Amalgams, connectifications, and homogeneous compacta (Q870246)

An example of \textit{J. van Mill} [Isr. J. Math. 133, 321-338 (2003; Zbl 1039.54003)] (generalized later by \textit{K. P. Hart} and \textit{G. J. Ridderbos} [Topology Appl. 150, No. 1--3, 207--211 (2005; Zbl 1066.54034)]) states that there exists a homogeneous compactum which is not homeomorphic to a product of dyadic compacta and first countable compacta, and the homogeneity of the space is independent of ZFC. In the paper under review the author constructs another example of such a space. To be precise, he shows that there exists a path-connected homogeneous compact Hausdorff space \(Y\) with cellularity and weight the continuum and small inductive dimension \(1\). Moreover, \(Y\) is not homeomorphic to a product of compacta that all have character less than continuum or have cf\((\mathfrak{c})\). In particular the space \(Y\) is not homeomorphic to a product of dyadic compacta and first countable compacta. The techniques used by the author permit him to obtain some results on connectifications. The following theorem is obtained: (a) If \(i\in \{1,2,3,4,5\}\), then every infinite product of infinite topological sums of \(T_{i}\) spaces has a \(T_{i}\) pathwise connectification, and (b) every countable infinite product of infinite topological sums of metrizable spaces has a metrizable pathwise connectification.