Reflection and Weakly Collectionwise Hausdorff Spaces

Abstract

We show that $\square (\theta )$ implies that there is a first countable $< \theta$-collectionwise Hausdorff space that is not weakly $\theta$-collectionwise Hausdorff. We also show that in the model obtained by Levy collapsing a weakly compact (supercompact) cardinal to ${\omega _2}$, first countable ${\aleph _1}$-collectionwise Hausdorff spaces are weakly ${\aleph _2}$-collectionwise Hausdorff (weakly collectionwise Hausdorff). In the last section we show that assuming $E_\theta ^\omega$, a certain $\theta$-family of integer-valued functions exists and that in the model obtained by Levy collapsing a supercompact cardinal to ${\omega _2}$, these families do not exist.