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It is believed in cosmology that the universe is homogeneous and isotropic. (I'm not qualified to comment on the plausibility of this belief, being a pure math topologist, but let's proceed from that current standard assumption.) Informally, that is to say that it looks the same from every point, and that from every point it looks the same in every direction....

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On very general grounds, the idea behind topological order is to classify a system by means of a topological, and as such non-local, invariant. The latter describes the bulk of your phase without needing to consider the system’s boundaries. However, this rather abstract invariant becomes of physical meaning when it comes to interfaces between topologically ...

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Spacetime topology certainly has importance in physics. In condensed matter physics, topological phases of matter are systems which are sensitive to spacetime topology. Many of the interesting ones are two dimensional, so they can in principle be fabricated into interesting surfaces in a lab. Alternatively, through clever tuning of non-local interactions, ...

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