It is relatively consistent with ZFC that every countable FUfin space of weight ℵ₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].
"Countable Fréchet Boolean groups: An independence result." J. Symbolic Logic 74 (3) 1061 - 1068, September 2009. https://doi.org/10.2178/jsl/1245158099