Abstract. We give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of the Wheeler-Shakmatov theorem, which states that if X is normal and dim(X) ≤ 1, then every quasi-measure on X extends to a measure.

We define several notions of forcing that allow us to manipulate the tightness of products of fans. Some consequences include: t(F θ × F ω ) = θ does not imply the existence of a (θ, ω)-gap, new examples of first countable < θ-cwH spaces that are not ≤ θ-cwH for singular cardinals θ, and for cardinals λ ≤ θ with cf (θ) ≥ ω 1 and either λ regular or λ ω ≤ θ, a first countable < θ-cwH not ≤ θ-cwH space that can be made cwH by removing a closed discrete set of cardinality λ. We also prove two theorems that characterize tightness of products of fans in terms of families of integer-valued functions.

Abstract. We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are ℵ 1 -additive. Finally, we construct an example that places strong restrictions on other forms of additivity.

Abstract. We show that 0(0) implies that there is a first countable < 6-collectionwise Hausdorff space that is not weakly (9-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to a>2, first countable ¡»^-collectionwise Hausdorff spaces are weakly ^-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming E% , a certain 0-family of integer-valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to u>i . these families do not exist.

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