Selection principles and the minimal tower problem

TL;DR

This paper investigates selection principles related to tau-covers and their variants, providing new characterizations, solving a topological problem connected to the Minimal Tower problem, and introducing tau^*-covers with tighter bounds.

Contribution

It introduces tau^*-covers, resolves a topological problem linked to the Minimal Tower problem, and offers new combinatorial and topological bounds.

Findings

Solved a topological problem related to the Minimal Tower problem.

Introduced tau^*-covers and established their properties.

Provided new lower bounds for the Minimal Tower problem.

Abstract

We study diagonalizations of covers using various selection principles, where the covers are related to linear quasiorderings (tau-covers). This includes: equivalences and nonequivalences, combinatorial characterizations, critical cardinalities and constructions of special sets of reals. This study leads to a solution of a topological problem which was suggested to the author by Scheepers (and stated in an earlier work) and is related to the Minimal Tower problem. We also introduce a variant of the notion of tau-cover, called tau^*-cover, and settle some problems for this variant which are still open in the case of $\tau$-covers. This new variant introduces new (and tighter) topological and combinatorial lower bounds on the Minimal Tower problem.