Minimal usco and minimal cusco maps and the topology of pointwise convergence
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Dušan Holý
Abstract
Let X be a Hausdorff topological space, MU(X, ℝ) and MC(X, ℝ) be the space of all minimal usco maps with values in ℝ and the space of all minimal cusco maps with values in ℝ, respectively. By τp is denoted the topology of pointwise convergence. We characterize some cardinal invariants of (MU(X, ℝ), τp) and (MC(ℝ,ℝ), τp). Let Q(ℝ,ℝ) be the space of all quasicontinuous functions on ℝ with values in ℝ and C(ℝ,ℝ) the space of all continuous functions on ℝ with values in ℝ. We compare cardinal invariants of (MU(ℝ,ℝ), τp), (MC(ℝ,ℝ), τp), (Q(ℝ,ℝ), τp) and (C(ℝ,ℝ), τp).
(Communicated by David Buhagiar)
References
[1] Beer, G: Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, 1993.10.1007/978-94-015-8149-3Suche in Google Scholar
[2] Borwein, J. M.: Minimal cuscos and subgradients of Lipschitz functions. In: Fixed point theory and applications (Marseille, 1989), Pitman Res. Notes Math. Ser. 252, Longman Sci. Tech., Harlow, 1991, pp. 57–81.Suche in Google Scholar
[3] Christensen, J. P. R.: Theorems of Namioka and R. E. Johnson type for upper semicontinuous and compact valued mappings, Proc. Amer. Math. Soc. 86 (1982), 649–655.10.1090/S0002-9939-1982-0674099-0Suche in Google Scholar
[4] Collingwood, E. F.—Lohwater, A. J.: The theory of cluster sets, Cambridge Tracts in Math. and Math. Phys. 56 (1989), 301–324.Suche in Google Scholar
[5] Drewnowski, L.—Labuda, I.: On minimal upper semicontinuous compact valued maps, Rocky Mt. J. Math. 20 (1990), 737–752.10.1216/rmjm/1181073096Suche in Google Scholar
[6] Engelking, R.: General Topology, Heldermann Verlag Berlin, 1989.Suche in Google Scholar
[7] Fuller, R. V.: Sets of points of discontinuity, Proc. Amer. Math. Soc. 38 (1973), 193–197.10.1090/S0002-9939-1973-0312457-9Suche in Google Scholar
[8] Hammer, S. T.—Mccoy, R. A.: Spaces of densely continuous forms, Set-Valued Anal. 5 (1997), 247–266.10.1023/A:1008666504767Suche in Google Scholar
[9] Holá, L’.: Spaces of densely continuous forms, USCO and minimal USCO maps, Set-Valued Anal. 11 (2003), 133–151.10.1023/A:1022982924962Suche in Google Scholar
[10] Holá, L’.—Holý, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math. 41 (2011), 1883–1894.10.1216/RMJ-2011-41-6-1883Suche in Google Scholar
[11] Holá, L’.—Holý, D.: New characterization of minimal CUSCO maps, Rocky Mountain J. Math. 44 (2014), 1851–1866.10.1216/RMJ-2014-44-6-1851Suche in Google Scholar
[12] Holá, L’.—Holý, D.: Relation betwen minimal USCO and minimal CUSCO maps, Port. Math. 70(3) (2013), 211–224.10.4171/pm/1931Suche in Google Scholar
[13] Holá, L’.—Holý, D.: Minimal usco and minimal cusco maps and compactness, J. Math. Anal. Appl. 439 (2016), 737–744.10.1016/j.jmaa.2016.03.007Suche in Google Scholar
[14] Holá, L’.—Holý, D.: Quasicontinuous functions and the topology of pointwise convergence, Topology Appl. 282 (2020), Art. 107301.10.1016/j.topol.2020.107301Suche in Google Scholar
[15] Holá, L’.—Holý, D.: Minimal usco maps and cardinal invariants of the topology of uniform convergence on compacta, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116(1) (2022), Art. No. 27.10.1007/s13398-021-01147-8Suche in Google Scholar
[16] Holá, L’.—Holý, D.: Minimal usco maps and the topology of uniform convergence on compacta, Filomat 37(13) (2023), 4249–4259.10.2298/FIL2313249HSuche in Google Scholar
[17] Holá, L’.—Holý, D.—Moors, W.: USCO and Quasicontinuous Mappings. De Gruyter Studies in Mathematics, Vol. 81, De Gruyter, Berlin, 2021.10.1515/9783110750188Suche in Google Scholar
