Or, take a torus, cut along the circle farthest from the center, and extend the new edges out to infinity?

That's what it's supposed to look like. The link just used smoke and mirrors to get the effect, and didn't get it right. For example, in a true manifold, you can see any point from any other point. This is because there is some shortest path between the points, and that path is a geodesic. In the link, you can only see one side of the other side of the portal.