Weak topologies in complete metric spaces

Author:
Bijan Ahmadi Kakavandi

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1029-1039

MSC (2010):
Primary 53C23; Secondary 54A10

DOI:
https://doi.org/10.1090/S0002-9939-2012-11743-5

Published electronically:
July 27, 2012

MathSciNet review:
3003694

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Abstract | References | Similar Articles | Additional Information

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 so-called half-space 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) 3689-3696]. 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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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:
https://doi.org/10.1090/S0002-9939-2012-11743-5

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