Developable hyperspaces are metrizable

L'Ubica Holá

Slovakia

Academy of Sciences

Institute of Mathematics

Jan Pelant

Czechia

Czech Academy of Sciences

Mathematical Institute

László Zsilinszky

United States

University of North Carolina at Pembroke

Department of Mathematics and Computer Science
|

Accepted: 2013-12-17

|

Published: 2003-10-01

DOI: https://doi.org/10.4995/agt.2003.2038
Funding Data

Downloads

Keywords:

Developable spaces, Vietoris topology, Fell topology, Locally finite topology, Bounded Vietoris topology, Gδ-diagonal

Supporting agencies:

GACR 201/00/1466.

Abstract:

Developability of hyperspace topologies (locally finite, (bounded) Vietoris, Fell, respectively) on the nonempty closed sets is characterized. Submetrizability and having a Gδ-diagonal in the hyperspace setting is also discussed.

Show more Show less

References:

H. Attouch, Variational convergence for functions and operators, Pitman, Boston (1984).

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993).

G. Beer, On the Fell Topology, Set-Valued Anal.1 (1993), 69-80. http://dx.doi.org/10.1007/BF01039292

G. A. Beer, C. J. Himmelberg, K. Prikry and F. S. Van Vleck, The locally finite topology on 2X, Proc. Amer. Math. Soc.101 (1987), 168-172. http://dx.doi.org/10.1090/S0002-9939-1987-0897090-2

G. Beer, A. Lechicki, S. Levi and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. (4) 162 (1992), 367-381. http://dx.doi.org/10.1007/BF01760016

H. Brandsma, Monolithic Hyperspaces; PhD. Thesis, Vrije Universiteit, Amsterdam (1998).

H. Brandsma and J. van Mill, Monotonically normal hyperspaces are metrisable.

M. Coban, Note sur topologie exponentielle, Fundam. Math.71 (1971), 27-42.

G. Di Maio, L'. Holá and J. Pelant, Properties related to the first countability of hyperspace topologies, Questions and Answers in General Topology 19 (2001), 139-157.

G. Di Miao and L'. Holá, On hit-and-miss topologies, Rend. Acc. Sc. Fis. Mat. Napoli 62 (1995), 103-124.

R. Engelking, General Topology, Helderman, Berlin (1989).

V. V. Fedorchuk, On some geometric properties of functors, Rend. Circ. Mat. Palermo (2) Suppl. 24 (1990), 73-78.

S. Fisher, P. Gartside, T. Mizokami and N. Shimane, Near metric properties of hyperspaces, Topol. Proc. 22 (1997), 197-211 .

S. L. Gulden, W. M. Fleischman and J. H. Weston, Linearly Ordered Topological Spaces, Proc. Amer. Math. Soc. 24 (1970), 760-766. http://dx.doi.org/10.1090/S0002-9939-1970-0253292-7

G. Gruenhage, Generalized metric spaces, in Handbook of Set-Theoretic topology edited by K. Kunen and J. Vaughan (1984), 423-501.

L'. Holá and S. Levi, Decomposition Properties of Hyperspace Topologies, Set-Valued Anal. 5 (1997), 309-321. http://dx.doi.org/10.1023/A:1008608209952

L'. Holá, S. Levi and J. Pelant, Normality and paracompactnees of the Fell topology, Proc. Amer. Math. Soc. 127 (1999), 2193-2197. http://dx.doi.org/10.1090/S0002-9939-99-04737-1

L'. Holá and H.P. Künzi, Properties related to compactness in hyperspaces, Topology Proceedings 23 (1998), 191-205.

J. Keesling, Normality and properties related to compactness in hyperspaces, Proc. Amer. Math. Soc. 24 (1970), 760-766. http://dx.doi.org/10.1090/S0002-9939-1970-0253292-7

J. Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J. Math. 33 (1970), 657-667. http://dx.doi.org/10.2140/pjm.1970.33.657

G. Matheron, Random sets and integral geometry, Wiley, New York (1975)

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. http://dx.doi.org/10.1090/S0002-9947-1951-0042109-4

T. Mizokami, Hyperspaces of a Moore space and a d-paracompact space, Glas. Mat. 30 (1995), 69-72.

T. Mizokami, On hyperspaces of spaces around Moore spaces, Houston J. Math. 22 (1996), 297-306 .

S. A. Naimpally and P. L. Sharma, Fine uniformity and the locally finite hyperspace topology, Proc. Amer. Math. Soc. 103 (1988), 641-646. http://dx.doi.org/10.1090/S0002-9939-1988-0943098-9

V. Popov, On the subspaces of expX, in: Colloquia Mathematica, Soc. J. B_olyai, Budapest 103 (1978), 977-984.

N. V. Velichko, On spaces of closed subsets, Sib. Math. J. 16 (1975), 484-486. http://dx.doi.org/10.1007/BF00967540

L. Zsilinszky, Topological games and hyperspace topologies, Set-Valued Anal. 6 (1998), 187- 207 . http://dx.doi.org/10.1023/A:1008669420995

Show more Show less