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

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 $\{z \in X | d(x,z) \leq d(z,y) \}$ 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.