×

\(C\)-epic compactifications. (English) Zbl 0993.54024

Let \(K\) be a compactification of the Tikhonov space \(X\), and \(\rho_K\colon C(K)\to C(X)\) the restriction operator. In case \(\rho_K\) is an epimorphism in the category of Archimedean \(l\)-groups with unit, then \(K\) is called a \(C\)-epic compactification of \(X\), or \(X\) is \(C\)-epic in \(K\). In this paper the authors study the question when \(X\) is \(C\)-epic in some compactification \(K\). They prove, for example, that if \(X\) is \(C\)-epic in \(K\) then \(K\) and \(\beta X\) have the same basically disconnected cover, and that \(\beta X\) is the only \(C\)-epic compactification of \(X\) if and only if \(X\) is pseudocompact. In addition, \(X\) is \(C\)-epic in each of its compactifications if and only if \(X\) is weakly Lindelöf. The paper contains many examples and there are several interesting open questions.

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
46A40 Ordered topological linear spaces, vector lattices
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54C45 \(C\)- and \(C^*\)-embedding
46J10 Banach algebras of continuous functions, function algebras
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
54C30 Real-valued functions in general topology
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, M.; Conrad, P., Epicomplete \(l\)-groups, Algebra Universalis, 12, 224-241 (1981) · Zbl 0457.06016
[2] Anderson, M.; Feil, T., Lattice-Ordered Groups (1988), Reidel: Reidel Dordrecht · Zbl 0636.06008
[3] B. Banaschewski, \(σ\); B. Banaschewski, \(σ\)
[4] B. Banaschewski, A uniform view of localic realcompactness, to appear; B. Banaschewski, A uniform view of localic realcompactness, to appear · Zbl 0953.54026
[5] B. Banaschewski, C. Gilmour, Realcompactness and the cozero part of a frame, to appear; B. Banaschewski, C. Gilmour, Realcompactness and the cozero part of a frame, to appear · Zbl 0978.54019
[6] B. Banaschewski, C. Gilmour, Cozero bases of frames, to appear; B. Banaschewski, C. Gilmour, Cozero bases of frames, to appear · Zbl 0964.54020
[7] Blair, R. L.; Hager, A. W., Extensions of zero-sets and of real-valued functions, Math. Z., 136, 41-58 (1974) · Zbl 0264.54011
[8] Ball, R. N.; Hager, A. W., Characterization of epimorphisms in Archimedean \(l\)-groups and vector lattices, (Glass, A.; Holland, W. C., Lattice-Ordered Groups, Advances and Techniques (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), Chapter 8
[9] Ball, R. N.; Hager, A. W., Epicomplete Archimedean \(l\)-groups, Trans. Amer. Math. Soc., 32, 459-478 (1990) · Zbl 0713.06006
[10] Ball, R. N.; Hager, A. W., Epicompletion of Archimedean \(l\)-group and vector lattices with weak unit, J. Austral. Math. Soc., 48, 25-56 (1990) · Zbl 0696.06011
[11] Ball, R. N.; Hager, A. W., Applications of spaces with filters to Archimedean \(l\)-groups with weak unit, (Martinez, J., Ordered Algebraic Structures (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 99-112
[12] Ball, R. N.; Hager, A. W., On the localic Yosida representation of an Archimedean lattice ordered group with weak order unit, J. Pure Appl. Algebra, 70, 17-43 (1991) · Zbl 0732.06009
[13] Ball, R. N.; Hager, A. W., Relative uniform density of the continuous functions in the Baire functions and of a divisible Archimedean \(l\)-group in any epicompletion, Topology Appl., 97, 109-126 (1999) · Zbl 0949.54016
[14] Ball, R. N.; Hager, A. W.; Molitor, A., Spaces with filter, (Proc. Symp. Cat. Top. Univ. Cape Town 1994 (1999), Dept. of Math. and Appl. Math. Univ. Cape Town), 21-36
[15] Ball, R. N.; Comfort, W. W.; Garcia-Ferreira, S.; Hager, A. W.; van Mill, J.; Robertson, L. C., \(ε\)-Spaces, Rocky Mountain J. Math., 25, 867-886 (1995) · Zbl 0849.54018
