Baire spaces and hyperspace topologies revisited

[EN] It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies generated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the base space is a 2nd countable Hausdorff Baire space.Bourquin, S.; Zsilinszky, L. (2014). Baire spaces and hyperspace topologies revisited. Applied General Topology. 15(1):85-92. doi:http://dx.doi.org/10.4995/agt.2014.1897.8592151Beer, G. (1993). Topologies on Closed and Closed Convex Sets. doi:10.1007/978-94-015-8149-3Beer, G., & Lucchetti, R. (1993). Weak topologies for the closed subsets of a metrizable space. Transactions of the American Mathematical Society, 335(2), 805-822. doi:10.1090/s0002-9947-1993-1094552-xCao, J. (2010). The Baire property in hit-and-miss hypertopologies. Topology and its Applications, 157(8), 1325-1334. doi:10.1016/j.topol.2009.03.053Cao, J., & Tomita, A. H. (2007). Baire spaces, Tychonoff powers and the Vietoris topology. Proceedings of the American Mathematical Society, 135(05), 1565-1574. doi:10.1090/s0002-9939-07-08855-7R. Engelking, General Topology, Helderman, Berlin, 1989.Fell, J. M. G. (1962). A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proceedings of the American Mathematical Society, 13(3), 472-472. doi:10.1090/s0002-9939-1962-0139135-6R. C. Haworth and R. A. McCoy, Baire spaces, Dissertationes Math. 141 (1977), 1-77.Holà, L., & Levi, S. (1997). Set-Valued Analysis, 5(4), 309-321. doi:10.1023/a:1008608209952Hou, J.-C., & Vitolo, P. (2008). Fell topology on the hyperspace of a non-Hausdorff space. Ricerche di Matematica, 57(1), 111-125. doi:10.1007/s11587-008-0032-yMcCoy, R. (1975). Baire spaces and hyperspaces. Pacific Journal of Mathematics, 58(1), 133-142. doi:10.2140/pjm.1975.58.133Michael, E. (1951). Topologies on spaces of subsets. Transactions of the American Mathematical Society, 71(1), 152-152. doi:10.1090/s0002-9947-1951-0042109-4J. C. Oxtoby, Cartesian products of Baire spaces, Fundam. Math. 49 (1961), 157-166.H. Poppe, Einige Bemerkungen über den Raum der abgeschlossenen Mengen, Fund. Math. 59 (1966), 159-169.Todd, A. (1981). Quasiregular, pseudocomplete, and Baire spaces. Pacific Journal of Mathematics, 95(1), 233-250. doi:10.2140/pjm.1981.95.233Zsilinszky, L. (1996). Baire spaces and hyperspace topologies. Proceedings of the American Mathematical Society, 124(8), 2575-2584. doi:10.1090/s0002-9939-96-03528-9Zsilinszky, L. (1998). Set-Valued Analysis, 6(2), 187-207. doi:10.1023/a:1008669420995L. Zsilinszky, Baire spaces and weak topologies generated by gap and excess functionals, Math. Slovaca 49 (1999), 357-366.Zsilinszky, L. (2004). Products of Baire spaces revisited. Fundamenta Mathematicae, 183(2), 115-121. doi:10.4064/fm183-2-3L. Zsilinszky, On Baireness of the Wijsman hyperspace, Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. (8) 10 (2007) 1071-1079.Zsilinszky, L. (1996). On separation axioms in hyperspaces. Rendiconti del Circolo Matematico di Palermo, 45(1), 75-83. doi:10.1007/bf0284509