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Domination by second countable spaces and Lindelöf \(\Sigma \)-property. (English) Zbl 1213.54033

Given a space \(X\), let \(\kappa(X)\) denote the family of all compact subsets of \(X\). Then \(X\) is called a Lindelöf \(\Sigma\)-space (or countably \(K\)-determined space) if there exists a second countable space \(M\) and a compact valued u.s.c. map \(\phi: M \to X\) such that \(\bigcup\{\phi(x) : x\in M\} = X\). Let \(F_K =\bigcup\{\phi(x) : x\in K\}\) for any compact set \(K\subset M\), then \(\mathcal I = \{F_K : K \in\kappa(M)\}\) is a compact cover of \(X\) and \(K\subset L\) implies \(F_K\subset F_L\). We say \(\mathcal I\) is an \(M\)-ordered compact cover of \(X\).
The notion of the class of spaces with an \(M\)-ordered compact cover for some second countable space was introduced by B. Cascales and J. Orihuela [J. Math. Anal. Appl. 156, No. 1, 86–100 (1991; Zbl 0760.54013)] and they proved that a Dieudonné complete space is Lindelöf \(\Sigma\) iff it belongs to this class.
In this paper, the authors prove that a space \(C_p(X)\) belongs to the above class iff it is a Lindelöf \(\Sigma\)-space. Under MA(\(\omega_1\)), if \(X\) is compact and \((X\times X)\setminus\Delta\) has a compact cover ordered by a Polish space, then \(X\) is metrizable; also if \(X\) is a compact space of countable tightness and \(X^2\setminus\Delta\) belongs to the above class, then \(X\) is metrizable in ZFC. Also the authors consider the class of spaces \(X\) which have a compact cover \(\mathcal I\) ordered by a second countable space with the additional property that, for every compact set \(P\subset X\), there exists \(F\in\mathcal I\) with \(P\subset F\). It is a ZFC result that if \(X\) is a compact space and \(X^2\setminus\Delta\) belongs to this class, then \(X\) is metrizable. Also it is proved that under CH, if \(X\) is compact and \(C_p(X)\) belongs to this class, then \(X\) is countable.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
03E50 Continuum hypothesis and Martin’s axiom
03E35 Consistency and independence results
54C35 Function spaces in general topology
54E35 Metric spaces, metrizability
54A35 Consistency and independence results in general topology

Citations:

Zbl 0760.54013
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References:

[1] Arhangel’skii, A. V., Structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk, 33, 6, 29-84 (1978), (in Russian) · Zbl 0414.54002
[2] Arhangel’skii, A. V., Topological Function Spaces (1992), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht · Zbl 0911.54004
[3] Arhangel’skii, A. V., \(C_p\)-theory, (Hušek, M.; van Mill, J., Modern Progress in General Topology (1992), Elsevier Science Publishers, B.V.: Elsevier Science Publishers, B.V. Amsterdam) · Zbl 0994.54020
[4] Baturov, D. P., On subspaces of function spaces, Vestnik Moskov. Univ. Ser. Mat. Mekh., 42, 4, 66-69 (1987), (in Russian) · Zbl 0626.54002
[5] Cascales, B., On \(K\)-analytic locally convex spaces, Arch. Math., 49, 232-244 (1987) · Zbl 0617.46014
[6] Cascales, B.; Ka̧kol, J.; Saxon, S. A., Weight of precompact subsets and tightness, J. Math. Anal. Appl., 269, 500-518 (2002) · Zbl 1012.46007
[7] B. Cascales M. Muñoz, J. Orihuela, Number of \(K\); B. Cascales M. Muñoz, J. Orihuela, Number of \(K\)
[8] Cascales, B.; Orihuela, J., On compactness in locally convex spaces, Math. Z., 195, 365-381 (1987) · Zbl 0604.46011
[9] Cascales, B.; Orihuela, J., A sequential property of set-valued maps, J. Math. Anal. Appl., 156, 1, 86-100 (1991) · Zbl 0760.54013
[10] Christensen, J. P.R., Topology and Borel Structure, Math. Stud., vol. 10 (1974), North-Holland: North-Holland Amsterdam · Zbl 0273.28001
[11] Engelking, R., General Topology (1977), PWN: PWN Warszawa
[12] Ferrando, J. C.; Ka̧kol, J.; López Pellicer, M.; Saxon, S. A., Tightness and distinguished Fréchet spaces, J. Math. Anal. Appl., 324, 2, 862-881 (2006) · Zbl 1114.46001
[13] Ferrando, J. C.; Ka̧kol, J.; López Pellicer, M.; Saxon, S. A., Quasi-Suslin weak duals, J. Math. Anal. Appl., 339, 2, 1253-1263 (2008) · Zbl 1151.46001
[14] Hodel, R. E., On a theorem of Arhangel’skii concerning Lindelöf \(p\)-spaces, Canad. J. Math., 27, 2, 459-468 (1975) · Zbl 0301.54010
[15] I. Juhász, Cardinal Functions in Topology — Ten Years Later, Mathematical Centre Tracts, vol. 123, Amsterdam, 1980.; I. Juhász, Cardinal Functions in Topology — Ten Years Later, Mathematical Centre Tracts, vol. 123, Amsterdam, 1980. · Zbl 0479.54001
[16] Juhász, I.; Szentmiklóssy, Z., Convergent free sequences in compact spaces, Proc. Amer. Math. Soc., 116, 4, 1153-1160 (1992) · Zbl 0767.54002
[17] Ka̧kol, J.; Saxon, S. A., Montel (DF)-spaces, sequential (LM)-spaces and the strongest locally convex topology, J. London Math. Soc. (2), 66, 2, 388-406 (2002) · Zbl 1028.46003
[18] Michael, E., \( \aleph_0\)-Spaces, J. Math. Mech., 15, 6, 983-1002 (1966) · Zbl 0148.16701
[19] Mazurkiewicz, S.; Sierpiński, W., Contribution à la topologie des ensembles dénombrables, Fund. Math., 1, 17-27 (1920) · JFM 47.0176.01
[20] M. Muñoz, Indice de \(Kσ\); M. Muñoz, Indice de \(Kσ\)
[21] Okunev, O. G., On Lindelöf \(Σ\)-spaces of continuous functions in the pointwise topology, Topology Appl., 49, 2, 149-166 (1993) · Zbl 0796.54026
[22] Orihuela, J., Pointwise compactness in spaces of continuous functions, J. London Math. Soc. (2), 36, 1, 143-152 (1987) · Zbl 0608.46007
[23] Rogers, C. A.; Jayne, J. E., \(K\)-analytic sets, (Analytic Sets (1980), Academic Press Inc.: Academic Press Inc. London), 1-181 · Zbl 0524.54028
[24] Roitman, J., Basic \(S\) and \(L\), (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 295-326
[25] Talagrand, M., Espaces de Banach faiblement \(K\)-analytiques, Ann. of Math., 110, 407-438 (1979) · Zbl 0393.46019
[26] Tkachuk, V. V., A space \(C_p(X)\) is dominated by irrationals if and only if it is \(K\)-analytic, Acta Math. Hungar., 107, 4, 253-265 (2005) · Zbl 1081.54012
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