Weak topologies in complete metric spaces
Author:
Bijan Ahmadi Kakavandi
Journal:
Proc. Amer. Math. Soc. 141 (2013), 10291039
MSC (2010):
Primary 53C23; Secondary 54A10
Published electronically:
July 27, 2012
MathSciNet review:
3003694
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Abstract: In this paper we consider some open questions concerning convergence in complete metric spaces (i.e. Hadamard spaces). Suppose is a Hadamard space such that the sets are convex for each . We introduce a socalled halfspace topology such that convergence in this topology is equivalent to convergence for any sequence in . For a major class of Hadamard spaces, our results answer positively open questions nos. 1, 2 and 3 in [W. A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal., 68 (2008) 36893696]. Moreover, we give a new characterization of convergence and a new topology that we call the weak topology via a concept of a dual metric space. The relations between these topologies and the topology which is induced by the distance function have been studied. The paper concludes with some examples.
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 R. Espínola, A. FernándezLeón, spaces, weak convergence and fixed points, J. Math. Anal. Appl. 353 (2009) 410427. MR 2508878 (2010d:47092)
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 M. Gromov, S.M. Bates, Metric structures for Riemannian and nonRiemannian spaces, with appendices by M. Katz, P. Pansu and S. Semmes, ed. by J. Lafontaine and P. Pansu, Progr. Math. 152, Birkhäuser, Boston (1999). MR 1699320 (2000d:53065)
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Additional Information
Bijan Ahmadi Kakavandi
Affiliation:
Department of Mathematics, Shahid Beheshti University G. C., P.O. Box 1983963113, Tehran, Iran
Email:
b{\textunderscore}ahmadi@sbu.ac.ir
DOI:
http://dx.doi.org/10.1090/S000299392012117435
Keywords:
$CAT(0)$space,
$Δ$convergence,
$w$convergence,
halfspace topology,
weak topology
Received by editor(s):
July 31, 2011
Published electronically:
July 27, 2012
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2012
American Mathematical Society
