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Trees, Gleason spaces, and coabsolutes of $ \beta {\bf N}\sim {\bf N}$


Author: Scott W. Williams
Journal: Trans. Amer. Math. Soc. 271 (1982), 83-100
MSC: Primary 54G05; Secondary 03E50, 04A30, 54D40, 54E30
DOI: https://doi.org/10.1090/S0002-9947-1982-0648079-X
MathSciNet review: 648079
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Abstract: For a regular Hausdorff space $ X$, let $ \mathcal{E}(X)$ denote its absolute, and call two spaces $ X$ and $ Y$ coabsolute ( $ \mathcal{G}$-absolute) when $ \mathcal{E}(X)$ and $ \mathcal{E}(Y)$ ( $ \beta \mathcal{E}(X)$ and $ \beta \mathcal{E}(Y)$) are homeomorphic. We prove $ X$ is $ \mathcal{G}$-absolute with a linearly ordered space iff the POSET of proper regular-open sets of $ X$ has a cofinal tree; a Moore space is $ \mathcal{G}$-absolute with a linearly ordered space iff it has a dense metrizable subspace; a dyadic space is $ \mathcal{G}$-absolute with a linearly ordered space iff it is separable and metrizable; if $ X$ is a locally compact noncompact metric space, then $ \beta X \sim X$ is coabsolute with a compact linearly ordered space having a dense set of $ P$-points; CH implies but is not implied by "if $ X$ is a locally compact noncompact space of $ \pi $-weight at most $ {2^\omega }$ and with a compatible complete uniformity, then $ \beta X \sim X$ and $ \beta N \sim N$ are coabsolute."


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0648079-X
Keywords: Tree, Gleason space, coabsolute, Stone-Cech remainder, Moore space
Article copyright: © Copyright 1982 American Mathematical Society

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