Quasi 𝐹-covers of Tychonoff spaces

Abstract

A Tychonoff topological space is called a quasi F F -space if each dense cozero-set of X X is C ∗ {C^{\ast }} -embedded in X X . In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi F F -cover" Q F ( X ) QF(X) of a compact space X X as an inverse limit space, and identify the ring C ( Q F ( X ) ) C(QF(X)) as the order-Cauchy completion of the ring C ∗ ( X ) {C^{\ast }}(X) . In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi F F -cover of an arbitrary Tychonoff space. In this paper the minimal quasi F F -cover of a compact space X X is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of X X . The relationship between Q F ( X ) QF(X) and Q F ( β X ) QF(\beta X) is studied in detail, and broad conditions under which β ( Q F ( X ) ) = Q F ( β X ) \beta (QF(X)) = QF(\beta X) are obtained, together with examples of spaces for which the relationship fails. (Here β X \beta X denotes the Stone-Čech compactification of X X .) The role of Q F ( X ) QF(X) as a "projective object" in certain topological categories is investigated.