Final compactness and separability in regular symmetrizable spaces

Abstract

In 1966 A. V. Arkhangel'skii posed the following question: Is it true that every regular finally compact symmetrizable space is separable? S. I. Nedev soon showed that a regular finally compact symmetrizable space is hereditarily finally compact. Consequently any counterexample to Arkhangel'skii's conjecture must be an L-space. Applying the technique of iterated forcing we prove that in the axiom systemZFC for set theory it is consistent to assume the existence of a regular (hereditarily) finally compact symmetrizable space X that is nonseparable. Thus it is impossible to prove using the axiom systemZFC that every regular finally compact symmetrizable space is separable. The space X has additional properties as well: it has a basis consisting of open/closed sets (i.e., it is zero-dimensional in the sense ofind, it can be mapped continuously and one-to-one onto a separable metric space, it is α-left and has cardinality ω1. Bibliography: 25 titles.