Reflection and weakly collectionwise Hausdorff spaces

LaBerge, Tim; Landver, Avner

doi:10.1090/S0002-9939-1994-1231038-X

Reflection and weakly collectionwise Hausdorff spaces
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by Tim LaBerge and Avner Landver PDF

Proc. Amer. Math. Soc. 122 (1994), 291-302 Request permission

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.

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