[16] Bigard, A.; Keimel, K.; Wolfenstein, S., Groups et Anneaux Reticules. Groups et Anneaux Reticules, Lecture Notes in Math., 608 (1977), Springer: Springer Berlin · Zbl 0384.06022
[17] Conrad, P., The Additive group of and \(f\)-ring, Canad. J. Math., 26, 1157-1168 (1974) · Zbl 0293.06019
[18] Comfort, W. W.; Negrepontis, S., Continuous Pseudometrics (1975), Dekker: Dekker New York · Zbl 0306.54004
[19] Dashiell, F.; Hager, A. W.; Henriksen, M., Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math., 32, 657-685 (1980) · Zbl 0462.54009
[20] Engelking, R., General Topology (1989), Heldermann: Heldermann Berlin · Zbl 0684.54001
[21] Gillman, L.; Jerison, M., Rings of Continuous Functions. Rings of Continuous Functions, Graduate Texts, 43 (1976), Van Nostrand: Van Nostrand Princeton, NJ: Springer: Van Nostrand: Van Nostrand Princeton, NJ: Springer Berlin, reprinted as:
[22] Hager, A. W., On inverse-closed subalgebras of \(C(X)\), Proc. London Math. Soc. (3), 19, 233-257 (1969) · Zbl 0169.54005
[23] Hager, A. W., Minimal covers of Topological Spaces, Ann. New York Acad. Sci., Papers on Gen. Topol. and Rel. Cat. Th. and Top. Alg., 552, 44-59 (1989) · Zbl 0881.54025
[24] Hager, A. W.; Johnson, D. G., A note on certain subalgebras of \(C(X)\), Canada J. Math., 20, 389-393 (1968) · Zbl 0162.26702
[25] Hager, A. W.; Martinez, J., Fraction-dense algebras and spaces, Canad. J. Math., 45, 977-996 (1993) · Zbl 0795.06017
[26] Hager, A. W.; Martinez, J., The laterally \(σ\)-complete reflection of an Archimedean lattice-ordered group, (Holland, W. C.; Martinez, J., Proc. Conf. Ordered Algebraic Structures, Curaçao, 1995 (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 217-236 · Zbl 0874.06013
[27] A.W. Hager, J. Martinez, Topological properties related to \(C\); A.W. Hager, J. Martinez, Topological properties related to \(C\)
[28] Hager, A. W.; Robertson, L. C., Representing and ringifying a Riesz space, Sympos. Math., 21, 411-431 (1977) · Zbl 0382.06018
[29] Hager, A. W.; Robertson, L. C., On the embedding into a ring of an Archimedean \(l\)-group, Canad. J. Math., 31, 1-8 (1979) · Zbl 0406.06007
[30] Henriksen, M.; Vermeer, J.; Woods, R. G., Quasi-\(F\) covers of Tychonoff spaces, Trans. Amer. Math. Soc., 303, 2, 779-803 (1987) · Zbl 0653.54025
[31] Herrlich, H.; Strecker, G., Category Theory (1973), Allyn and Bacon: Allyn and Bacon Boston, MA · Zbl 0265.18001
[32] Isbell, J., Atomless parts of spaces, Math. Scand., 31, 5-32 (1972) · Zbl 0246.54028
[33] Johnstone, P. T., Stone Spaces. Stone Spaces, Cambridge Studies in Advanced Math., 3 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0499.54001
[34] Madden, J. J., Frames associated with an Abelian \(l\)-group, Trans. Amer. Math. Soc., 331, 265-278 (1992) · Zbl 0765.54029
[35] Madden, J. J., \(κ\)-frames, J. Pure Appl. Algebra, 70, 107-127 (1991) · Zbl 0721.06006
[36] Madden, J. J.; Vermeer, J., Lindelöf locales and realcompactness, Math. Proc. Cambridge Philos. Soc., 99, 473-480 (1986) · Zbl 0603.54021
[37] Madden, J. J.; Vermeer, J., Epicomplete Archimedean \(l\)-groups via a localic Yosida theorem, J. Pure Appl. Algebra, 68, 243-252 (1990) · Zbl 0718.06005
[38] Porter, J. R.; Woods, R. G., Extensions and Absolutes of Hausdorff Spaces (1988), Springer: Springer Berlin · Zbl 0652.54016
[39] Sikorski, R., Boolean Algebras (1969), Springer: Springer Berlin · Zbl 0191.31505
[40] Vermeer, J., The smallest basically disconnected preimage of a space, Topology Appl., 17, 217-232 (1984) · Zbl 0584.54033
[41] Zakarov, V. K.; Koldunov, A. V., Characterization of the \(σ\)-cover of a compact, Math. Nachr., 107, 7-16 (1982) · Zbl 0544.54033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.