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

In this paper we consider some open questions concerning Δconvergence 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 Δ-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) 36893696]. Moreover, we give a new characterization of Δ-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.