Trees, Gleason spaces, and coabsolutes of 𝛽𝑁∼𝑁

Williams, Scott W.

doi:10.1090/S0002-9947-1982-0648079-X

Trees, Gleason spaces, and coabsolutes of $\beta \textbf {N}\sim \textbf {N}$
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Trans. Amer. Math. Soc. 271 (1982), 83-100 Request permission

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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