Baire spaces and hyperspace topologies revisited

Steven Bourquin

United States

University of North Carolina at Pembroke

László Zsilinszky

United States

University of North Carolina at Pembroke

Submitted: 2013-11-16

|

Accepted: 2014-02-13

|

Published: 2014-04-01

DOI: https://doi.org/10.4995/agt.2014.1897
Funding Data

Downloads

Keywords:

Baire spaces, Approach spaces, (Proximal) hit-and-miss topologies, Weak hypertopologies, Banach-Mazur game.

Supporting agencies:

This research was not funded

Abstract:

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.

Show more Show less

References:

G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht, 1993.

(http://dx.doi.org/10.1007/978-94-015-8149-3)

G. Beer and R. Lucchetti, Weak topologies on the closed subsets of a metrizable space, Trans. Amer. Math. Soc. 335 (1993), 805-822.

(http://dx.doi.org/10.1090/S0002-9947-1993-1094552-X)

J. Cao, The Baire property in hit-and-miss hypertopologies, Topology Appl. 157 (2010), 1325-1334.

(http://dx.doi.org/10.1016/j.topol.2009.03.053)

J. Cao and A. H. Tomita, Baire spaces, Tychonoff powers and the Vietoris topology, Proc. Amer. Math. Soc. 135 (2007), 1565-1573.

(http://dx.doi.org/10.1090/S0002-9939-07-08855-7)

R. Engelking, General Topology, Helderman, Berlin, 1989.

J. Fell, A Hausdorff topology for the closed subsets of locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472-476.

(http://dx.doi.org/10.1090/S0002-9939-1962-0139135-6)

R. C. Haworth and R. A. McCoy, Baire spaces, Dissertationes Math. 141 (1977), 1-77.

L. Holá and S. Levi, Decomposition properties of hyperspaces topologies, Set-Valued Anal. 5 (1997), 309-321. (http://dx.doi.org/10.1023/A:1008608209952)

J. Hou and P. Vitolo, Fell topology on the hyperspace of a non-Hausdorff space, Ricerche mat. 57 (2008) 111-125. (http://dx.doi.org/10.1007/s11587-008-0032-y)

R. McCoy, Baire spaces and hyperspaces, Pacific J. Math. 58 (1975),133-142.

(http://dx.doi.org/10.2140/pjm.1975.58.133)

E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.

J. 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.

A. R. Todd, Quasiregular, pseudocomplete, and Baire spaces, Pacific J. Math. 95 (1981), 233-250.

(http://dx.doi.org/10.2140/pjm.1981.95.233)

L. Zsilinszky, Baire spaces and hyperspace topologies, Proc. Amer. Math. Soc. 124 (1996), 2575-2584.

(http://dx.doi.org/10.1090/S0002-9939-96-03528-9)

L. Zsilinszky, Topological games and hyperspace topologies, Set-Valued Anal. 6 (1998), 187-207.

(http://dx.doi.org/10.1023/A:1008669420995)

L. Zsilinszky, Baire spaces and weak topologies generated by gap and excess functionals, Math. Slovaca 49 (1999), 357-366.

L. Zsilinszky, Products of Baire spaces revisited, Fundam. Math. 183 (2004), 115-121.

(http://dx.doi.org/10.4064/fm183-2-3)

L. Zsilinszky, On Baireness of the Wijsman hyperspace, Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. (8) 10 (2007) 1071-1079.

L. Zsilinszky, On separation axioms in hyperspaces, Rendiconti del Circolo Matematico di Palermo (2) 45 (1996), 75-83. http://dx.doi.org/10.1007/BF02845090

Show more Show less