Diamond on ladder systems and countably metacompact topological spaces

TL;DR

This paper investigates the properties of ladder systems and countably metacompact spaces, providing ZFC examples of ladder systems where the associated space is not countably metacompact, thus advancing understanding of these topological concepts.

Contribution

The paper constructs ZFC examples of ladder systems over certain cardinals where the associated spaces are not countably metacompact, answering an open question.

Findings

Existence of ladder systems with non-countably metacompact spaces at specific cardinals

Construction of examples under assumptions like $eth_ ext{omega}= ext{aleph}_ ext{omega}$

The ladder system $L$ can be $ extomega$-bounded in the constructed examples.

Abstract

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\Delta$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $\Delta$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\Delta$-space. We prove that an affirmative answer holds for the cardinal $\kappa=cf(\beth_{\omega+1})$. Assuming $\beth_\omega=\aleph_\omega$, we get an example at a much lower cardinal, namely $\kappa=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $\omega$-bounded.