On middle box products and paracompact cardinals

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.

The version after the referee report