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An example of a space which is countably compact whose square is countably paracompact but not countably compact


Author: Lee Parsons
Journal: Proc. Amer. Math. Soc. 65 (1977), 351-354
MSC: Primary 54G20; Secondary 54D20
DOI: https://doi.org/10.1090/S0002-9939-1977-0451218-0
MathSciNet review: 0451218
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Abstract: A subspace P of $ \beta N - N$ is obtained whose square is disjoint from the graph, G, of a pre-selected homeomorphism $ f:\beta N \to \beta N$ that has no fixed points. The construction is performed in such a way that, for $ X = P \cup N$, all countable subsets of $ {X^2} - G$ will have a limit point in $ {X^2}$. We use the following lemma: If $ K \subset {(\beta N)^2} - G$ is countably infinite, then $ \vert{\text{cl}_{{{(\beta N)}^2}}}K - G\vert = {2^c}$.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0451218-0
Keywords: Countably compact, countably paracompact, extremally disconnected, M-space, pseudocompact
Article copyright: © Copyright 1977 American Mathematical Society

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