Complete systems, elementary submodels and the tightness of upper hyperspaces (Q2502986)

The paper can be roughly divided into two parts. In the first part the author studies some strengthenings of Frolík's notion of complete systems of open covers for topological spaces. The new notions of c-complete, F-complete (for Frolík-complete) and SF-complete (strongly-Frolík-complete) systems are introduced and their Čech numbers defined. It is shown that a Hausdorff space is regular if and only if it admits a SF-complete system, and that the Čech numbers all coincide for a regular space. However, it is noted that not every F-complete system is SF-complete, and a method to construct such counterexamples is provided. The second part is where the main contribution of the paper lies. In this part the author considers upper hyperspaces following the work of Costantini, Holá and Vitolo, and computes the tightness and the Lindelöf number of these spaces from cardinal functions of the base spaces. In particular the SF-Čech number and the Lindelöf number of the base space and some hereditary versions of them all play a role in this computation. The main tool in this computation is a powerful set-theoretic method known as elementary submodels, and the author gives a clear account of how the technique is used in general. The results obtained generalize those of Costantini, Holá and Vitolo.