Quasi 𝐹-covers of Tychonoff spaces

Henriksen, M.; Vermeer, J.; Woods, R. G.

doi:10.1090/S0002-9947-1987-0902798-0

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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