Countable products of absolute Cδ spaces

AbstractIt is shown that countable products of completely regular absolute Cδ spaces inherit each of the following properties from their factors: paracompactness, Lindelöfness, metacompactness and ultra-paracompactness. This generalizes previous results by Z. Frolik, by K. Alster, by M.E. Rudin and S. Watson, by L.M. Friedler, H.W. Martin and S.W. Williams, and by A. Hohti and Y. Ziqui. The proof uses ideas from localic topology.Completely regular absolute Cδ's (spaces which are Cδ subspaces of every extension) are characterized as the partition complete spaces of R. Telgarsky and H.H. Wicke. A subspace is called Cδ if it is a countable intersection of complemented subspaces, and a subspace C is complemented if for every nonempty closed subspace F not both C ∩ F and F are dense in F. The class of completely regular absolute Cδ's shares many properties with the class of absolute Gδ's (closed under countable products, direct and inverse images along perfect maps) and even improves on some, for instance, it is closed under open images