The Uniform Box Product Π Μ

The uniform box product Πμ of countably many copies of the one-point compactification of the discrete space of cardinality אμ is not weakly δθ-refinable. This is shown with the help of a closed subspace T which has a natural tree structure and is naturally homeomorphic to the space F∗ of all functions from ω1 to ω ∪ {−1} that are constantly equal to −1 on [α, ω1) and are unequal to it and one-to-one on [0, α). The following basic result seems to be new: Theorem. In the topology of pointwise convergence, F∗ is collectionwise normal, ω1compact and hereditarily realcompact, but not Lindelöf, nor even weakly δθ-refinable, but it has the property that every countable subset has metrizable closure. Failure of weak δθ-refinability is shown using an Aronszajn subtree A of T and a finer topology on T that is introduced here, called the truncated coarse wedge topology. With this topology, A has the properties of T listed in the theorem. The concept of a Ξ-product is introduced, and it is shown that the Ξ-product of any number of copies of the one-point compactification of the discrete space of cardinality אμ is Fréchet-Urysohn.