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Tytuł książki

Decomposition of topologies on lattices and hyperspaces

Seria
Rozprawy Matematyczne tom/nr w serii: 381 wydano: 1999
Zawartość
Warianty tytułu
Abstrakty
EN
Abstract
The notion of decomposable topology is introduced in a partially ordered set and, in particular, in the lattice C(X) of all closed subsets (ordered by reverse inclusion) of a topological space X, which is also called the hyperspace of X. This notion is closely related to the concepts, defined in the same framework, of lower, upper and strong upper topology.
We investigate decomposability and unique decomposability of the main hyperspace topologies, and of topologies which are defined on some quite natural lattices or semilattices.
EN
CONTENTS
Introduction.................................................................................5
1. Decomposable topologies.......................................................6
2. Locally convex topologies......................................................10
3. Semilattices. Strong decomposability.....................................13
4. Convex topologies.................................................................15
5. Topologies on linearly ordered sets.......................................18
6. Topologies on lattices............................................................20
7. The Scott topology.................................................................26
8. Uniqueness of decomposition................................................28
9. Hyperspace topologies..........................................................32
10. The Vietoris topology...........................................................35
11. The Hausdorff metric topology.............................................37
12. The proximal topology..........................................................39
13. The Kuratowski convergence..............................................40
14. Uniqueness of decomposition for hypertopologies..............44
References................................................................................47
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 381
Liczba stron
48
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXXI
Daty
wydano
1999
otrzymano
1998-06-16
poprawiono
1999-02-22
Twórcy
  • Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy , costanti@dm.unito.it
autor
  • Dipartimento di Matematica, Università della Basilicata, via Nazario Sauro 85, 85100 Potenza, Italy, vitolo@unibas.it
Bibliografia
  • [1] G. A. Beer, C. J. Himmelberg, K. Prikry and F. S. van Vleck, The locally finite topology on $2^X$, Proc. Amer. Math. Soc. 101 (1987), 168-172.
  • [2] G. A. Beer, A. Lechicki, S. Levi and S. A. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pura Appl. (4) 162 (1992), 367-381.
  • [3] G. A. Beer and R. Lucchetti, Weak topologies for the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993), 805-822.
  • [4] C. Costantini, S. Levi and J. Pelant, Infima of hyperspace topologies, Mathematika 83 (1995), 67-86.
  • [5] C. Costantini, S. Levi and J. Zieminska, Metrics that generate the same hyperspace convergence, Set-Valued Anal. 1 (1993), 141-157.
  • [6] C. Costantini and P. Vitolo, On the infimum of the Hausdorff metric topologies, Proc. London Math. Soc. 70 (1995), 441-480.
  • [7] D. E. Dobbs, Posets admitting a unique order-compatible topology, Discrete Math. 41 (1982), 235-240.
  • [8] S. Dolecki, G. H. Greco and A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc. 347 (1995), 2869-2884.
  • [9] S. Francaviglia, A. Lechicki and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370.
  • [10] G. Gierz et al., A Compendium of Continuous Lattices, Springer, New York, 1980.
  • [11] L. Holá and R. Lucchetti, Equivalence among hypertopologies, Set-Valued Anal. 3 (1995), 339-350.
  • [12] S. Levi, R. Lucchetti and J. Pelant, On the infimum of the Hausdorff and Vietoris topologies, Proc. Amer. Math. Soc. 118 (1993), 971-978.
  • [13] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-183.
  • [14] J. van Mill and G. M. Reed (eds.), Open Problems in Topology, North-Holland, Amsterdam, 1991.
  • [15] T. Nogura and D. B. Shakhmatov, When does the Fell topology on a hyperspace of closed sets coincide with the meet of the upper Kuratowski and the lower Vietoris topologies?, Topology Appl. 70 (1996), 213-243.
  • [16] H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 24 (1972), 507-530.
  • [17] P. Vitolo, Strutture d'ordine e topologie sugli iperspazi, PhD thesis, Università degli studi di Milano, Biblioteca Nazionale, Roma-Firenze, 1991.
  • [18] P. Vitolo, Scott topology and Kuratowski convergence on the closed subsets of a topological space, in: Fifth Conf. on Topology (Lecce-Otranto, 1990), Rend. Circolo Mat. Palermo (2) Suppl. 29 (1992), 593-603.
  • [19] H. Weber, Uniform lattices. I: A generalization of topological Riesz spaces and topological boolean rings, Ann. Mat. Pura Appl. (4) 160 (1991), 347-370.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 54B20; Secondary 06A12, 54A10.
Identyfikator YADDA
bwmeta1.element.zamlynska-7e320ad6-fffd-428f-8383-035e35f24602
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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