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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1997 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.