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Transactions of the American Mathematical Society

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Quasi $ F$-covers of Tychonoff spaces


Authors: M. Henriksen, J. Vermeer and R. G. Woods
Journal: Trans. Amer. Math. Soc. 303 (1987), 779-803
MSC: Primary 54G05
DOI: https://doi.org/10.1090/S0002-9947-1987-0902798-0
MathSciNet review: 902798
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Abstract: A Tychonoff topological space is called a quasi $ F$-space if each dense cozero-set of $ X$ is $ {C^{\ast}}$-embedded in $ X$. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi $ F$-cover" $ QF(X)$ of a compact space $ X$ as an inverse limit space, and identify the ring $ C(QF(X))$ as the order-Cauchy completion of the ring $ {C^{\ast}}(X)$. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi $ F$-cover of an arbitrary Tychonoff space.

In this paper the minimal quasi $ F$-cover of a compact space $ X$ is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of $ X$. The relationship between $ QF(X)$ and $ QF(\beta X)$ is studied in detail, and broad conditions under which $ \beta (QF(X)) = QF(\beta X)$ are obtained, together with examples of spaces for which the relationship fails. (Here $ \beta X$ denotes the Stone-Čech compactification of $ X$.) The role of $ QF(X)$ as a "projective object" in certain topological categories is investigated.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0902798-0
Keywords: Quasi $ F$-space, cover, projective cover
Article copyright: © Copyright 1987 American Mathematical Society

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