  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.

General Topology;Combinatorics;Logic