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When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?


Authors: Szymon Dolecki, Gabriele H. Greco and Alojzy Lechicki
Journal: Trans. Amer. Math. Soc. 347 (1995), 2869-2884
MSC: Primary 54B20; Secondary 06B30, 54A20
DOI: https://doi.org/10.1090/S0002-9947-1995-1303118-7
MathSciNet review: 1303118
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Abstract: A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Čechcomplete topologies are consonant and that consonance is not preserved by passage to $ {G_\delta }$-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some $ \Gamma $-convergences.


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  • [1] R. Arens, A topology for spaces of transformations, Ann. of Math. (2) 47 (1946), 480-495. MR 0017525 (8:165e)
  • [2] R. Arens and J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951), 5-31. MR 0043447 (13:264d)
  • [3] E. Čech, Topological spaces, Academia, Prague, 1959.
  • [4] G. Choquet, Convergences, Ann. Univ. Grenoble 23 (1947-48), 57-112. MR 0025716 (10:53d)
  • [5] B. J. Day and G. M. Kelly, On topological quotient maps, Proc. Cambridge Philos. Soc. 67 (1970), 553-559. MR 0254817 (40:8024)
  • [6] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei 58 (1975), 842-850. MR 0448194 (56:6503)
  • [7] E. De Giorgi, Generalized limits in calculus of variations, Topics in Functional Analysis, 1980-81, Quaderni della Scuola Norm. Sup. di Pisa, 1982. MR 671756 (84b:49018)
  • [8] S. Dolecki and G. H. Greco, Cyrtologies of convergences. II: Sequential convergences, Math. Nachr. 127 (1986), 317-334. MR 861735 (88b:54002b)
  • [9] -, Topologically maximal pretopologies, Studia Math. 77 (1983), 265-281. MR 745283 (85j:54003)
  • [10] S. Dolecki, G. H. Greco and A. Lechicki, Compactoid and compact filters, Pacific J. Math. 117 (1985), 69-98. MR 777438 (86h:54031)
  • [11] -, Sur la topologie de la convergence supérieure de Kuratowski, C. R. Acad. Sci. Paris 312 (1991), 923-926. MR 1111329 (92c:54007)
  • [12] R. Engelking, General topology, PWN, Warszawa, 1977. MR 0500780 (58:18316b)
  • [13] J. Fell, A Hausdorff topology for closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476. MR 0139135 (25:2573)
  • [14] K. H. Hofmann and J. D. Lawson, The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285-310. MR 515540 (80c:54045)
  • [15] S. P. Franklin, Solution of problem 5468 (S. W. Williams), Amer. Math. Monthly 74 (1967), 207.
  • [16] J. R. Isbell, Meet-continuous lattices, Sympos. Math. 16 (1975), 41-54. MR 0405339 (53:9133)
  • [17] E. Lowens-Colebunders, On the convergence of closed and compact sets, Pacific J. Math. 108 (1983), 133-139. MR 709705 (84g:54004)
  • [18] D. Scott, Continuous lattices, Lecture Notes in Math., vol. 274, Springer, New York, 1972, pp. 97-136. MR 0404073 (53:7879)
  • [19] F. Topsoe, Compactness in spaces of measures, Studia Math. 36 (1970). MR 0268347 (42:3245)
  • [20] A. S. Ward, Problem in "Topology and its applications" (Proc. Herceg., Nov. 1968), Belgrade, 1969, p. 352.
  • [21] P. Wilker, Adjoint product and HOM functors in general topology, Pacific J. Math. 34 (1970), 269-283. MR 0270329 (42:5218)
  • [22] R. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)
  • [23] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic, 1993. MR 1269778 (95k:49001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1303118-7
Keywords: Kuratowski convergence, co-compact topology, continuous convergence, Gamma convergence, compact families
Article copyright: © Copyright 1995 American Mathematical Society

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