Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
continuous selections; Vietoris topology; linearly orderable space; weakly orderable space; space of continuous functions; dyadic spaces
continuous selections; Vietoris topology; linearly orderable space; weakly orderable space; space of continuous functions; dyadic spaces
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.
References:
[Arh1] Arhangel'skii A.V.: On mappings of everywhere dense subsets of topological product. Soviet Math. Dokl. 2 (1971), 520-524.
[Arh2] Arhangel'skii A.V.: Topological Function Spaces. Kluwer Academic Publishers, Mathematics and its Applications, vol. 78 Dordrecht, Boston, London (1992). MR 1144519
[Č] Čoban M.: Many-valued mappings and Borel sets, I. Trans. Moscow Math. Soc. 22 (1970), 258-280. MR 0372812
[C] Contreras A.: Espacios de funciones continuas del tipo $C(X,E)$. Tesis doctoral, Facultad de Ciencias, UNAM México (2003).
[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
[E] Engelking R.: General Topology. Heldermann Verlag Berlin (1989). MR 1039321 | Zbl 0684.54001
[EE] Efimov B., Engelking R.: Remarks on dyadic spaces, II. Colloq. Math. 13 (1965), 181-197. MR 0188964 | Zbl 0137.16104
[EHM] Engelking R., Heath R.W., Michael E.: Topological well-ordering and continuous selections. Invent. Math. 6 (1968), 150-158. MR 0244959 | Zbl 0167.20504
[EP] Engelking R., Pelczyński A.: Remarks on dyadic spaces. Colloq. Math. 11 (1963), 55-63. MR 0161296
[GS] García-Ferreira S., Sanchis M.: Weak selections and pseudocompactness. Proc. Amer. Math. Soc. 132 (2004), 1823-1825. MR 2051146 | Zbl 1048.54012
[H] Herrlich H.: Ordnungsfähigkeit total-diskontinuierlicher Räumen. Math. Ann. 159 (1965), 77-80. MR 0182944
[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. MR 0564104 | Zbl 0472.54020
[M] Michael E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 | Zbl 0043.37902
[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
[vMW] van Mill J., Wattel E.: Selections and orderability. Proc. Amer. Mat. Soc. 83 3 (1981), 601-605. MR 0627702 | Zbl 0473.54010
[vMPP] van Mill J., Pelant J., Pol R.: Selections that characterize topological completeness. Fund. Math. 149 (1996), 127-141. MR 1376668 | Zbl 0861.54016
[VRS] Venkataraman M., Rajagopalan M., Soundararajan T.: Orderable topological spaces. General Topology Appl. 2 (1972), 1-10. MR 0298631 | Zbl 0238.54029
[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. MR 0630539 | Zbl 0473.54018
Search
Partner of
