Quasi $F$covers of Tychonoff spaces
HTML articles powered by AMS MathViewer
 by M. Henriksen, J. Vermeer and R. G. Woods PDF
 Trans. Amer. Math. Soc. 303 (1987), 779803 Request permission
Abstract:
A Tychonoff topological space is called a quasi $F$space if each dense cozeroset of $X$ is ${C^{\ast }}$embedded in $X$. In Canad. J. Math. 32 (1980), 657685 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 orderCauchy completion of the ring ${C^{\ast }}(X)$. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173189, 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 StoneČech compactification of $X$.) The role of $QF(X)$ as a "projective object" in certain topological categories is investigated.References

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.
 Robert L. Blair and Anthony W. Hager, Extensions of zerosets and of realvalued functions, Math. Z. 136 (1974), 41–52. MR 385793, DOI 10.1007/BF01189255
 Henry B. Cohen, The $k$extremally disconnected spaces as projectives, Canadian J. Math. 16 (1964), 253–260. MR 161294, DOI 10.4153/CJM19640249
 W. W. Comfort, Neil Hindman, and S. Negrepontis, $F^{\prime }$spaces and their product with $P$spaces, Pacific J. Math. 28 (1969), 489–502. MR 242106, DOI 10.2140/pjm.1969.28.489
 Frederick K. Dashiell Jr., Nonweakly compact operators from orderCauchy complete $C(S)$ lattices, with application to Baire classes, Trans. Amer. Math. Soc. 266 (1981), no. 2, 397–413. MR 617541, DOI 10.1090/S00029947198106175417 —, The quasi $F$cover of a compact space and strongly irreducible surjections, Abstracts Amer. Math. Soc. 3(1982), 96. —, The quasi $F$cover of a compact space and strongly irreducible surjections, unpublished manuscript.
 Alan Dow and Ortwin Förster, Absolute $C^{\ast }$embedding of $F$spaces, Pacific J. Math. 98 (1982), no. 1, 63–71. MR 644938, DOI 10.2140/pjm.1982.98.63
 F. Dashiell, A. Hager, and M. Henriksen, OrderCauchy completions of rings and vector lattices of continuous functions, Canadian J. Math. 32 (1980), no. 3, 657–685. MR 586984, DOI 10.4153/CJM19800520
 Jürgen Flachsmeyer, Topologische Projektivräume, Math. Nachr. 26 (1963), 57–66 (German). MR 161298, DOI 10.1002/mana.19630260106
 N. J. Fine and L. Gillman, Extension of continuous functions in $\beta N$, Bull. Amer. Math. Soc. 66 (1960), 376–381. MR 123291, DOI 10.1090/S000299041960104600
 Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.TorontoLondonNew York, 1960. MR 0116199, DOI 10.1007/9781461578192
 M. Henriksen, A summary of results on orderCauchy completions of rings and vector lattices of continuous functions, Topology Proc. 4 (1979), no. 1, 239–263 (1980). Edited by Ross Geoghegan. MR 583707
 Anthony W. Hager, The projective resolution of a compact space, Proc. Amer. Math. Soc. 28 (1971), 262–266. MR 271907, DOI 10.1090/S00029939197102719075
 Melvin Henriksen and J. R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958), 83–105. MR 96196
 C. B. Huijsmans and B. de Pagter, On $z$ideals and $d$ideals in Riesz spaces. II, Nederl. Akad. Wetensch. Indag. Math. 42 (1980), no. 4, 391–408. MR 597997, DOI 10.1016/13857258(80)900402
 C. B. Huijsmans and B. de Pagter, On $z$ideals and $d$ideals in Riesz spaces. II, Nederl. Akad. Wetensch. Indag. Math. 42 (1980), no. 4, 391–408. MR 597997, DOI 10.1016/13857258(80)900402 —, Maximal $d$ideals in a Riesz space, Canad. J. Math. 35 (19830, 10101029.
 S. Iliadis, Absolutes of Hausdorff spaces, Dokl. Akad. Nauk SSSR 149 (1963), 22–25 (Russian). MR 0157354 W. Luxemburg and A. Zaanen, Riesz spaces, NorthHolland, Amsterdam, 1971.
 Jan van Mill, An introduction to $\beta \omega$, Handbook of settheoretic topology, NorthHolland, Amsterdam, 1984, pp. 503–567. MR 776630
 Fredos Papangelou, Order convergence and topological completion of commutative latticegroups, Math. Ann. 155 (1964), 81–107. MR 174498, DOI 10.1007/BF01344076
 Young Lim Park, The quasi$F$ cover as a filter space, Houston J. Math. 9 (1983), no. 1, 101–109. MR 699052
 Jack R. Porter and R. Grant Woods, Extensions of Hausdorff spaces, Pacific J. Math. 103 (1982), no. 1, 111–134. MR 687966, DOI 10.2140/pjm.1982.103.111
 G. L. Seever, Measures on $F$spaces, Trans. Amer. Math. Soc. 133 (1968), 267–280. MR 226386, DOI 10.1090/S00029947196802263865
 J. Vermeer, On perfect irreducible preimages, Proceedings of the 1984 topology conference (Auburn, Ala., 1984), 1984, pp. 173–191. MR 781560 —, Expansions of $H$closed spaces, Doctoral Dissertation, Vrije Universiteit, Amsterdam, The Netherlands, 1983.
 J. Vermeer, The smallest basically disconnected preimage of a space, Topology Appl. 17 (1984), no. 3, 217–232. MR 752272, DOI 10.1016/01668641(84)900439 A. K. Veksler, $P’$points, $P’$sets, $P’$spaces. A new class of order continuous measures and functionals, Soviet Math. Dokl. 4 (1973), 14431450.
 A. I. Veksler, Absolutes and vector lattices, Proceedings of the conference Topology and measure, IV, Part 2 (Trassenheide, 1983) Wissensch. Beitr., ErnstMoritzArndt Univ., Greifswald, 1984, pp. 217–235. MR 824033
 Russell C. Walker, The StoneČech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, SpringerVerlag, New YorkBerlin, 1974. MR 0380698, DOI 10.1007/9783642619359
 Nancy M. Warren, Properties of StoneČech compactifications of discrete spaces, Proc. Amer. Math. Soc. 33 (1972), 599–606. MR 292035, DOI 10.1090/S0002993919720292035X
 R. Grant Woods, A survey of absolutes of topological spaces, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp. 323–362. MR 565852
 Valeriĭ Konstantinovich Zakharov and A. V. Koldunov, Sequential absolute and its characterization, Dokl. Akad. Nauk SSSR 253 (1980), no. 2, 280–284 (Russian). MR 581394
Additional Information
 © Copyright 1987 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 303 (1987), 779803
 MSC: Primary 54G05
 DOI: https://doi.org/10.1090/S00029947198709027980
 MathSciNet review: 902798