The cardinality of compact spaces satisfying the countable chain condition

We prove that for a compact Hausdorff space X, if λ c ( X ) < w ( X ) for every infinite cardinal λ < w ( X ) and λ c ( X ) < cf ( w ( X ) ) for every infinite cardinal λ < cf ( w ( X ) ) , then Tikhonov cube [ 0 , 1 ] w ( X ) is a continuous image of X, in particular the cardinality of X is just 2 w ( X ) . As an application of this result, we consider elementary submodel spaces and improve Tallʹs result in [17].