Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T16:20:18.901Z Has data issue: false hasContentIssue false
  • English
  • Français

FRAGMENTABILITY BY THE DISCRETE METRIC

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  05 January 2015

WARREN B. MOORS
WARREN B. MOORS*
Affiliation:
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent paper, topological spaces $(X,{\it\tau})$ that are fragmented by a metric that generates the discrete topology were investigated. In the present paper we shall continue this investigation. In particular, we will show, among other things, that such spaces are ${\it\sigma}$-scattered, that is, a countable union of scattered spaces, and characterise the continuous images of separable metrisable spaces by their fragmentability properties.

Keywords

fragmentablesigma-scatteredtopological game

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 303 - 310
Copyright
Copyright © 2015 Australian Mathematical Publishing Association Inc. 

References

Bledsoe, W. W., ‘Neighbourly functions’, Proc. Amer. Math. Soc. 3 (1972), 114–115.Google Scholar
Cao, J. and Moors, W. B., ‘A survey on topological games and their applications in analysis’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 100 (2006), 39–49.Google Scholar
Davis, W. J. and Phelps, R. R., ‘The Radon–Nikodým property and dentable sets in Banach spaces’, Proc. Amer. Math. Soc. 45 (1974), 119–122.Google Scholar
Fabian, M. and Finet, C., ‘On Stegall’s smooth variational principle’, Nonlinear Anal. 66 (2007), 565–570.CrossRefGoogle Scholar
Giles, J. R. and Moors, W. B., ‘A selection theorem for quasi-lower semi-continuous set-valued mappings’, J. Nonlinear Convex Anal. 2 (2001), 345–350.Google Scholar
Glasner, E. and Megrelishvili, M., ‘On fixed point theorems and nonsensitivity’, Israel J. Math. 190 (2012), 289–305.Google Scholar
Heydari, F., Behmardi, D. and Behroozi, F., ‘On weak fragmentability of Banach spaces’, J. Aust. Math. Soc. 97 (2014), 251–256.Google Scholar
Huff, R. E., ‘Dentability and the Radon–Nikodým property’, Duke Math. J. 41 (1974), 111–114.CrossRefGoogle Scholar
Jayne, J., Namioka, I. and Rogers, C. A., ‘𝜎-fragmentable Banach spaces’, Mathematika 39 (1992), 161–188 and 197–215.Google Scholar
Jayne, J., Namioka, I. and Rogers, C. A., ‘Fragmentability and 𝜎-fragmentability’, Fund. Math. 143 (1993), 207–220.Google Scholar
Jayne, J., Namioka, I. and Rogers, C. A., ‘Topological properties of Banach spaces’, Proc. Lond. Math. Soc. (3) 66 (1993), 651–672.Google Scholar
Jayne, J. and Rogers, C. A., ‘Borel selectors for upper semi-continuous set-valued maps’, Acta Math. 155 (1985), 41–79.Google Scholar
Kalenda, O. F. K., ‘Weak Stegall spaces’, unpublished manuscript, 1997.Google Scholar
Kalenda, O. F. K., ‘A weak Asplund space whose dual is not in Stegall’s class’, Proc. Amer. Math. Soc. 130 (2002), 2139–2143.Google Scholar
Kelley, J. L., General Topology, Graduate Texts in Mathematics (Springer, New York–Berlin, 1975).Google Scholar
Kempisty, S., ‘Sur les fonctions quasi-continues’, Fund. Math. 19 (1931), 184–197.CrossRefGoogle Scholar
Kenderov, P. S., Kortezov, I. and Moors, W. B., ‘Topological games and topological groups’, Topology Appl. 109 (2001), 157–165.Google Scholar
