Abstract. In this paper we prove the existence of a completion of a T0-pre-uniform space (X, U), with the property that each Cauchy filter in (X, U) contains a weakly round filter.2000 AMS Classification: 54A05, 54A20, 54E15.

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ABSTRACT. A Tychonoff topological space is called a quasi F-space if each dense cozero-set of X is C*-embedded in X. In Canad. 1. 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( f3 X) is studied in detail, and broad conditions under which f3(QF( X)) = QF(f3 X) are obtained, together with examples of spaces for which the relationship fails. (Here f3 X 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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