Vietoris topology on spaces dominated by second countable ones

• Autorzy: Carlos Islas , Daniel Jardon
• Języki publikacji: EN

Abstrakty

EN

For a given space X let C(X) be the family of all compact subsets of X. A space X is dominated by a space M if X has an M-ordered compact cover, this means that there exists a family F = {FK : K ∈ C(M)} ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ∈ C(M). A space X is strongly dominated by a space M if there exists an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . Let K(X) D C(X)\{Ø} be the set of all nonempty compact subsets of a space X endowed with the Vietoris topology. We prove that a space X is strongly dominated by a space M if and only if K(X) is strongly dominated by M and an example is given of a σ-compact space X such that K(X) is not Lindelöf. It is stablished that if the weight of a scattered compact space X is not less than c, then the spaces Cp(K(X)) and K(Cp(X)) are not Lindelöf Σ. We show that if X is the one-point compactification of a discrete space, then the hyperspace K(X) is semi-Eberlein compact.

Słowa kluczowe

EN

Strong domination by second countable spaces; Hemicompact space; Lindelöf p-space; Lindelöf -space; Vietoris topology; One-point compactification; Eberlein compact; Scattered spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2014-08-14

zaakceptowano

2014-12-06

online

2015-01-07

Twórcy

autor

Carlos Islas

• Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, San Lorenzo 290 Colonia del Valle Sur,
  CP 03100, Mexico City, Mexico

autor

Daniel Jardon

• Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Calzada Ermita
  Iztapalapa 4163 Colonia Lomas de Zaragoza, CP 09620, Mexico City, Mexico

Bibliografia

• [1] Amir D., Lindenstrauss J., The structure of the weakly compact sets in Banach spaces, Annals of Math., 1968, 88(1), 35–46 [Crossref]
• [2] Arkhangel’skii A. V., Topological Function Spaces, Kluwer Academic Publishers, Dordrecht, 1992
• [3] Buhagiar D., Yoshioka I., Sums and products of ultracomplete topological spaces, Topology Appl., 2002, 122, 77–86 [WoS]
• [4] Cascales B., Orihuela J., A sequential property of set-valued maps, J. Math. Anal. Appl., 1991, 156, 86–100
• [5] Cascales B., Orihuela J., Tkachuk V., Domination by second countable spaces and Lindelöf - property, Topology Appl., 2011, 158, 204–214 [WoS]
• [6] Engelking R., General Topology, Heldermann, Berlin, 1989
• [7] Fedorchuk V., Filippov V., General Topology. Basic Constructions, Moscow University Press, Moscow, 1988 (in Russian)
• [8] Guerrero Sánchez D., Domination by metric spaces, Topology Appl., 2013, 160, 1652–1658
• [9] Kubís W., Leiderman A., Semi-Eberlein spaces, Topology Proc., 2004, 28, 603–616
• [10] Mamatelashvili A., Tukey order on sets of compact subsets of topological spaces, PhD thesis, University of Pittsburgh, Pittsburgh, USA, July 2014
• [11] Nagami K., -spaces, Fund. Math., 1969, 65, 169–192
• [12] Ntantu I., Cardinal functions on hyperspaces and function spaces, Topology Proc., 1985, 10, 357–375
• [13] Ponomarev V., Tkachuk V., The countable character of X in βX compared with the countable character of the diagonal in XxX, Vestnik Mosk. Univ., 1987, 42(5), 16–19 (in Russian)
• [14] Tkachuk V., A Cp-Theory Problem Book, Springer, Heidelberg, 2010
• [15] Tkachuk V., Lindelöf -spaces: an omnipresent class, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., 2010, 104(2), 221–244 [Crossref]
• [16] Zenor L., On the completeness of the space of compact sets, Proc. Amer. Math. Soc., 1970, 26, 190–192

Identyfikatory

DOI

10.1515/math-2015-0018