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Some ``almost-Dowker'' spaces

Author: Brian M. Scott
Journal: Proc. Amer. Math. Soc. 68 (1978), 359-364
MSC: Primary 54D15
MathSciNet review: 0467668
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Abstract: Call X an AD-space (for ``almost-Dowker") if it is $ {T_3}$ but not countably metacompact. We construct, without set-theoretic assumptions, a class of zero-dimensional, orthocompact, nonnormal AD-spaces. Using the same techniques, we simplify an example due to Hayashi by showing that if $ \exp (\exp (\omega )) = \exp ({\omega _1})$, (e.g., if the continuum hypothesis holds), the ``Cantor tree of height $ {\omega _1}$'' is also such a space.

Since $ X \times [0,1]$ is orthocompact iff X is orthocompact and countably metacompact, we now have ``absolute'' examples of orthocompact Tikhonov spaces whose products with [0, 1] are not orthocompact.

References [Enhancements On Off] (What's this?)

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Keywords: Dowker space, orthocompact, countably metacompact
Article copyright: © Copyright 1978 American Mathematical Society