30
$\begingroup$

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?

enter image description here

The question was motivated by my own answer here.

  • 1
    It would probably be a quotient topology of some sort.2012-04-28
  • 1
    It was always a matter of time until this question appeared here on MSE. The Portal world, in terms of playability, is not quite $\Bbb R^3$ topologically; you can't fly (well...) or go through walls after all. I figure the best analogy here would be devising a "playspace" as an open subset of a topological space $X$, selecting two open subsets of $X$, intersecting each with the boundary of the playspace for the "portals," and then identifying the portals on the closure of the playspace. This is rather generic and I don't think amenable to a specific answer in the general case.2012-04-28
  • 3
    @anon: The game (with its not quite $\mathbb{R}^3$ world) is only the motivation for the question. I could formulate the question without any reference to Portal$^{\text{TM}}$: 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?2012-04-28
  • 1
    You could, for intuition, think of this in dimension two, i.e. take the plane and glue intervals. At the very beginning you will see, that the topology (in a strict view) will depend on whether you glue open or closed intervals, but I'm pretty sure it won't affect the standard algebraic invariants. By the way, the standard glueing in topology is not the same as in those games I think: in the quotient space a point could choose whether to pass a portal or ignore it.2012-04-28
  • 1
    Your description of portaling isn't quite correct; portals are attached to a 2D boundary in a 3D space, and each portal is one-sided. You might ask instead, for example, about a solid ball $B^3$ with $n$ pairs of discs on its boundary identified.2012-04-28
  • 0
    @QiaochuYuan: Couldn't we just guess, that if the portals wouldn't stick to the wall, from the backside of the portal you come to the backside of the other portal?2012-04-28
  • 1
    @Nick: but that isn't accurately described by identifying two free-standing disks either. If you identify two free-standing disks you could use either side of one portal to get to either side of _both_ portals.2012-04-29

2 Answers 2