Countable Box Products of Ordinals 123

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If píJxgA is a family of topological spaces, a box in ILeA X\ is a set llxeA U\ where each Ux is open in Xx. The box product of {Ax}AeA is IL\eA X\ topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IIxeA ß\ is used to mean the ordinary Cartesian product of the ßx's and never the cardinal or ordinal arithmetic product. Similarly a? means the set of all functions from ß into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that ß < a implies there is a y G A with ß < y. Observe that a is a a-compact ordinal if and only if a is compact or cf (a) = «q. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classifications (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30,02K25.