K)\Q, C) has the K + -C.C, but does not have the K-C.C. EXAMPLE 13.12. A topological space (X, r) has the c.c.c. if and only if the p.o. (r\{0}, C) has the c.c.c. EXERCISE 13.20 (PG). Let K be an infinite cardinal. Suppose (P, ) is a p.o. with the Av-c.c. and F is a ^-complete ultrafilter in (P, ). Show that there exists p G F with no incompatible elements below p, i.e., such that Vg, r p{q jLr). 3 Of course, this should be called the countable antichain condition, but our inappropriate terminology is used throughout the literature.