6 13. FILTERS AND IDEALS IN PARTIAL ORDERS THEOREM 13.7 (Tychonoff's Theorem). If {Xi : i e 1} is any family of com- pact topological spaces, then Y\ieI Xi is compact. PROOF. Let {Xi : i e 1} be as in the assumption, and let F be an ultrafilter of closed subsets of f] i G / Xi. By Theorem 13.6 and Exercise 13.11, it suffices to show that f ) F T^ 0- F o r each i e I, let F, = {Y C Xi : ir~lY G F}. Since the projection iXi on the i-th coordinate is a continuous function and since inverse images preserve set-theoretic operations, F is a filter of closed subsets of Xi. By compactness of Xi, f]¥i ^ 0 for every i G /. Now the following exercise concludes the proof of Theorem 13.7. EXERCISE 13.13 (R). Show that if / e YlieIXi is such that f{i) e f]¥i for every i E J, then / £ f]¥. n REMARK 13.8. In Chapter 9 we showed that Tychonoff's Theorem implies the Axiom of Choice. Note that (AC) is necessary in the proof of Theorem 13.7 in order to guarantee the existence of a function / as in Exercise 13.13. But if each of the spaces Xi is Hausdorff, then the function / is uniquely determined. EXERCISE 13.14 (G). Show that if X{ is Hausdorff and is defined as in the proof of Theorem 13.7, then | f]¥i\ = 1. However, this does not mean that the restriction of Tychonoff's Theorem to the class of Hausdorff spaces is provable in ZF alone. We also used the fact that there exists an ultrafilter of closed sets in n ^ e / ^ ' an( ^ ^n^s ^ac^ *s no ^ Provable in ZF alone, although it follows from the so-called Prime Ideal Theorem2 which is weaker than the full Axiom of Choice (see T. Jech, The Axiom of Choice, North Holland, Amsterdam, 1973). The lack of "intersections" in arbitrary p.o.'s also dictates some caution in generalizing the notion of K-closedness. Let K be a regular uncountable cardinal. A p.o. (P, ) is K-closed if every decreasing sequence (p$ : £ A) of elements of P of length A K has a lower bound in P. A p.o. (P, ) is ^-directed closed if every filter base 5 C P o f size less than K has a lower bound in P. Since every chain in a p.o. is a filter base in this p.o., ^-directed closed p.o.'s are Ac-closed. However, the converse is not true. EXERCISE 13.15 (PG). (a) Consider the following strict partial order relation on [wi]-No: X Y iff Y is a proper subset of X and Y is finite. Show that the corresponding p.o. ([CJI]- H °, ) is ^-closed but not ^-directed closed. (b) Show that a p.o. is countably closed (i.e., Ki-closed) if and only if it is Ki-directed closed. A filter F in a p.o. (P, ) will be called K-closed if the p.o. (F, ) is ^-closed. EXERCISE 13.16 (PG). Show that if F is a filter in a p.o. (P, ), then (F, ) is ^-closed iff (F, ) is ^-directed closed. Let us conclude this section by introducing a few concepts that will play an important role in Chapters 18 and 19. A subset A C P is an antichain in a p.o. (P, ) if every two elements of A are incompatible. The set A is a maximal 2 The Prime Ideal Theorem is a statement very similar to our Theorem 13.4. It states that every Boolean algebra has a prime ideal.
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