Additivity of the Gerlits--Nagy property and concentrated sets

TL;DR

This paper resolves open problems about the additivity of the Gerlits--Nagy property, computes minimal concentrated sets related to Rothberger's property, and constructs spaces with specific product properties involving Menger's property.

Contribution

It settles all open problems on the additivity of the Gerlits--Nagy property and introduces new constructions of spaces with particular product properties.

Findings

Resolved all open problems on the additivity of the Gerlits--Nagy property.

Computed the minimal number of concentrated sets needed to alter Rothberger's property.

Constructed a large family of spaces whose product with every Hurewicz space has Menger's property.

Abstract

We settle all problems posed by Scheepers, in his tribute paper to Gerlits, concerning the additivity of the Gerlits--Nagy property and related additivity numbers. We apply these results to compute the minimal number of concentrated sets of reals (in the sense of Besicovitch) whose union, when multiplied with a Gerlits--Nagy space, need not have Rothberger's property. We apply these methods to construct a large family of spaces, whose product with every Hurewicz space has Menger's property.