Developable hyperspaces are metrizable
Submitted: 2013-12-17
|Accepted: 2013-12-17
|Published: 2003-10-01
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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.
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
