ON PRODUCTS OFCOUNTABLYCOMPACTSPACES

2 c Ginsburg and Saks have proved that if the power X is countably compact, X is a Hausdorff space, then every power of X is countably compact ([1]). Some natural ques 2ctions were raised. Is the cardinal numbE~r minimal in this context? Does there exist a family of countably 2ccompact spaces {X.: i E I} such that (1) III =, 1. (2) ITiEIX is not countab1y compact and (3) if J c I withi o < IJ I < 2c then ITiEJX i is countably conlpact? (These two questions were first raised by w. W. Comfort in Mathe matical Review 52 # 1613.) Saks discussed the second ques tion in [2]. He raised a conjecture: w * is not the union 2cof < cluster sets (a set C is called a cluster set if there exist an x E w * and a sequence {x: n < w} in Sw n such that C = {p E w*: x = p- lim x and {n: x ~ x} E p}). n n n He indicated that the conjecture implies the affirmative answer of the second question. The conjecture is consistent with ZFC. Is it true in ZFC? Let us consider some propositions in set theory. It is well known that there are points in w * such2c that they are pairwise incomparable in Rudin-Keisler order ([3]). But given a point p E w * does there exist a point q E w * such that poq are incomparable? This is another open problem (see [4]) and the answer is unknown in ZFC. This also related to work of Neil Hindman (Number Theory) 2cof the last 5 years. Taking < points instead of one point such as 222 Yang Proposition 1. For every I c w * if III < 2c ~ then there exists p E w * such that p is incomparable in Rudin-Keisler order with all q E I. It appears to be difficult to prove the proposition is true in ZFC. However, in this paper our main result is following. Theorem 2. The following are equivalent. 2c(1) There exist a Hausdorff space X such that x is 2c Xanot countable compact and for every a < is countably compact. (2) The proposition 1. 2 c(3) Saks ' conjecture: w * is not the union of < cluster set. We are now going to construct an example to sho