# Article

 Title: Continuous selections on spaces of continuous functions  (English) Author: Tamariz-Mascarúa, Ángel Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 47 Issue: 4 Year: 2006 Pages: 641-660 . Category: math . Summary: For a space $Z$, we denote by $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\leq 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for $\Cal{K}(C_p(X,E))$, the existence of a continuous selection for $\Cal{F}_2(C_p(X,E))$ and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\Bbb{N}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space. Keyword: continuous selections Keyword: Vietoris topology Keyword: linearly orderable space Keyword: weakly orderable space Keyword: space of continuous functions Keyword: dyadic spaces MSC: 54B20 MSC: 54C35 MSC: 54C65 MSC: 54F05 idZBL: Zbl 1150.54021 idMR: MR2337419 . Date available: 2009-05-05T17:00:18Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119625 . Reference: [Arh1] Arhangel'skii A.V.: On mappings of everywhere dense subsets of topological product.Soviet Math. Dokl. 2 (1971), 520-524. Reference: [Arh2] Arhangel'skii A.V.: Topological Function Spaces.Kluwer Academic Publishers, Mathematics and its Applications, vol. 78 Dordrecht, Boston, London (1992). MR 1144519 Reference: [Č] Čoban M.: Many-valued mappings and Borel sets, I.Trans. Moscow Math. Soc. 22 (1970), 258-280. MR 0372812 Reference: [C] Contreras A.: Espacios de funciones continuas del tipo $C(X,E)$.Tesis doctoral, Facultad de Ciencias, UNAM México (2003). Reference: [CT] Contreras-Carreto A., Tamariz-Mascarúa A.: On some generalizations of compactness in spaces $C_p(X,2)$ and $C_p(X,\Bbb Z)$.Bol. Soc. Mat. Mex. 9 (2003), 291-308. MR 2029278 Reference: [E] Engelking R.: General Topology.Heldermann Verlag Berlin (1989). Zbl 0684.54001, MR 1039321 Reference: [EE] Efimov B., Engelking R.: Remarks on dyadic spaces, II.Colloq. Math. 13 (1965), 181-197. Zbl 0137.16104, MR 0188964 Reference: [EHM] Engelking R., Heath R.W., Michael E.: Topological well-ordering and continuous selections.Invent. Math. 6 (1968), 150-158. Zbl 0167.20504, MR 0244959 Reference: [EP] Engelking R., Pelczyński A.: Remarks on dyadic spaces.Colloq. Math. 11 (1963), 55-63. MR 0161296 Reference: [GS] García-Ferreira S., Sanchis M.: Weak selections and pseudocompactness.Proc. Amer. Math. Soc. 132 (2004), 1823-1825. Zbl 1048.54012, MR 2051146 Reference: [H] Herrlich H.: Ordnungsfähigkeit total-diskontinuierlicher Räumen.Math. Ann. 159 (1965), 77-80. MR 0182944 Reference: [L] Lutzer D.J.: Ordered topological spaces.Surveys in General Topology, edited by G.M. Reed, Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980, pp.247-295. Zbl 0472.54020, MR 0564104 Reference: [M] Michael E.: Topologies on spaces of subsets.Trans. Amer. Math. Soc. 71 (1951), 152-182. Zbl 0043.37902, MR 0042109 Reference: [MH] Maurice M.A., Hart K.P.: Some general problems on generalized metrizability and cardinal invariant in ordered topological spaces.Topology and Order Structures, part 1, edited by H.R. Bennet and D.J. Lutzer, Mathematical Centre Tracts, 142, Mathematisch Centrum, Amsterdam, 1981, pp.51-57. MR 0630540 Reference: [vMW] van Mill J., Wattel E.: Selections and orderability.Proc. Amer. Mat. Soc. 83 3 (1981), 601-605. Zbl 0473.54010, MR 0627702 Reference: [vMPP] van Mill J., Pelant J., Pol R.: Selections that characterize topological completeness.Fund. Math. 149 (1996), 127-141. Zbl 0861.54016, MR 1376668 Reference: [VRS] Venkataraman M., Rajagopalan M., Soundararajan T.: Orderable topological spaces.General Topology Appl. 2 (1972), 1-10. Zbl 0238.54029, MR 0298631 Reference: [W] Williams S.W.: Spaces with dense orderable subspaces.Topology and Order Structures, part 1, edited by H.R. Bennet and D.J. Lutzer, Mathematical Centre Tracts, 142, Mathematisch Centrum, Amsterdam, 1981, pp.27-49. Zbl 0473.54018, MR 0630539 .

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