Forcing tightness in products of fans
Abstract
The theta--fan Ftheta is the quotient space obtained by identifying the non--isolated points of the product theta × (omega+1) with a single point ∞. (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 (Ftheta × Fomega) = theta does not imply the existence of a (theta,omega)--gap, new examples of first countable <theta--collectionwise Hausdorff (cwH) spaces that are not ≤ theta--cwH for singular cardinals theta, and for cardinals lambda ≤ theta with cf(theta) ≥ omega1 and lambda regular, a first countable <theta--cwH not ≤ 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.
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