Cardinal functions on lexicographic products (Q6558884)

The authors consider lexicographic products of GO-spaces. In the case of a product \(\prod_{\alpha<\gamma}X_\alpha\) of LOTS's one takes the order topology. If some (or all) of the factors are GO-spaces one needs to take into account that the factors carry topologies that extend the order topologies. The authors use the method of \textit{N. Kemoto} [Topology Appl. 232, 267--280 (2017; Zbl 1382.54020)]: first embed (each) \(X_\alpha\) densely into a canonical LOTS \(X_\alpha^*\) and then let the topology on \(\prod_{\alpha<\gamma}X_\alpha\) be the subspace topology inherited from the order topology on the lexicographic product \(\prod_{\alpha<\gamma}X_\alpha^*\).\par The authors express the weight, density, and spread of the product in terms of the values of the corresponding functions on the factors \emph{and} the cardinalities of initial subproducts. When \(\gamma\)~is a limit then the values on the factors do not really matter because every partial point \(\langle x_\alpha:\alpha<\beta\rangle\) (with \(\beta<\gamma\)) determines a non-trivial interval in the final product and this leads to the outcome \(\sup_{\beta<\gamma}\left|\prod_{\alpha<\beta}X_\alpha\right|\) for the functions. In case \(\gamma\) is a successor the calculations become a bit messier; one deals essentially with \((\prod_{\alpha<\delta}X_\alpha)\times X_\delta\) (where \(\gamma=\delta+1\)), a lexicographic product of two spaces: a union of \(\left|\prod_{\alpha<\delta}X_\alpha\right|\) many intervals that are all copies of~\(X_\delta\). In this case the weight, density, and spread of~\(X_\delta\) enter into the calculations as well. The messiness alluded to above is due to the existence or absence of \(\min X_\delta\) and \(\max X_\delta\) as well as the nature of left- and right-hand limits in \(\left|\prod_{\alpha<\delta}X_\alpha\right|\). This leads to a number of cases that need to be considered.