On (strong) α-favorability of the Wijsman hyperspace

AbstractThe Banach–Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach–Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6]

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.SWORD8592151Beer, 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