Compact Spaces with a $P$-base

Abstract

In the paper, we investigate (scattered) compact spaces with a P -base for some poset P . More specifically, we prove that any compact space with an ω ω -base is metrizable and any scattered compact space with an ω ωbase is countable under the assumption ω 1 < b. These give positive solutions to Problems 8.6.9 and 8.7.7 in [2]. Using forcing, we also prove that in a model of ω 1 < b, there is a non-first-countable compact space with a P -base for some poset P with calibre ω 1 .