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Consider a universe as a curved 1D line looped onto itself. The second dimension is time. On one hand, this line is easy to visualize as a circle embedded in a flat 2D plane. However, there is only one spatial dimension, so this 2D plane is not real. On the other hand, a 1D space has no intrinsic curvature, but without a curvature, how can this space be looped in a circle? Can anyone please clarify if a 2D universe with the metric signature of (1,1) can be closed and if so how would it be described mathematically?

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    $\begingroup$ Open/closed is a topological concept, that is, independent of the metric or signature. You are essentially asking whether there are two-dimensional closed manifolds (and the answer is quite trivially yes). $\endgroup$ – AccidentalFourierTransform Jan 29 '18 at 1:08
  • $\begingroup$ @AccidentalFourierTransform Hmm... Has I managed to ask such a simple question in such a way that it is unclear? Obviously I did not imply any dependence of topology on the metric signature. You pretty much can ignore time in my question, which becomes about space only. Now, space here is one-dimensional, so it is not a two-dimensional closed manifold, which would have an intrinsic curvature. In contrast, a one-dimensional manifold does not have an intrinsic curvature. We can imagine it closed by embedding, but what if the embedding space does not exist? Can a 1D manifold still be closed? $\endgroup$ – safesphere Jan 29 '18 at 1:20
  • $\begingroup$ Again, open/closed is a topological concept. It has nothing to do with the geometry of the manifold. It is independent of the curvature, or any other metric notion. Yes: you can have a 1D closed manifold (e.g., the circle). $\endgroup$ – AccidentalFourierTransform Jan 29 '18 at 1:25
  • $\begingroup$ @AccidentalFourierTransform Alright, let me put it this way (there is a reason I'm asking this on the physics site instead of math). If normally space is curved by gravity and with enough gravity can be curved enough to be looped onto itself and closed, this works, because gravity creates a local intrinsic curvature that adds up globally to make the space closed. With no intrinsic curvature in 1D, gravity is not possible. There is nothing to add up globally to make the space closed. Then what could cause this space to be closed? The initial conditions perhaps? $\endgroup$ – safesphere Jan 29 '18 at 1:35
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    $\begingroup$ @safesphere: 'circle' can be defined without any embeddings. For example, 'fractional part' function induces circular topology on a real line with $x \sim \mathop{\mathrm{frac}} (x)$ equivalence without any appeal to higher dimensions. Your thinking about 2D (or 3D) embedding is an artifact of intuition formed with a visualization. An example of concept that does require embedding into higher dimension is a 'winding number'. $\endgroup$ – A.V.S. Jan 29 '18 at 3:07

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