3-dim positively curved Alexandrov space What is the classification of 3-dim positively curved Alexandrov space?
And if a 3-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification?     
 A: I guess you are interested in topological classification (?).
Given a 3-dimensional Alexandrov space $M$,
you can always find an other Alexandrov space $\bar M$ with isometric involution $J$
such that


*

*$M$ is isometric to $\bar M/J$

*$\bar M$ is topological manifold without boundary.


This is almost an answer to your first question.
If $M$ is compact and simply connected then $M$
has to be homeomorphic to one of the following (I might miss something):


*

*$\mathbb S^3$

*spherical suspension over $\mathbb R\mathrm P^2$

*$\mathbb D^3$

*ball in the cone over $\mathbb R\mathrm P^2$.


If noncompact then you get in addition $\mathbb R^3$ and cone over $\mathbb R\mathrm P^2$.
If the space has a totally quasigeodesic surface then cutting along this surface should give you an Alexandrov space with boundary.
Since the curvature is positive it should be disc or ball in cone over $\mathbb R\mathrm P^2$.
So you original space is described by an isometric involution of the boundary of one of these spaces.
