Examples of CAT(0)-spaces that are no manifolds I´m trying to think of some examples of CAT(0)-spaces that are not manifolds. I haven´t found any examples in the book of Bridson and Haefliger, so I have tried to come up with some own examples. I suppose there are some involving Graph Theory, but since I´ver never studied graph theory, I try to avoid it for now.
The only example I could think that migth work is the following: Let $A=\{(x,y) \in \mathbb{R}^2 \;\vert\; x \cdot y =0\}$ equipped with the taxicab metric $d_1$. We have shown in my Differential Geometry 1 course that this is not a manifold by the usual argument involving connected components. When thinking and drawing about it, it feels like a CAT(0)-space, but I was not able to prove it. I can't find any connection between a triangle in $A$ and a corresponding comparison triangle in $\mathbb{R}^2$, except the length of the sides.
$\textbf{EDIT:}$ I have come to the conclusion that the above $(A,d_1)$ is in fact an $\mathbb{R}$-tree and this answer Geometric realisations of trees are Cat(0) proved that it is a CAT(0)-space. I´m still interested in different examples, if anyone knows something!
Does anyone know some other examples of such spaces?
 A: Exercise - Euclidean geometry : In $\mathbb{E}^2$, consider a triangle $[ABC]$. If $\theta=\angle BAC$, then $|B-C|^2=|A-B|^2+|A-C|^2-2|A-B||A-C|\cos\theta$. Hence when $|A-B|,\ |A-C|$ are fixed, then $\frac{d}{d\theta}|B-C|\geq 0$.
Example : $Y=\bigcup_i\ Y_i$, union of closed half planes, where $Y_1=\{ (x,y,z)| z=0,\ x\geq 0\},\ Y_2= \{ (x,y,z)
  | z=0,\ x\leq 0\} ,\ Y_3=
   \{(x,y,z)|x=0,\ z\geq 0\} $
Proof : To show that $Y$ is ${\rm CAT}[0]$ space, define a
triangle $
\Delta = [y_1y_2y_3],\ y_i\in Y_i$
If $[\overline{y}_1\overline{y}_2\overline{y}_3]$ is a comparison
triangle, we have to show that $|y_3-p|\leq
|\overline{y}_3-\overline{p}|$ for all $p\in [y_1y_2]$.
If $P$ is an intersection point between a line segment $[y_1y_2]$
and $y$-axis, and if we rotate a point $y_3$ $-\frac{\pi}{2}$ degree along
$y$-axis so that we have $Y_3$, then we consider the triangle
$[y_1PY_3]$. If $p\in [y_1P]$, then we have $ |p-Y_3|\leq
|\overline{p}-\overline{y}_3|$ by Exercise. Remainings are trivial.
Example : $ \{ (x,y,z)| z\leq 0\} \bigcup \{
 (x,y,z)| y=0\}$. The proof is almost same to the above.
Remark : Circle is not a ${\rm CAT}[0]$ space since a shortest path
between two points is not unique.
