Weakly Boolean maximal chains of homeomorphic topologies

TL;DR

This paper characterizes weakly Boolean maximal chains of homeomorphic topologies based on their order and weight properties, and applies these findings to topology.

Contribution

It introduces a new characterization of weakly Boolean maximal chains in relation to infinite cardinals and their weights, with applications in topology.

Findings

Characterization of maximal chains in terms of weakly Boolean property and weights.

Conditions relating chain structure to cardinalities and weights.

Application of chain characterization to topological spaces.

Abstract

If $\lambda <\kappa$ are infinite cardinals, a linear order $L$ is isomorphic to a maximal chain in $[\kappa ]^{\kappa |\kappa }$ (resp. $[\kappa ]^{\lambda |\kappa }$; $[\kappa ]^{\kappa |\lambda }$) iff $L$ is weakly Boolean, the weight of all initial segments of $L$ is equal to $\kappa$ (resp. $\lambda$, $\lambda ^+$) and the weight of all final segments of $L$ is equal to $\kappa$ (resp. $\lambda ^+$, $\lambda $). We also provide an application of this result in topology.