The cardinality of compact spaces satisfying the countable chain condition

摘要

We prove that for a compact Hausdorff space X, if λ~(c(x)) ＜ ω(X) for every infinite cardinal λ ＜ ω(X) and λ~(c(X) ＜ cf(ω(X)) for every infinite cardinal λ ＜ cf(ω(X)), then Tikhonov cube (ω(x)) is a continuous image of X, in particular the cardinality of X is just 2~(ω(x)). As an application of this result, we consider elementary submodel spaces and improve Tail's result in [17].