Small dense sets in products
by
Gary Gruenhage
Auburn University
Coauthors: Tomasz Natkaniec and Zbigniew Piotrowski

The notions of thin and very thin dense subsets of a product space were introduced by Piotrowski, and here we also introduce the notion of a slim dense set, which is a dense set in the product whose intersection with any cross-section is nowhere dense in that cross-section. We obtain a number of results concerning the existence and non-existence of these types of small dense sets, and we study the relations among them. Many open questions remain, including the following: (1) Is there in ZFC a separable dense-in-itself space X such that there is no slim dense set in X² (i.e., for any dense subset D of X², there exists x ∈ X such that the projection of D∩({x}×X)) or D∩(X×{x}) on X is somewhere dense in X)? (2) Is there a consistent example of a dense-in-itself space X whose countable power has no slim dense set? Any counterexample for (2) must contain a strongly irresolvable Baire subspace.

Date received: February 27, 2006