[18] Holá, L’.—Holý, D.—Novotný, B.: Spaces of minimal usco and minimal cusco maps as Fréchet topological vector spaces. In: Advances in Topology and Their Interdisciplinary Applications, Chapter 1, Springer, 2023, pp. 1–18.10.1007/978-981-99-0151-7_1Suche in Google Scholar
[19] Holá, L’.—Mccoy, R. A.: Cardinal invariants of the topology of uniform convergence on compact sets on the space of minimal usco maps, Rocky Mountain J. Math. 37 (2007), 229–246.10.1216/rmjm/1181069328Suche in Google Scholar
[20] Holý, D.—Vadovič, P.: Densely continuous forms, poinwise topology and cardinal functions, Czechoslovak Math. J. 58(133) (2008), 79–92.10.1007/s10587-008-0006-ySuche in Google Scholar
[21] Holá, L’.—Novotný, B.: Subcontinuity, Math. Slovaca 62 (2012), 345–362.10.2478/s12175-012-0013-1Suche in Google Scholar
[22] Holá, L’.—Novotný, B.: New characterizations of completeness and Hausdorff uniformity, preprint.Suche in Google Scholar
[23] Holá, L’.—Pelant, J.: Recent progress in hyperspaces. In: Recent Progress in General Topology, Chapter 10, North Holland, Elsevier, 2002, pp. 253–285.10.1016/B978-044450980-2/50010-8Suche in Google Scholar
[24] Lechicki, A.—Levi, S.: Extensions of semicontinuous multifunctions, Forum Math. 2 (1990), 341–360.10.1515/form.1990.2.341Suche in Google Scholar
[25] Moors, W. B.: A characterization of minimal subdifferential mappings of locally Lipschitz functions, Set-Valued Anal. 3 (1995), 129–141.10.1007/BF01038595Suche in Google Scholar
[26] Mccoy, R. A.—Ntantu, I.: Topological Properties of Spaces of Continuous Functions. Lecture Notes in Math., Vol. 1315, Springer-Verlag Berlin, 1988.10.1007/BFb0098389Suche in Google Scholar
[27] Moors, W. B.—Somasundaram, S.: USCO selections of densely defined set-valued mappings, Bull. Austral. Math. Soc. 65 (2002), 307–321.10.1017/S0004972700020347Suche in Google Scholar
[28] Moors, W. B.—Somasundaram, S.: A Gateaux differentiability space that is not weak Asplund, Proc. Amer. Math. Soc. 134 (2006), 2745–2754.10.1090/S0002-9939-06-08402-4Suche in Google Scholar
[29] Neubrunn, T. Quasi-continuity, Real Anal. Exchange 14 (1988), 259–306.10.2307/44151947Suche in Google Scholar
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Artikel in diesem Heft
- A note on the coprime power graph of groups
- Simplified axiomatic system of DRl-semigroups
- Special filters in bounded lattices
- Reichenbach’s causal completeness of quantum probability spaces
- A construction of magmas and related representation
- Extensions of the triangular D(3)-Pair {3, 6}
- Hermite-Hadamard type inequalities for new class h-convex mappings utilizing weighted generalized fractional integrals
- Divergence operator of regular mappings
- Monotonicity of the ratio of two arbitrary gaussian hypergeometric functions
- Oscillation and asymptotic criteria for certain third-order neutral differential equations involving distributed deviating arguments
- Multiplicity results for a fourth-order elliptic equation of p(x)-kirchhoff type with weights
- Singular discrete dirac equations
- Convergence of bivariate exponential sampling series in logarithmic weighted spaces of functions
- Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application
- Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations
- On almost cosymplectic generalized (k, μ)ʹ-spaces
- On some recent selective properties involving networks
- Minimal usco and minimal cusco maps and the topology of pointwise convergence
- Corrigendum to: Every positive integer is a sum of at most n + 2 centered n-gonal numbers
