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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weak topologies in complete $ CAT(0)$ metric spaces


Author: Bijan Ahmadi Kakavandi
Journal: Proc. Amer. Math. Soc. 141 (2013), 1029-1039
MSC (2010): Primary 53C23; Secondary 54A10
Published electronically: July 27, 2012
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Abstract: In this paper we consider some open questions concerning $ \Delta $-convergence in complete $ CAT(0)$ metric spaces (i.e. Hadamard spaces). Suppose $ (X,d)$ is a Hadamard space such that the sets $ \{z \in X \vert \, d(x,z) \leq d(z,y) \}$ are convex for each $ x,y \in X$. We introduce a so-called half-space topology such that convergence in this topology is equivalent to $ \Delta $-convergence for any sequence in $ X$. 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) 3689-3696]. Moreover, we give a new characterization of $ \Delta $-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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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/S0002-9939-2012-11743-5
PII: S 0002-9939(2012)11743-5
Keywords: $CAT(0)$-space, $Δ$-convergence, $w$-convergence, half-space 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