BOXES OF COMPACT ORDINALS

If {X: nEw} is a family of spaces, then DE X, n-n w n called the box product of those spaces, denotes the cartesian product of the sets with the topology generated by all sets of the form ITnEwG, where each G need only be open in the n n wfactor space X. If X X V nEw, we denote DnEwX by U X. n n n M. E. Rudin [5] and K.,Kunen [3 and 6, pg. 58] have shown that CH implies UnEw(A + 1) is paracompact for every counta n ble collection of ordinals {An: nEw}. At the 1976 Auburn University Tbpology Conference I demonstrated [7] that the paracompactness of UW(w + 1) is implied by the existence of a k-scale in ww, a set-theoretic axiom which is a consequence of, but not equivalent to, Martin's Axiom, and hence CH. In addition, I proved uW(w + 1) is paracompact iff uW(a + 1)l is paracompact V countable ordinals a. If this is coupled with E. van Douwen ' s (3 a k-scale in ww) ~ E X is para-n w n compact for all collections {X: nEw} of compact metrizable n spaces [1], we have LJw((J)l + 1) is paracompact if 3 a k-scale in ww. However, none of the proofs generalize to higher ordinals (Uw(W 2 + 1), for example). We conjecture: If UW(w + 1) is paracompact, then UW(A + 1) is paracom lpact V ordinals A. lIt is unknown whether it is consistent for Ow(w + 1) not to be paracompact; however, 3 compact spaces X such that n U X is not normal. Moreover, irrationals x(Uw(w + 1)) nEw n is not normal [6, pg. 58]