Forcing tightness in products of fans

Abstract

The theta--fan F_theta is the quotient space obtained by identifying the non--isolated points of the product theta \times (omega+1) with a single point \infty. (Here, theta has the discrete topology and omega+1 has the order topology.) We define several notions of forcing that allow us to manipulate the tightness t of products of fans. Some consequences include: t (F_theta \times F_omega) = theta does not imply the existence of a (theta,omega)--gap, new examples of first countable <theta--collectionwise Hausdorff (cwH) spaces that are not \leq theta--cwH for singular cardinals theta, and for cardinals lambda \leq theta with cf(theta) \geq omega_1 and lambda regular, a first countable <theta--cwH not \leq theta--cwH space that can be made cwH by removing a closed discrete set of cardinality lambda. We also prove two theorems that characterize tightness of products of fans in terms of families of integer-- valued functions.