Cardinal invariants for $κ$-box products

W. W. Comfort, Ivan S. Gotchev

Published 2013-11-11Version 1

Definition. Let $\kappa$ be an infinite cardinal, let {X(i)} be a (not necessarily faithfully indexed) set of topological spaces, and let X be the product of the spaces X(i). The $\kappa$-box product topology on X is the topology generated by those products of sets U(i) for which (a) for each i, U(i) is open in X(i); and (b) U(i) = X(i) with fewer than $\kappa$-many exceptions. (Thus, the usual Tychonoff product topology on X is the $\omega$-box topology.) With emphasis on weight, density character, and Souslin number, the authors study and determine the value of several "cardinal invariants" on the space X with its $\kappa$-box topology, in terms of the corresponding invariants of the individual spaces X(i). To the authors' knowledge, this work is the first systematic study of its kind. Some of the results are axiom-sensitive, and some duplicate (and extend, and make precise) earlier work of Hewitt-Marczewski-Pondiczery, of Englking-Karlowicz, of Comfort-Negrepontis, and of Cater-Erdos-Galvin.