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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Quasi $F$-covers of Tychonoff spaces
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by M. Henriksen, J. Vermeer and R. G. Woods PDF
Trans. Amer. Math. Soc. 303 (1987), 779-803 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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