Portal is a video game, where you can create 2 disks $D\in\mathbb{R}^3$, which then are identified. The world is glued together at these points.
http://www.thebuzzmedia.com/wp-content/uploads/2007/10/hl2_portal_gun.jpg
This kind of reminds me of some procedures to construct spaces for CW complex and whatnot in algebraic topology. I don't know if that kills properties like smoothness, but that doesn't really matter here.
My question:
What is the topology of a world with one or $n$ portals?
If you take topological $\mathbb{R}^3$ and two or even $2n$ therein seperated two-dimensional discs $D_1,D_2$ which you identify (say pairwise), what is the resulting topology?
The question was motivated by my own answer here.