The cardinality of compact spaces satisfying the countable chain condition

We prove that for a compact Hausdorff space X, if lambda(c(X)) < w(X) for every infinite cardinal lambda < w(X) and lambda(c(X)) < cf(w(X)) for every infinite cardinal lambda < cf(w(X)), then Tikhonov cube (w(x))[0,1] 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 Tall's result in [17]. (C) 2014 Elsevier B.V. All rights reserved.