Absorbing spaces for \(C\)-compacta (Q1295356)

Recall that a space \(X\) is a \(C\)-space if each sequence \(\{\alpha_n\mid n\in\mathbb N\}\) of open covers admits an open cover which can be written as the countable union of pairwise disjoint collections \(\beta_n\) where each \(\beta_n\) refines \(\alpha_n\) [\textit{D. Addis} and \textit{J. Gresham}, Fundam. Math. 101, 195-205 (1978; Zbl 0397.54051)]. There is a notion \(\dim_C\), which is the transfinite extension of covering dimension that classifies \(C\)-compacta (spaces here are separable and metrizable). This concept was first defined by P. Borst and is recaptured in the current paper. The author proves that for every countable ordinal \(\alpha\), there exists a \(C\)-compactum which is universal for the class of all compacta \(X\) with \(\dim_C X\leq\alpha\). The result is applied to show that for uncountably many ordinals \(\beta\), there exist noncountable-dimensional pre-Hilbert spaces \(D_\beta\) which are absorbing spaces [\textit{M. Bestvina} and \textit{J. Mogilski}, Mich. Math. J. 33, 291-313 (1986; Zbl 0629.54011)] for the class of compacta with \(\dim_C<\beta\).