The weight of lexicographic products (Q2216583)

Let \(\gamma\) be an ordinal \(\geq 2\) and \(X_\alpha\) be a GO-space for \(\alpha<\gamma\). The authors calculate the weight \(w(X)\) of the lexicographic product \(X=\prod_{\alpha<\gamma}X_\alpha.\) (For a lexicographic product of LOTS this was done by \textit{D. Buhagiar} et al. [Eur. J. Math. 4, 1505--1514(2018; Zbl 1405.54015)].) The authors then calculate the weight of special types of lexicographic products. In particular, \(w(2^\gamma)=2^{<\gamma}:=\sup\{2^\mu : \mu \text{ is a cardinal and }\mu<\gamma\}\). They deduce that \(w(2^{\omega_1})=\aleph_1\) is equivalent to the Continuum Hypothesis, i.e, \(2^{\aleph_0}=\aleph_1\). Moreover, the condition that the lexicographic product \(2^\gamma\) is homeomorphic to the Tychonoff product \(2^\gamma\) is equivalent to \(\gamma\leq\omega\).