Reading Perelman's preprint(1991) Alexandrov space II now. Got confused about the last section 6.4, which contains an example which indicate that the statement ".... manifold is diffeomorphic to the normal bundle over soul" in Cheeger-Gromoll's Soul Theory won't hold for Alexandrov spaces. I also read BBI's book(A course in Metric Geometry) (page 400-401) which also contains the description of this example. I am using the notation in BBI's book, the example goes as follows:

Let $\pi: K_0(\mathbb{CP}^2)\to K_0(\mathbb{CP}^1)$ be the projection. $\bar{B}_0(1)$ be the unit ball in $\mathbb{CP}^1$ (Note: here should be $K_0(\mathbb{CP}^1$), right?). Let $X^5=\pi^{-1}(\bar{B}_0(1))$. Take double of $X^5$ and it will be the example

The picture in my mind is $K_0(\mathbb{CP}^1)$ is sub-cone of $K_0(\mathbb{CP}^2)$, so the projection is the projection on the second factor if we write the coordinate in cone as $(t, x)$ for $t\in \mathbb R$ and $x\in \mathbb{CP}^2$.

My question is what is the topology of $X^5$?

1) Is $\bar{B}_0(1)$ a close ball? If so then $\bar{B}_0(1)$ will have boundary $\mathbb{CP}^1$, right? and $X^5$ will be a closed cone over $\mathbb{CP}^2$, right?

2) Is $X^5$ compact?