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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spaces homeomorphic to $ (2\sp{a})\sb{a}$. II


Authors: H. H. Hung and S. Negrepontis
Journal: Trans. Amer. Math. Soc. 188 (1974), 1-30
MSC: Primary 54A25; Secondary 02K15, 02K35, 54D35, 54G10
MathSciNet review: 0370463
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Abstract: Topological characterizations and properties of the spaces $ {({2^\alpha })_\alpha }$, where $ \alpha $ is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces $ {({2^\alpha })_\alpha }$ in terms of a simple set of conditions, and (b) the $ \alpha $-Baire category property of $ {({2^\alpha })_\alpha }$ and the stability of the class of spaces homeomorphic to $ {({2^\alpha })_\alpha }$ (or to $ {({\alpha ^\alpha })_\alpha }$) when taking intersections of at most $ \alpha $ open and dense subsets of $ {({2^\alpha })_\alpha }$. Among the applications of these results are the following. Assuming $ {\alpha ^ + } = {2^\alpha }$, the class of spaces homeomorphic to $ {({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}$ includes the space of uniform ultrafilters on $ \alpha $ with the $ {P_{{\alpha ^ + }}}$-topology $ {(U(\alpha ))_{{\alpha ^ + }}}$, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming $ {\omega ^ + } = {2^\omega }$ (or in some cases only Martin's axiom), the class of spaces homeomorphic to $ {({2^{({\omega ^ + })}})_{{\omega ^ + }}}$ includes the following: The space $ {(\beta X\backslash X)_{{\omega ^ + }}}$ where X is a noncompact locally compact realcompact space such that $ \vert C(X)\vert \leq {2^\omega }$ and its subspaces of $ {P_{{\omega ^ + }}}$-points of $ \beta X\backslash X$ and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0370463-8
PII: S 0002-9947(1974)0370463-8
Keywords: Spaces $ {({2^\alpha })_\alpha }$, $ {P_\alpha }$-spaces, $ \mathcal{B}$-compact, ramification system, open partition, weakly compact cardinals, $ \alpha $-Baire category, uniform, good, Rudin-Keisler minimal ultrafilter, remote points
Article copyright: © Copyright 1974 American Mathematical Society