WEAK TOPOLOGIES IN COMPLETE CAT(0) METRIC SPACES.

In this paper we consider some open questions concerning ?- convergence in complete CAT(0) metric spaces (i.e. Hadamard spaces). Suppose (X, d) is a Hadamard space such that the sets {z ? X| d(x, z) = d(z, y)} are convex for each x, y ? X. We introduce a so-called half-space topology such that convergence in this topology is equivalent to ?-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 ?-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. [ABSTRACT FROM AUTHOR]