In a different thread, JesseM raised ( https://www.physicsforums.com/showpost.php?p=2858281&postcount=37 ) what I thought was an interesting question: can a manifold (over the real numbers) contain points that are infinitely far apart? Since a bare manifold doesn't come equipped with a metric, it's not clear exactly what notion of distance to apply here. It can't be metric distance. It probably can't even be an affine parameter, because I don't think you can define that without a connection, and in any case you can only define an affine parameter along a geodesic, and I think manifolds can contain points that don't have any geodesic connecting them (e.g., events inside the event horizons of two black holes?). I wonder if the mathematically precise way to say this is that the space has to be paracompact: http://en.wikipedia.org/wiki/Paracompact_space Spaces that aren't paracompact don't admit metrics. Carroll's text has some good examples of things that aren't metrics: "With all of these examples, the notion of a manifold may seem vacuous; what isn't a manifold? There are plenty of things which are not manifolds, because somewhere they do not look locally like Rn. Examples include a one-dimensional line running into a two-dimensional plane, and two cones stuck together at their vertices. (A single cone is okay; you can imagine smoothing out the vertex.)" I think a manifold can admit a metric that changes signature (because the definition of a bare manifold is more basic than manifold+metric), but the standard formulation of GR can't handle changes of signature. Anyone got any other illuminating examples? I wonder if it's possible/useful to define a manifold over the hyperreals rather than the reals? http://en.wikipedia.org/wiki/Hyperreal_number Then you could certainly have points infinitely far apart.