On middle box products and paracompact cardinals

TL;DR

This paper investigates conditions under which certain complex box product spaces are paracompact, revealing deep connections between paracompactness, generalized metrisability, and large cardinal properties.

Contribution

It provides new sufficient conditions for paracompactness of box products, especially involving products of the form ${}^{< ext{ extkappa}}oxempty 2^ ext{ extlambda}$, linking topological properties to large cardinal hypotheses.

Findings

Paracompactness of ${}^{< ext{ extkappa}}oxempty 2^ ext{ extlambda}$ implies $ ext{ extkappa}$ is at least inaccessible.

Results on paracompactness related to generalized metrisability and Sikorski spaces.

Partial answers to the question of paracompactness for products with singular $ ext{ extkappa}$.

Abstract

The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of many factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type ${}^{<\kappa}\square 2^\lambda$, where we prove that for a regular uncountable cardinal $\kappa$, if ${}^{<\kappa}\square 2^\lambda$ is paracompact for every $\lambda\ge\kappa$, then $\kappa$ is at least inaccessible. The case of the products of the type ${}^{<\kappa}\square X^\lambda$ for $\kappa$ singular has not been studied much in the literature and we offer various results. The question if ${}^{<\kappa}\square 2^\lambda$ can be paracompact for all $\lambda$ when $\kappa$ is singular has been partially answered but remains open in general.