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I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This incompleteness may be not the incompleteness of geodesics, because, in the case of spacetially flat FRW universe with only positive cosmological constant that is half of the De-Sitter spacetime, every timelike and null geodesic is complete. Therefore what's the accurate definition of this incompleteness and which book has this definition?

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Incompleteness of a coordinate system is not a canonical definition as that, for instance, of geodesical (in)completeness. It simply means that the domain of the coordinate system does not cover the whole manifold (and perhaps there are several inequivalent extensions of the initial manifold represented by the given domain of the coordinate system). If a coordinate system is incomplete in this sense, there are geodesics in the domain of the coordinates which are not complete (the domain of their affine parameter is not the whole $\mathbb R$), but the converse may be false. In that case there is a true metrical singularity.

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