Quasi -covers of Tychonoff spaces

Authors:
M. Henriksen, J. Vermeer and R. G. Woods

Journal:
Trans. Amer. Math. Soc. **303** (1987), 779-803

MSC:
Primary 54G05

MathSciNet review:
902798

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Abstract: A Tychonoff topological space is called a quasi -space if each dense cozero-set of is -embedded in . In Canad. J. Math. **32** (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi -cover" of a compact space as an inverse limit space, and identify the ring as the order-Cauchy completion of the ring . *In On perfect irreducible preimages*, Topology Proc. **9** (1984), 173-189, Vermeer constructed the minimal quasi -cover of an arbitrary Tychonoff space.

In this paper the minimal quasi -cover of a compact space is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of . The relationship between and is studied in detail, and broad conditions under which are obtained, together with examples of spaces for which the relationship fails. (Here denotes the Stone-Čech compactification of .) The role of as a "projective object" in certain topological categories is investigated.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0902798-0

Keywords:
Quasi -space,
cover,
projective cover

Article copyright:
© Copyright 1987
American Mathematical Society