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A Tychonoff topological space is called a quasi F-space if each dense cozero-set of X is C*-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*(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(βX) is studied in detail, and broad conditions under which β(QF(X)) = QF(βX) are obtained, together with examples of spaces for which the relationship fails. (Here βX denotes the Stone-Cech compactification of X.) The role of QF(X) as a "projective object" in certain topological categories is investigated.


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