A short note on hit-and-miss hyperspaces
Submitted: 2013-12-16
|Accepted: 2013-12-16
|Published: 2003-10-01
Downloads
Keywords:
Hit-and-miss topology, Compactness, Relative completeness, Relative compact unions, Upper Vietoris topology
Supporting agencies:
Abstract:
Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “κ-Lindelöfness” and “κ-compactness” for cardinals κ. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of Som Naimpally, to who this article is hearty dedicated.
References:
Bartsch, R., Dencker, P., Poppe, H., Ascoli-Arzelà-Theory based on continuous convergence in an (almost) non-Hausdorff setting, in "Categorical Topology"; Dordrecht (1996).
Beer, G., Tamaki, R.,On hit-and-miss hyperspace topologies, Commentat. Math. Univ. Carol. 34 (1993), No.4, 717-728.
Beer, G., Tamaki, R., The infimal value functional and the uniformization of hit-and-miss hyperspace topologies, Proc. Am. Math. Soc. 122, No.2 (1994), 601-612. http://dx.doi.org/10.1090/S0002-9939-1994-1264804-5
Comfort, W.W., Negrepontis, S., The Theory of Ultrafilters, Berlin (1974). http://dx.doi.org/10.1007/978-3-642-65780-1
Klein, E., Thompson, A.C., Theory of correspondences. Including applications to mathematical economics. Canadian Mathematical Society Series of Monographs and Advanced Texts (1984).
Michael, E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. http://dx.doi.org/10.1090/S0002-9947-1951-0042109-4
Naimpally, S., Hyperspaces and Function Spaces, Q & A in General Topology 9 (1991), 33-60.
Naimpally, S., All Hypertopologies are Hit-and-Miss, Appl. Gen. Topol. 3, No.1 (2001), 45-53.
Poppe, H., Eine Bemerkung über Trennungsaxiome in Räumen von abgeschlossenen Teilmengen topologischer Räume, Arch.Math. 16 (1965), 197-199. http://dx.doi.org/10.1007/BF01220021
Poppe, H., Einige Bemerkungen über den Raum der abgeschlossenen Mengen, Fund.Math. 59 (1966), 159-169.
Poppe, H., Compactness in General Function Spaces, Berlin (1974).
