Suppose $X$ is a complete CAT(0) space.Then is it true that the cone-topology as, it is defined in Bridson-Haefliger's book metrizable? Also what can we say about Hausdorffness? What if we take a complete CAT(0) $M_k$ simplicial complex with countably many simplices?
I think that the topology will be Hausdorff as in hyperbolic space situation it is Hausdorff but there to prove this they used slim triangle but here we don't have this property. So i am stuck. Thanks.