Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quasi $ F$-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
DOI: https://doi.org/10.1090/S0002-9947-1987-0902798-0
MathSciNet review: 902798
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [B] B. Banaschewski, Projective covers in certain categories, General Topology and its Relation to Modern Analysis and Algebra. II (Prague, 1966), Academic Press, New York, 1967.
  • [BH] R. L. Blair and A. W. Hager, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41-57. MR 0385793 (52:6652)
  • [C] H. Cohen, The $ k$-extremally disconnected spaces as projectives, Canad. J. Math. 16 (1964), 253-260. MR 0161294 (28:4502)
  • [CHN] W. W. Comfort, N. Hindman, and S. Negrepontis, $ F' $-spaces and their products with $ P$-spaces, Pacific J. Math. 28 (1969), 489-502. MR 0242106 (39:3440)
  • [D$ _{1}$] F. Dashiell, Non-weakly compact operators from $ C(S)$ lattices with applications to Baire classes, Trans. Amer. Math. Soc. 266 (1981), 397-413. MR 617541 (83d:47043)
  • [D$ _{2}$] -, The quasi $ F$-cover of a compact space and strongly irreducible surjections, Abstracts Amer. Math. Soc. 3(1982), 96.
  • [D$ _{3}$] -, The quasi $ F$-cover of a compact space and strongly irreducible surjections, unpublished manuscript.
  • [DF] A. Dow and O. Forster, Absolute $ {C^{\ast}}$-embedding of $ F$-spaces, Pacific J. Math. 98 (1982), 63-71. MR 644938 (83c:54019)
  • [DHH] F. Dashiell, A. W. Hager and M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math. 32 (1980), 657-685. MR 586984 (81k:46020)
  • [F] J. Flaschmeyer, Topologische Projektivräume, Math. Nachr. 26 (1963), 57-66. MR 0161298 (28:4506)
  • [FG] N. J. Fine and L. Gillman, Extensions of continuous functions in $ \beta N$, Bull. Amer. Math. Soc. 66 (1960), 376-381. MR 0123291 (23:A619)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
  • [H] M. Henriksen, A summary of results on order-Cauchy completions of rings and vector lattices of continuous functors, Topology Proc. 4 (1979), 239-263. MR 583707 (82f:54024)
  • [Ha] A. Hager, The projective resolution of a compact space, Proc. Amer. Math. Soc. 28 (1971), 262-266. MR 0271907 (42:6788)
  • [HI] M. Henriksen and J. Isbell, Some properties of compactifications, Duke Math. J. 25 (1957), 83-105. MR 0096196 (20:2689)
  • [HdP$ _{1}$] C. B. Huijsmans and B. de Pagter, On $ z$-ideals and $ d$-ideals in Riesz spaces. II, Indag. Math. 42 (1980), 391-408. MR 597997 (83c:46004a)
  • [HdP$ _{2}$] -, On $ z$-ideals and $ d$-ideals in Riesz spaces. III, Indag. Math. 43 (1981), 409-422. MR 639858 (83c:46004b)
  • [HdP$ _{3}$] -, Maximal $ d$-ideals in a Riesz space, Canad. J. Math. 35 (19830, 1010-1029.
  • [I] S. Iliadis, Absolutes of Hausdorff spaces, Soviet Math. Dokl. 4 (1963), 295-298. MR 0157354 (28:589a)
  • [LZ] W. Luxemburg and A. Zaanen, Riesz spaces, North-Holland, Amsterdam, 1971.
  • [vM] J van Mill, An introduction to $ \beta \omega $, Handbook of Set-Theoretic Topology (Chapter 11), Elsevier, Amsterdam, 1984. MR 776630 (86f:54027)
  • [Pap] F. Papengelou, Order convergence and topological completion of commutative groups, Math. Ann. 155 (1964), 81-107. MR 0174498 (30:4699)
  • [P] Y. L. Park, The quasi $ F$-cover as a filter space, Houston J. Math. 9 (1983), 101-109. MR 699052 (84h:54011)
  • [PW] J. R. Porter and R. G. Woods, Absolutes and extensions of Hausdorff spaces, Springer Universitext series (to appear). MR 687966 (85c:54018)
  • [S] G. L. Seever, Measures on $ F$-spaces, Trans. Amer. Math. Soc. 133 (1968), 269-280. MR 0226386 (37:1976)
  • [V$ _{1}$] J. Vermeer, On perfect irreducible preimages, Topology Proc. 9 (1984). MR 781560 (86f:54024)
  • [V$ _{2}$] -, Expansions of $ H$-closed spaces, Doctoral Dissertation, Vrije Universiteit, Amsterdam, The Netherlands, 1983.
  • [V$ _{3}$] -, The smallest basically disconnected pre-image of a space, Topology Appl. 17 (1984), 217-232. MR 752272 (85m:54039)
  • [Vek$ _{1}$] A. K. Veksler, $ P' $-points, $ P' $-sets, $ P' $-spaces. A new class of order continuous measures and functionals, Soviet Math. Dokl. 4 (1973), 1443-1450.
  • [Vek$ _{2}$] -, Absolutes and vector lattices, Proc. Conf. Topology and Measure. IV (Trassenheide, G. D. R. 1983), 1984, pp. 217-235. MR 824033 (87e:46013)
  • [Wa] R. C. Walker, The Stone-Čech compactification, Springer-Verlag, New York, 1974. MR 0380698 (52:1595)
  • [War] N. M. Warren, Properties of Stone-Čech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599-606. MR 0292035 (45:1123)
  • [W] R. G. Woods, A survey of absolutes of topological spaces, Topological Structures. II, Math. Centrum Tract no. 116, 1979, pp. 323-362. MR 565852 (81d:54019)
  • [ZK] V. K. Zakharov and A. V. Koldunov, The sequential absolute and its characterizations, Soviet Math. Dokl. 22 (1980), 70-74. MR 581394 (82g:54018)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54G05

Retrieve articles in all journals with MSC: 54G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0902798-0
Keywords: Quasi $ F$-space, cover, projective cover
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society