Kenderov, P. S., Kortezov, I. and Moors, W. B., ‘Continuity points of quasi-continuous mappings’, Topology Appl. 109 (2001), 321–346.Google Scholar
Kenderov, P. S., Kortezov, I. and Moors, W. B., ‘Norm continuity of weakly continuous mappings into Banach spaces’, Topology Appl. 153 (2006), 2745–2759.Google Scholar
Kenderov, P. S. and Moors, W. B., ‘Game characterisation of fragmentability of topological spaces’, Proc. 25th Spring Conf. Union of Bulgarian Mathematicians, Kazanlak, Bulgaria, 1996, 8–18.Google Scholar
Kenderov, P. S. and Moors, W. B., ‘Fragmentability and sigma-fragmentability of Banach spaces’, J. Lond. Math. Soc. (3) 60 (1999), 203–223.Google Scholar
Kenderov, P. S. and Moors, W. B., ‘A dual differentiation space without an equivalent locally uniformly rotund norm’, J. Aust. Math. Soc. Ser. A 77 (2004), 357–364.Google Scholar
Kenderov, P. S. and Moors, W. B., ‘Fragmentability of groups and metric-valued function spaces’, Topology Appl. 159 (2012), 183–193.Google Scholar
Kenderov, P. S., Moors, W. B. and Sciffer, S., ‘A weak Asplund space whose dual is not weak∗ fragmentable’, Proc. Amer. Math. Soc. 129 (2001), 3741–3747.Google Scholar
Megrelishvili, M., ‘Fragmentability and continuity of semigroup actions’, Semigroup Forum 57 (1998), 101–126.Google Scholar
Megrelishvili, M., ‘Fragmentability and representations of flows’, Topology Proc. 27 (2003), 497–544.Google Scholar
Moors, W. B., ‘Some more recent results concerning weak Asplund spaces’, Abstr. Appl. Anal. 2005 (2005), 307–318.Google Scholar
Moors, W. B. and Giles, J. R., ‘Generic continuity of minimal set-valued mappings’, J. Aust. Math. Soc. Ser. A 63 (1997), 238–262.Google Scholar
Moors, W. B. and Somasundaram, S., ‘A weakly Stegall space that is not a Stegall space’, Proc. Amer. Math. Soc. 131 (2003), 647–654.Google Scholar
Moors, W. B. and Somasundaram, S., ‘A Gâteaux differentiability space that is not weak Asplund’, Proc. Amer. Math. Soc. 134 (2006), 2745–2754.Google Scholar
Namioka, I. and Pol, R., ‘Mappings of Baire spaces into function spaces and Kadeč renormings’, Israel J. Math. 78 (1992), 1–20.Google Scholar
Oxtoby, J. C., ‘Measure and category’, in: A Survey of the Analogies Between Topological and Measure Spaces, Graduate Texts in Mathematics, II (Springer, New York–Berlin, 1971).Google Scholar
Rieffel, M., ‘Dentable subsets of Banach spaces, with applications to a Radon–Nikodým theorem’, in: Functional Analysis (Proc. Conf., Irvine, CA, 1966) (ed. Gelbaum, B. R.) (Academic Press–Thompson, London–Washington, DC, 1967), 71–77.Google Scholar
Royden, H. L., Real Analysis, 3rd edn (Macmillan, New York, 1988).Google Scholar
Ryll-Nardzewski, C., ‘Generalized random ergodic theorem and weakly almost periodic functions’, Bull. Acad. Polon. Sci. 10 (1962), 271–275.Google Scholar
Stegall, C., ‘The Radon–Nikodým property in conjugate Banach spaces’, Trans. Amer. Math. Soc. 206 (1975), 213–223.Google Scholar
Stegall, C., ‘The duality between Asplund spaces and spaces with the Radon–Nykodým property’, Israel J. Math. 29 (1978), 408–412.CrossRefGoogle Scholar
Stegall, C., ‘Optimization of functions on certain subsets of Banach spaces’, Math. Ann. 236 (1978), 171–176.Google Scholar
Stegall, C., ‘Optimization and differentiation in Banach spaces’, Linear Algebra Appl. 84 (1986), 191–211.Google Scholar