Nonshrinking open covers and K. Moritaʹs duality conjectures

For every uncountable cardinal κ we construct, in ZFC, a space Xκ such that (a) oduct of Xκ with every metrizable space is normal; an increasing ω1-cover by open sets that has no refinement by ⩽κ many closed sets. answers a question of M.E. Rudin. It also proves a conjecture of K. Morita on a characterization of metrizability in terms normality of products.