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BY proving the theorem stated in the title, we show that local compactness in the factor spaces is not necessary for paracompactness in the box product. 0. Introduction and theorems. History. Over twenty years ago, A. H. Stone first raised the issue of normality and paracompactness in box products ([Kn]-1964). The problem remains in large part unsolved. Suppose A is a separable metric space. Is [lW(A) paracompact?l The answer is yes if A is locally compact and the Continuum Hypothesis holds (M. E. Rudin, 1972-[Ru]); but if we take A to be the irrationals, then OW(A) is nonnormal in ZFC (E. K. van Douwen, 1975-[vDl]). Is local compactness in A a necessary condition for paracompactness in OW(A)? Our answer is no in the presence of either of two combinatorial statements each of which is known to be consistent with ZFC. (In addition to the papers cited in the preceding paragraph, see [Ku, M, Rol, Ro2, vD2, W1 and W2] for the main results on normality and paracompactness in OW(A).) Cardinal numbers. For , 71 E wu, define E n) (6m < 71m) With respect to <*, let b be the minimal cardinality of an unbounded family and let d be the minimal cardinality of a dominant (cofinal) family. These two cardinal numbers have received much attention in recent years. First note that in ZFC: (X1 < b < d < c (c is the cardinality of the continuum of real numbers); there is a <* well-ordered unbounded family of order type b, so b is regular; b < cof d and the value of d is the same for eventual domination as it is for strict domination where we require Em < 71m on all indices m. It is consistent with ZFC to simultaneously change any of the above relations to either strict equality or strict inequality. In particular, both the Continuum Hypothesis and Martin's Axiom imply b = d = c. The equality b = d holds iff there Received by the editors February 18, 1987 and, in revised form, August 13, 1987. Results of this paper were announced at the annual meeting of the American Mathematical Society in January, 1987. 1980 Mathematics Subgect Classification (1985 Revision). Primary 54D18; Secondary 54A35, 54B10, 54B20.

The box product of countably many copies of the rationals is consistently paracompact