Abstract
We introduce a cardinal function that assigns to each topological space Y a cardinal number ℓΣ (Y) that measures how the space is determined by its compact subsets via upper semicontinuous compact valued maps defined on metric spaces. By doing so we extend and take to a different dimension the study of the so-called countably K-determined spaces (or Lindelöf Σ-spaces) and their associates Gul’ko compacta. We study the behaviour of ℓΣ(·) with respect to the usual operations for topological spaces as well as some of the standard operations within the category of Banach spaces. We study the relationship of ℓΣ(·) with regard to other cardinal functions like for instance the weight w(·) of spaces, for which we observe that although for any compact space K we always have \({\ell\Sigma (C(K),\tau_p)\leq w (C(K),\tau_p)}\) there is a space \({\mathbb Y}\) such that \({w (\mathbb Y) < \ell\Sigma (\mathbb Y)}\) : the example \({\mathbb Y}\) is a subspace of \({\beta\mathbb{N}}\) of cardinality \({2^{2^{\omega}}}\) whose compact subsets are finite. We also study some weakening of G δ -conditions for diagonal of compact spaces that still imply metrizability of the underlying space and that have numerous applications in functional analysis. We close the paper establishing the relationship between ℓΣ(·), the Σ-degree introduced by Hödel and the class of strong Σ-spaces studied by Nagami and others.
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References
Arhangel′skiĭ A.V.: On some topological spaces that occur in functional analysis. Russ. Math. Surv. 31(5), 14–30 (1976)
Arhangel′skiĭ A.V.: Structure and classification of topological spaces and cardinal invariants. Russ. Math. Surv. 33(6), 33–96 (1978)
Arhangel′skiĭ, A.V.: Topological function spaces. In: Mathematics and its Applications (Soviet Series), vol. 78. Kluwer, Dordrecht (1992). Translated from the Russian by R.A.M. Hoksbergen
Avilés A.: The number of weakly compact sets which generate a Banach space. Isr. J. Math. 159, 189–204 (2007)
Cascales B., Oncina L.: Compactoid filters and USCO maps. J. Math. Anal. Appl. 282(2), 826–845 (2003)
Cascales B., Orihuela J.: On compactness in locally convex spaces. Math. Z. 195(3), 365–381 (1987)
Cascales B., Orihuela J.: On pointwise and weak compactness in spaces of continuous functions. Bull. Soc. Math. Belg. Sér. B 40(3), 331–352 (1988)
Cascales B., Orihuela J.: A sequential property of set-valued maps. J. Math. Anal. Appl. 156(1), 86–100 (1991)
Cascales B., Orihuela J., Tkachuk V.: Domination by second countable spaces and lindelöf σ-property. Topol. Appl. 158(2), 204–214 (2011)
Christensen, J.P.R.: Topology and Borel structure. North-Holland, Amsterdam (1974). Descriptive topology and set theory with applications to functional analysis and measure theory. North-Holland Mathematics Studies, vol. 10. Notas de Matemática, No. 51
Deville, R., Godefroy, G., Zizler, V.: Smoothness and renormings in Banach spaces. In: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64. Longman Scientific & Technical, Harlow (1993)
Drewnowski L., Labuda I.: On minimal upper semicontinuous compact-valued maps. Rocky Mount. J. Math. 20(3), 737–752 (1990)
Engelking, R.: General Topology. PWN—Polish Scientific Publishers, Warsaw (1977). Translated from the Polish by the author, Monografie Matematyczne, Tom 60. [Mathematical Monographs, vol. 60]
Floret, K.: Weakly compact sets. In: Lecture Notes in Mathematics, vol. 801. Springer, Berlin (1980). Lectures held at S.U.N.Y., Buffalo, in Spring 1978
Hödel R.: On a theorem of Arhangel′skii concerning Lindelöf p-spaces. Can. J. Math. 27, 459–468 (1975)
Hödel, R.E.: Cardinal functions I. In: Handbook of Set-Theoretic Topology, pp. 1–61. North-Holland, Amsterdam (1984)
Jayne, J.E., Rogers, C.A.: k-analytic sets. In: Analytic Sets. Lectures delivered at a Conference held at University College, University of London, London, July 16–29, 1978, pp. 1–181. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1980)
Ka̧kol, J., Kubiś, W., López-Pellicer, M.: Descriptive topology in selected topics of functional analysis. Developments in Mathematics, vol. 24. Springer, New York (2011)
Kelley, J.L.: General topology. Springer, New York (1975). Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27
Kortezov I.: Fragmentability and cardinal invariants. Topol. Appl. 101(2), 93–106 (2000)
Köthe, G.: Topological vector spaces I. Translated from the German by D.J.H. Garling. Die Grundlehren der mathematischen Wissenschaften, Band 159. Springer-Verlag New York Inc., New York (1969)
Lechicki A., Levi S.: Extensions of semicontinuous multifunctions. Forum Math. 2(4), 341–360 (1990)
van Mill, J.: The infinite-dimensional topology of function spaces. North-Holland Mathematical Library, vol. 64. North-Holland, Amsterdam (2001)
Muñoz, M.: Indice de K-determinación de espacios topológicos y σ-fragmentabilidad de aplicaciones. Ph.D. thesis, Universidad de Murcia (2004)
Nagami K.: Σ-Spaces. Fundam. Math. 65, 169–192 (1969)
Orihuela, J.: Pointwise compactness in spaces of continuous functions. J. Lond. Math. Soc. (2) 36(1), 143–152 (1987)
Pytkeev E.G.: Tightness of spaces of continuous functions. Uspekhi Mat. Nauk. 37(1(223)), 157–158 (1982)
Srivastava, S.M.: A course on Borel sets. Graduate Texts in Mathematics, vol. 180. Springer, New York (1998)
Talagrand, M.: Espaces de Banach faiblement \({{\mathcal K}}\) -analytiques. Ann. Math. (2) 110(3), 407–438 (1979)
Tkachuk V.V.: Growths over discretes: some applications. Mosk. Univ. Math. Bull. 45(4), 19–21 (1990)
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) doi:10.1007/s10474-005-0194-y
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We dedicate this paper to our friend and colleague Gabriel Vera who retired this year.
The research of B. Cascales and J. Orihuela was supported by FEDER and MEC grant MTM2008-05396 and by Fundación Séneca (CARM), grant 08848/PI/08.
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Cascales, B., Muñoz, M. & Orihuela, J. The number of K-determination of topological spaces. RACSAM 106, 341–357 (2012). https://doi.org/10.1007/s13398-012-0058-6
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DOI: https://doi.org/10.1007/s13398-012-0058-6