Quasi $F$covers of Tychonoff spaces
Authors:
M. Henriksen, J. Vermeer and R. G. Woods
Journal:
Trans. Amer. Math. Soc. 303 (1987), 779803
MSC:
Primary 54G05
DOI:
https://doi.org/10.1090/S00029947198709027980
MathSciNet review:
902798
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Abstract  References  Similar Articles  Additional Information
Abstract: A Tychonoff topological space is called a quasi $F$space if each dense cozeroset of $X$ is ${C^{\ast }}$embedded in $X$. In Canad. J. Math. 32 (1980), 657685 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 orderCauchy completion of the ring ${C^{\ast }}(X)$. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173189, 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 StoneČech compactification of $X$.) The role of $QF(X)$ as a "projective object" in certain topological categories is investigated.

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Keywords:
Quasi <IMG WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$F$">space,
cover,
projective cover
Article copyright:
© Copyright 1987
American Mathematical Society