Small Valdivia compacta and trees (Q2833672)

In this nicely written paper the authors study ``small'' Valdivia compact spaces (i.e. Valdivia compact spaces of weight \(\omega_1\)). The authors prove in Theorem 3.4 that small Valdivia compacta are exactly those compact spaces which are obtained as an inverse limit of certain continuous inverse sequence of path spaces on trees. Such a result encourages one to construct small Valdivia compact spaces by manipulating the properties of the corresponding tree.NEWLINENEWLINEThe authors use this technique to construct a consistent counterexample to a conjecture raised by the authors in [J. Funct. Anal. 270, No. 2, 842--853 (2016; Zbl 1347.46014)], by showing the existence of a small Valdivia compactum \(K\) which has ccc, does not have \(G_\delta\) points and does not have a nontrivial convergent sequence in the complement of a dense \(\Sigma\)-subset, see Theorem 4.1. The construction is done assuming the axiom \(\diamond\).NEWLINENEWLINESuch an example of a small Valdivia compactum is related to the following problem: given a Valdivia compact space \(K\), does there exist a nontrivial twisted sum of \(c_0\) and \(C(K)\)? Let us note that this problem has been very recently consistently answered in the negative in a recent paper by \textit{W. Marciszewski} and \textit{G. Plebanek} [``Extension operators and twisted sums of \(c_0\) and \(C(K)\) spaces'', \url{arXiv:1703.02139}].