Nonshrinking open covers and K. Moritaʹs Third Conjecture

For every regular cardinal κ we construct a hereditarily normal, countably paracompact space Xκ which has an increasing open cover W = {Wμ: μ ϵ ω1} such that W has no refinement by ⩽κ closed sets. By work of Chiba, Przymusiński and Rudin, the existence of such a space proves a conjecture of Morita that σ-locally compact metrizable spaces are precisely the spaces whose product with every normal, countably paracompact space is normal. The spaces Xκ are the first known examples of hereditarily normal ω1-Dowker spaces.